bout the first edition . . . ... a definitive source-book. "
-General Relativity and Gravitation
bout the second edition . . .
his fully revised and updated Second Edition of an incomparable referenceltext 5dges the gap between modem differential geometry and the mathematical ~ysics of general relativity by providing an invariant treatment of Lorentzian 2ometry-reflecting the more complete understanding of Lorentzian geometry zhieved since the publication of the previous edition.
arefully comparing and contrasting Lorentzian geometry with Riemannian geo- ietry throughout, Global Lorentzian Geometry, Second Edition offers a compre- znsive treatment of the space-time distance function not available in other books ... :cent results on the general instability in the space of Lorentzian metrics for a wen manifold of both geodesic completeness and geodesic incompleteness ... new laterial on geodesic connectibili ty... a more in-depth explication of the behavior I the sectional curvature function in a neighborhood of a degenerate two-plane .and more.
bout the authors . . . 3HN K. BEEM is a Professor of Mathematics at the University of Missouri, olumbia. The coauthor or coeditor of three books and the author or coauthor of ver 60 professional papers, he is a member of the American Mathematical ociety, the Mathematical Association of America, and the International Society )r General Relativity and Gravitation. Dr. Beem received the Ph.D. degree !968) in mathematics from the University of Southern California, Los Angeles.
AUL E. EHRLICH is a Professor of Mathematics at the University of Florida, iainesville. A member of the American Mathematical Society, the Mathematical .ssociation of America, and the International Society for General Relativity and iravitation, he is the author or coauthor of numerous professional papers that :flect his research interests in differential geometry and general relativity. Dr. .hrlich received the Ph.D. degree (1974) in mathematics from the State University f New York at Stony Brook.
~ V I N L. EASLEY is an Associate Professor of Mathematics at Truman State Jniversity, Kirksville, Missouri. He is a member of the American Mathematical ociety, the Mathematical Association of America, and the American Association 3r the Advancement of Science. Dr. Easley received the Ph.D. degree (1991) in lathematics from the University of Missouri, Columbia, where he studied ~rentzian geometry under Professors Beem and Ehrlich.
' R E A N D A P P L I E D M A T H E M A T I C S
A Series of Monographs and Textbooks
GLOBAL LORENTZ GEOMETRY Second Edition
John K. Beem Paul E. Ehrlich Kevin L. Faslev
GLOBAL LORENTZIAN GEOMETRY
PURE AND APPLIED MATHEMATICS
A Program of Monographs, Textbooks, and Lecture Notes
EXECUTIVE EDITORS
Earl J . Taft Rutgers University
New Brunswick, New Jersey
EDITORIAL BOARD
Zuhair Nashed University of Delaware
Newark, Delaware
M. S. Baouendi Anil Nerode University of California, Cornell University
Sun Diego Donald Passman
Jane Cronin University of Wisconsin, Rugers University Madison
Jack K. Hale Fred S. Roberts Georgia Institute of Technology Rugers University
S. Kobayashi Gian-Carlo Rota University of California, Massachusetts Institute of
Berkeley Technology
Marvin Marcus David L. Russell University of California, Virginia Polytechnic Institute
Santa Barbara and State University
W. S. Massey Walter Schempp Yale University Universitiit Siegen
Mark Teply University of Wisconsin,
Milwaukee
MONOGRAPHS AND TEXTBOOKS IN PURE AND APPLIED MATHEMATICS
1. K. Yano, Integral Formulas in Riemannian Geometry (1 970) 2. S. Kobayashi, Hyperbolic Manifolds and Holomorphic Mappings (1970) 3. V. S. Vladimirov, Equations of Mathematical Physics (A. Jeffrey, ed.; A. Littlewood,
trans.) (1 970) 4. B. N. Pshenichnyi, Necessary Conditions for an Extremum (L. Neustadt, translation
ed.; K. Makowski, trans.) (1971) 5. L. Narici et a/., Functional Analysis and Valuation Theory (1 971 ) 6. S. S. Passman, Infinite Group Rings (1 971 ) 7. L. Dornhoff, Group Representation Theory. Part A: Ordinary Representation Theory.
Part B: Modular Representation Theory (1 971, 1972) 8. W. Boothby and G. L. Weiss, eds., Symmetric Spaces (1 972) 9. Y. Matsushima, Differentiable Manifolds (E. T. Kobayashi, trans.) (1 972)
10. L. E. Ward, Jr., Topology (1972) 1 1. A. Babakhanian, Cohomological Methods in Group Theory (1 972) 12. R. Gilmer, Multiplicative Ideal Theory (1 972) 13. J. Yeh, Stochastic Processes and the Wiener lntegral (1 973) 14. J. Barros-Nero, lntroduction to the Theory of Distributions (1 973) 15. R. Larsen, Functional Analysis (1 973) 16. K. Yano and S. lshihara, Tangent and Cotangent Bundles (1 973) 17. C. Procesi, Rings with Polynomial Identities (1 973) 18. R. Hermann, Geometry, Physics, and Systems (1 973) 19. N. R. Wallach, Harmonic Analysis on Homogeneous Spaces (1 973) 20. J. Dieudonn&, lntroduction to the Theory of Formal Groups (1 973) 21. 1. Vaisman, Cohomology and Differential Forms (1 973) 22. B.-Y. Chen, Geometry of Submanifolds (1973) 23. M. Marcus, Finite Dimensional Multilinear Algebra (in two parts) (1973, 1975) 24. R. Larsen, Banach Algebras (1 973) 25. R. 0. Kujala andA. 1. Vitrer, eds., Value Distribution Theory: Part A; Part 8: Deficit
and Bezout Estimates by Wilhelm St011 (1 973) 26. K. B. Stolarsky, Algebraic Numbers and Diophantine Approximation (1974) 27. A. R. Magid, The Separable Galois Theory of Commutative Rings (1 974) 28. B. R. McDonald, Finite Rings with Identity (1 974) 29. J. Satake, Linear Algebra (S. Koh et al., trans.) (1975) 30. J. S. Golan, Localization of Noncommutative Rings (1 975) 31. G. Klambauer, Mathematical Analysis (1 975) 32. M . K. Agoston, Algebraic Topology (1976) 33. K. R. Goodearl, Ring Theory (1 976) 34. L. E. Mansfield, Linear Algebra with Geometric Applications (1 976) 35. N. J. Pullman, Matrix Theorv and Its ADDlications (1 976) 36. B. R. ~ c ~ o n a l d , ~eometr ic Algebra over Local Rings (1 976) 37. C. W. Groetsch, Generalized Inverses of Linear Operators (1 977)
J. E. Kuczkowskiand J. L. Gersting, Abstract Algebra (1977) C. 0. Christenson and W. L. Voxman, Aspects of Topology (1 977) M. Nagata, Field Theory (1 977) R. L. Long, Algebraic Number Theory (1 977) W. F. Pfeffer, lntegrals and Measures (1 977) R. L. Wheeden and A. Zygmund, Measure and Integral (1 977) J. H. Curtiss, lntroduction to Functions of a Complex Variable (1 978) K. Hrbacek and T. Jech, lntroduction to Set Theory (1978) W. S. Massey, Homology and Cohomology Theory (1 978) M. Marcus, lntroduction to Modern Algebra (1 978) E. C. Young, Vector and Tensor Analysis (1978) S. B. Nadler, Jr., Hyperspaces of Sets ( 1 978) S. K. Segal, Topics in Group Kings (1 978) A. C. M. van Rooij, Non-Archimedean Functional Analysis (1 978) L. Corwin and R. Szczarba, Calculus in Vector Spaces (1 979) C. Sadosky, interpolation of Operators and Singular Integrals (1 979)
54. J. Cronin, Differential Equations (1 980) 55. C. W. Groetsch, Elements of Applicable Functional Analysis (1980) 56. 1. Vaisman, Foundations of Three-Dimensional Euclidean Geometry (1 980) 57. H. I. Freedan, Deterministic Mathematical Models in Population Ecology (1 980) 58. S. B. Chae, Lebesgue Integration (1980) 59. C. S. Rees etal., Theory and Applications of Fourier Analysis (1 981) 60. L. Nachbin, lntroduction to Functional Analysis (R. M. Aron, trans.) (1981) 61. G. Onech and M. Orzech, Plane Algebraic Curves (1 981) 62. R. Johnsonbaugh and W. E. Pfaffenberger, Foundations of Mathematical Analysis
(1 981) 63. W. L. Voxman and R. H. Goetschel, Advanced Calculus (1 981) 64. L. J. Corwin and R. H. Szczarba, Multivariable Calculus (1982) 65. V. I. Ist@tescu, lntroduction t o Linear Operator Theory (1981) 66. R. D. Jsirvinen, Finite and Infinite Dimensional Linear Spaces (1981) 67. J. K. Beem and P. E. Ehrlich, Global Lorentzian Geometry (1981) 68. D. L. Armacost, The Structure of Locally Compact Abelian Groups (1 981) 69. J. W. Brewer and M. K. Smith, eds., Emmy Noether: A Tribute (1 981) 70. K. H. Kim, Boolean Matrix Theory and Applications (1 982) 71 . T. W. Wieting, The Mathematical Theory of Chromatic Plane Ornaments (1 982) 72. D. B.Gauld, Differential Topology (1 982) 73. R. L. Faber, Foundations of Euclidean and Non-Euclidean Geometry (1983) 74. M. Carmeli, Statistical Theory and Random Matrices (1 983) 75. J. H. Carruth et a/., The Theory of Topological Semigroups (1 983) 76. R. L. Faber, Differential Geometry and Relativity Theory (1983) 77. S. Bamett, Polynomials and Linear Control Systems (1983) 78. G. Karpilovsky, Commutative Group Algebras (1983) 79. F. Van Oystaeyen and A. Verschoren, Relative lnvariants of Rings (1 983) 80. 1. Vaisman, A First Course in Differential Geometry (1984) 81. G. W. Swan, Applications of Optimal Control Theory in Biomedicine (1 984) 82. T. Petrie and J. D. Randall, Transformation Groups on Manifolds (1 984) 83. K. Goebeland S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive
Mappings (1 984) 84. T. Albu and C. Ngstasescu, Relative Finiteness in Module Theory (1 984) 85. K. Hrbacek and T. Jech, lntroduction t o Set Theory: Second Edition (1 984) 86. F. Van Oystaeyen and A. Verschoren, Relative lnvariants of Rings (1 984) 87. B. R. McDonald, Linear Algebra Over Commutative Rings (1984) 88. M. Namba, Geometry of Projective Algebraic Curves 11 984) 89. G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics (1985) 90. M. R. Bremner et a/., Tables of Dominant Weight Multiplicities for Representations of
Simple Lie Algebras (1 985) 91. A. E. Fekete, Real Linear Algebra (1 985) 92. S. B. Chae, Holomorphy and Calculus in Normed Spaces (1 985) 93. A. J. Jerri, lntroduction t o Integral Equations w i th Applications (1 985) 94. G. Karpilovsky, Projective Representations of Finite Groups (1985) 95. L. Nariciand E. Beckenstein, Topological Vector Spaces (1 985) 96. J. Weeks, The Shape of Space (1985) 97. P. R. Gribik and K. 0. Kortanek, Extremal Methods of Operations Research (1 985) 98. J.-A. Chao and W. A. Woyczynski, eds., Probability Theory and Harmonic Analysis
(1986) 99. G. D. Crown etal., Abstract Algebra (1986)
100. J. H. Carruth etal., The Theory of Topological Semigroups, Volume 2 (1 986) 101. R. S. Doran and V. A. Belfi, Characterizations of C*-Algebras (1986) 102. M. W, Jeter, Mathematical Programming (1 986) 103. M. Altman, A Unified Theory of Nonlinear Operator and Evolution Equations wi th
Applications (1 986) 104. A. Verschoren, Relative lnvariants of Sheaves (1 987) 105. R. A. Usmani, Applied Linear Algebra (1 987) 106. P. Blass and J. Lang, Zariski Surfaces and Differential Equations in Characteristic p
> 0 (1 987) 107. J. A. Reneke eta/., Structured Hereditary Systems (1 987) 108. H. Busemann and B. B. Phadke, Spaces wi th Distinguished Geodesics (1987) 109. R. Harre, lnvertibility and Singularity for Bounded Linear Operators (1 988)
110. G. S. Ladde e t a/., Oscillation Theory of Differential Equations w i th Deviating Arguments (1 987)
1 1 1. L. Dudkin et al., Iterative Aggregation Theory (1 987) 1 12. T. Okubo, Differential Geometry (1 987) 113. D. L. Stancland M. L. Stancl, Real Analysis wi th Point-Set Topology (1 987) 114. T. C. Gard, lntroduction t o Stochastic Differential Equations (1988) 115. S. S. Abhyankar, Enumerative Combinatorics of Young Tableaux (1 988) 1 16. H. Strade and R. Farnsteiner, Modular Lie Algebras and Their Representations (1 988) 1 17. J. A. Huckaba, Commutative Rings wi th Zero Divisors (1 988) 1 18. W. D. Wallis, Combinatorial Designs (1 988) 1 19. W. Wipslaw, Topological Fields (1 988) 1 20. G. Karpilovsky, Field Theory (1 988) 121. S. Caenepeel and F. Van Oystaeyen, Brauer Groups and the Cohomology of Graded
Rings (1 989) 122. W. Kozlowski, Modular Function Spaces (1 988) 123. E. Lowen-Colebunders, Function Classes of Cauchy Continuous Maps (1 989) 124. M. Pavel, Fundamentals of Pattern Recognition (1989) 125. V. Lakshmikantham eta/., Stability Analysis of Nonlinear Systems (1 989) 126. R. Sivaramakrishnan, The Classical Theory of Arithmetic Functions (1 989) 127. N. A. Watson, Parabolic Equations on an lnfinite Strip (1 989) 128. K. J. Hastings, lntroduction t o the Mathematics of Operations Research (1 989) 129. B. fine, Algebraic Theory of the Bianchi Groups (1 989) 130. D. N. Dikranjan et a/., Topological Groups (1 989) 131. J. C. Morgan 11, Point Set Theory (1 990) 132. P. Biler and A. Witkowski, Problems in Mathematical Analysis (1 990) 133. H. J. Sussmann, Nonlinear Controllability and Optimal Control (1 990) 134. J.-P. Florens et a/., Elements of Bayesian Statistics (1 990) 135. N. Shell, Topological Fields and Near Valuations (1 990) 136. B. F. Doolin and C. F. Martin, lntroduction t o Differential Geometry for Engineers
(1 990) 137. S. S. Holland, Jr., Applied Analysis by the Hilbert Space Method (1 990) 138. J. Oknirfski, Semigroup Algebras (1 990) 139. K. Zhu, Operator Theory in Function Spaces (1 990) 140. G. B. Price, An lntroduction t o Multicomplex Spaces and Functions (1 991 ) 141 . R. B. Darst, lntroduction t o Linear Programming (1 99 1 ) 142. P. L. Sachdev, Nonlinear Ordinary Differential Equations and Their Applications
(1991) 143. T. Husain, Orthogonal Schauder Bases (1 991 ) 144. J. Foran, Fundamentals of Real Analysis (1 991 ) 145. W. C. Brown, Matrices and Vector Spaces (1991) 146. M. M. Rao and Z. D. Ren, Theory of Orlicz Spaces (1 991 ) 147. J. S. Golan and T. Head, Modules and the Structures of Rings (1 991) 148. C. Small, Arithmetic of Finite Fields (1 991) 149. K. Yang, Complex Algebraic Geometry (1 991) 150. D. G. Hoffman etal., Coding Theory (1991) 151. M. 0. Gonzdlez, Classical Complex Analysis ( 1 992) 152. M. 0. Gonzdlez, Complex Analysis (1 992) 153. L. W. Baggett, Functional Analysis (1 992) 154. M. Sniedovich, Dynamic Programming (1 992) 155. R. P. Agarwal, Difference Equations and Inequalities (1 992) 156. C. Brezinski, Biorthogonality and Its Applications t o Numerical Analysis (1 992) 157. C. Swarrz, An lntroduction to Functional Analysis (1 992) 158. S. B. Nadler, Jr., Continuum Theory (1 992) 159, M. A. Al-Gwaiz, Theory of Distributions (1 992) 160. E. Perry, Geometry: Axiomatic Developments wi th Problem Solving (1 992) 161. E. Castillo and M. R. Ruiz-Cobo, Functional Equations and Modelling in Science and
Engineering (1 992) 162. A. J. Jerri, Integral and Discrete Transforms with Applications and Error Analysis
(1 992) 163. A. Charlier e t a/., Tensors and the Clifford Algebra (1 992) 164. P. Biler and T. Nadzieja, Problems and Examples in Differential Equations (1 992) 165. E. Hansen, Global Optimization Using Interval Analysis (1 992)
S. Guerre-Delabridre, Classical Sequences in Banach Spaces (1 992) Y. C. Wong, Introductory Theory of Topological Vector Spaces (1 992) S. H. Kulkarniand B. V. Limaye, Real Function Algebras (1 992) W. C. Brown, Matrices Over Commutative Rings (1 993) J. Loustau and M. Dillon, Linear Geometry wi th Computer Graphics (1 993) W. V. Petryshyn, Approximation-Solvability of Nonlinear Functional and Differential Equations (1 993) E. C. Young, Vector and Tensor Analysis: Second Edition (1 993) T. A. Bick, Elementary Boundary Value Problems (1 993) M. Pavel, Fundamentals of Pattern Recognition: Second Edition (1 993) S. A. Albeverio eta/., Noncommutative Distributions (1 993) W. Fulks, Complex Variables (1 993) M. M. Rao, Conditional Measures and Applications (1 993) A. Janicki and A. Weron, Simulation and Chaotic Behavior of a-Stable Stochastic Processes (1 994) P. Neittaanmakiand D. Tiba, Optimal Control of Nonlinear Parabolic Systems (1 994) J. Cronin, Differential Equations: lntroduction and Qualitative Theory, Second Edition (1 994) S. Heikkila and V. Lakshmikantham, Monotone Iterative Techniques for Discontinu- ous Nonlinear Differential Equations (1 994) X. Mao. Exoonential Stabilitv of Stochastic Differential Eauations 11 994)
183. B. S. ~homion, Symmetric Properties of Real Functions ( i 994) 184. J. E. Rubio. Ootimization and Nonstandard Analvsis (1 994) 185. J. L. Bueso eral., Compatibility, Stability, and sheaves (1 995) 186. A. N. Micheland K. Wang, Qualitative Theory of Dynamical Systems (1 995) 187. M. R. Darnel, Theory of Lattice-Ordered Groups (1 995) 188. Z. Naniewicz and P. D. Panagiotopoulos, Mathematical Theory of Hemivariational
Inequalities and Applications (1 995) 189. L. J. Convin and R. H. Szczarba, Calculus in Vector Spaces: Second Edition (1995) 190. L. H. Erbe eta/., Oscillation Theory for Functional Differential Equations (1995) 191. S. Aaaian et a/.. Binaw Polvnomial Transforms and Nonlinear Diaital Filters 11 995) 192. M. l .v~i/ ' , ~orm '~s t ima t i ons for Operation-Valued Functions and-~pplications (1 995) 193. P. A. Grillet, Seminroups: An lntroduction t o the Structure Theorv (1995) 194. S. ~ichenassam~,- onl linear Wave Equations (1 996) 195. V. F. Krotov, Global Methods in Optimal Control Theory (1 996) 196. K. I. Beidar eta/., Rings with Generalized Identities (1 996) 197. V. I. Arnautov et a/., lntroduction t o the Theory of Topological Rings and Modules
(1 996) 198. G. Sierksma, Linear and Integer Programming (1 996) 199. R. Lasser, lntroduction to Fourier Series (1 996) 200. V. Sima, Algorithms for Linear-Quadratic Optimization (1 996) 201. D. Redmond, Number Theory (1 996) 202. J. K. Beem eta/. Global Lorentzian Geometry: Second Edition (1 996)
Additional Volumes in Preparation
GLOBAL LORENTZIAN GEOMETRY Second Edition
John K. Beem Department of Mathematics University of Missouri- Columbia Columbia, Missouri
Paul E. Ehrlich Department of Mathematics University of Florida- Gainesville Gainesville, Florida
Kevin L. Easlev d
Department of Mathematics Truman State University Kirks ville, Missouri
Marcel Dekker, Inc. New YorkeBasel Hong Kong
PREFACE TO THE SECOND EDITION Library of Congress Cataloging-in-Publication Data
Beem, John K. Global Lorentzian geometry. - 2nd ed. /John K. Beern, Paul E. Ehrlich,
Kevin L. Easley. p. cm. - (Monographs and textbooks in pure and applied mathematics ;
202) Includes bibliographical references and index. ISBN 0-8247-9324-2 (pbk. : alk. paper) 1. Geometry, Differential. 2. General relativity (Physics).
I. Ehrlich, Paul E. 11. Easley, Kevin L. 111. Title. IV. Series. QA649.B42 1996 5 1 6 . 3 ' 7 6 ~ 2 0 96-957
CIP
The publisher offers discounts on this book when ordered in bulk quantities. For more information, write to Special Sales/Professional Marketing at the address below.
This book is printed on acid-free paper.
Copyright O 1996 by MARCEL DEKKER, INC. All Rights Reserved.
Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage and retrieval system, without permission in writing from the publisher.
MARCEL DEKKER, INC. 270 Madison Avenue, New York, New York 10016
Current printing (last digit): 1 0 9 8 7 6 5 4 3 2 1
PRINTED IN THE UNITED STATES OF AMERICA
The second edition of this book continues the study of Lorentzian geometry,
the mathematical theory used in general relativity. Chapters 3 through 12 con-
tain material, slightly revised in some cases, which was discussed in Chapters 2
through 11 of the first edition. Much new material has been added to Chapters
7 and 11, and new Chapters 13 and 14 have been written reflecting the more
complete and detailed understanding that has been gained in the intervening
years on many of the topics treated in the first edition. Inspired by an example
of P. Williams (1984), additional material on the instability of both geodesic
completeness and geodesic incompleteness for general space-times has been
provided in Section 7.1. Section 7.4 has been added giving sufficient condi-
tions on a spacetime to guarantee the stability of geodesic completeness for
metrics in a neighborhood of a given complete metric. New material has also
been added to Section 11.3 on the topic of geodesic connectibility. Appendixes
A, B, and C of the first edition have now been expanded into Chapter 2,
which also contains new material on the generic condition as well as Section
2.3, which gives a proof that the null cone determines the metric up to a con-
formal factor in any semi-Riemannian manifold which is neither positive nor
negative definite. Also, a deeper treatment of the behavior of the sectional
curvature function in a neighborhood of a degenerate two-plane is given in
Chapter 2.
In the concluding Chapter 11 of the first edition, which is now Chapter 12,
we showed how the Lorentzian distance function and causally disconnecting
sets could be used to obtain the Hawking-Penrose Singularity Theorem con-
cerning geodesic incompleteness of the space-time manifold. Around 1980,
S. T. Yau suggested that the concept of "curvature rigidity," well known for
iv PREFACE TO THE SECOND EDITION
the differential geometry of complete Riemannian manifolds since the publica-
tion of Cheeger and Ebin (1975), might be applied to the seemingly unrelated
topic of singularity theorems in space-time differential geometry. (Earlier. Ge-
roch (1970b) had suggested that spatially closed space-times should fail to be
timelike geodesically incomplete only under special circumstances.) As a step
toward this conjectured rigidity of geodesic incompleteness, the Lorentzian
analogue of the Cheeger-Gromoll Splitting Theorem for complete Riemann-
ian manifolds of nonnegative Ricci curvature needed to be obtained. This
was accomplished in a series of research papers published between 1984 and
1990. Aspects of the proof of the Lorentzian Splitting Theorem are discussed
in the new Chapter 14. Another new chapter in the second edition, Chapter
13, draws upon investigations of Ehrlich and Emch (1992a,b,c, 1993) and is
devoted to a study of the geodesic behavior and causal structure of a class
of geodesically complete Ricci flat space-times, the gravitational plane waves,
which were introduced into general relativity as astrophysical models. These
space-times provide interesting and nontrivial examples of astigmatic conju-
gacy [cf. Penrose (1965a)l and of the nonspacelike cut locus, a concept dis-
cussed in Chapter 8 of the first edition.
As for the first edition, this book is written using the notations and meth-
ods of modern differential geometry. Thus the basic prerequisites remain a
working knowledge of general topology and differential geometry. This book
should be accessible to advanced graduate students in either mathematics or
mathematical physics.
The list of works to which we are indebted for the two editions is quite
extensive. In particular, we would like to single out the following five im-
portant sources: The Large Scale Stmcture of Space-time by S. W. Hawk-
ing and G. I?. R. Ellis; Techniques of Diflerential Topology in Relativity by
R. Penrose; Riemannsche Geometric im Grossen by D. Gromoll, W. Klin-
genberg, and W. Meyer; the 1977 Diplomarbeit a t Bonn University, Exzstenz
und Bedeutung von konjugierten Werten in der Raum-Zeit, by G. Bolts; and
Semi-Riemannian Geometry by B. O'Neill.
In the time from the late 1970's when we wrote the first edition to the
PREFACE TO THE SECOND EDlTION v
present, it has been interesting to observe the enormous expansion in the
journal literature on space-time differential geometry. This is reflected in the
substantial growth of the list of references for the second edition. However,
this wealth of new material has precluded our treating many interesting new
developments in space-time geometry since 1980 which are less closely tied in
with the overall viewpoint and selection of topics originally discussed in the
first edition.
The authors would like to thank all those who have provided encouraging
comments about the first edition and urged us to issue a second edition after
the first edition had gone out of print, especially Gregory Galloway, Steven
Harris, Andrzej Kr6lak, Philip Parker, and Susan Scott. We thank Gerard
Emch for insisting that the second edition be undertaken, and the first two
authors thank Stephen Summers and Maria Allegra for independently sug-
gesting that a third author be added to the team to share the duties of the
completion of this revision. It is also a pleasure to thank Maria Allegra and
Christine McCafferty at Marcel Dekker, Inc., for working with us to see the
second edition into print. We are also indebted to Lia Petracovici for much
helpful proofreading and to Todd Hammond for valuable technical advice con-
cerning A M S W .
John K. Beem
Paul E. Ehrlich
Kevin L. Easley
PREFACE TO THE FIRST EDITION
This book is about Lorentzian geometry, the mathematical theory used in
general relativity, treated from the viewpoint of global differential geometry.
Our goal is to help bridge the gap between modern differential geometry and
the mathematical physics of general relativity by giving an invariant treat-
ment of global Lorentzian geometry. The growing importance in physics of
this approach is clearly illustrated by the recent Hawking-Penrose singularity
theorems described in the text of Hawking and Ellis (1973).
The Lorentzian distance function is used as a unifying concept in our book.
Furthermore, we frequently compare and contrast the results and techniques
of Lorentzian geometry to those of Riemannian geometry to alert the reader
to the basic differences between these two geometries.
This book has been written especially for the mathematician who has a
basic acquaintance with Riemannian geometry and wishes to learn Lorentzian
geometry. Accordingly, this book is written using the notation and methods
of modern differential geometry. For readers less familiar with this notation,
we have included Appendix A which gives the local coordinate representations
for the symbols used.
The basic prerequisites for this book are a working knowledge of general
topology and differential geometry. Thus this book should be accessible to
advanced graduate students in either mathematics or mathematical physics.
In writing this monograph, both authors profited greatly from the oppor-
tunity to lecture on part of this material during the spring semester, 1978, a t
the University of Missouri-Columbia. The second author also gave a series of
lectures on this material in Ernst Ruh's seminar in differential geometry a t
Bonn University during the summer semester, 1978, and would like to thank
... vlll PREFACE TO THE FIRST EDITION
Professor Ruh for giving him the opportunity to speak on this material. We
would like to thank C. Ahlbrandt, D. Carlson, and M. Jacobs for several help
ful conversations on Section 2.4 and the calculus of variations. We would like
to thank M. Engman, S. Harris, K. Nomizu, T. Powell, D. Retzloff, and H. Wu
for helpful comments on our preliminary version of this monograph. We also
thank S. Harris for contributing Appendix D to this monograph and J.-H. Es-
chenburg for calling our attention to the Diplomarbeit of Bolts (1977). To
anyone who has read either of the excellent books of Gromoll, Klingenberg,
and Meyer (1975) on Riemannian manifolds or of Hawking and Ellis (1973)
on general relativity, our debt to these authors in writing this work will be
obvious. It is also a pleasure for both authors to thank the Research Council
of the University of Missouri-Columbia and for the second author to thank
the Sonderforschungsbereich Theoretische Mathematik 40 of the Mathematics
Department, Bonn University, and to acknowledge an NSF Grant MCS77-
18723(02) held at the Institute for Advanced Study, Princeton, New Jersey,
for partial financial support while we were working on this monograph. Fi-
nally it is a pleasure to thank Diane Coffman, DeAnna Williamson, and Debra
Retzloff for the patient and cheerful typing of the manuscript.
John K. Beem
Paul E. Ehrlich
CONTENTS
Preface to the Second Edition iii
Preface to the First Edition vii
List of Figures xiii
1. Introduction: Riemannian Themes in Lorentzian Geometry 1
2. Connections and Curvature 15
2.1 Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Semi-Riemannian Manifolds 20
. . . . . . . . . . . . . . . . . . . . 2.3 Null Cones and Semi-Riemannian Metrics 25
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Sectional Curvature 29
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 The Generic Condition 33
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 The Einstein Equations 44
3. Lorentzian Manifolds and Causality 49
3.1 Lorentzian Manifolds and Convex Normal Neighborhoods . . . . . . 50
3.2 Causality Theory of Spacetimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
. . . . . . . . . . . . . . . . . 3.3 Limit Curves and the C" Topology on Curves 72
3.4 Two-Dimensional Spacetimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.5 The Second Fundamental Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.6 WarpedProducts .......................................... 94
3.7 Semi-Riemannian Local Warped Product Splittings . . . . . . . . . . . 117
x CONTENTS
4 . Lorentzian Distance 135
............................. 4.1 Basic Concepts and Definitions 135
4.2 Distance Preserving and Homothetic Maps .................. 151
. . . . . . . . . . . . . 4.3 The Lorentzian Distance Function and Causality 157
. . . . . . . . . . . . . 4.4 Maximal Geodesic Segments and Local Causality 166
5 . Examples of Space-times 173
5.1 Minkowski Space-time .................................... 174
5.2 Schwarzschild and Kerr Space-times ........................ 179
5.3 Spaces of Constant Curvature .............................. 181
5.4 Robertson-Walker Space-times ............................ 185 I 5.5 Bi-Invariant Lorentzian Metrics on Lie Groups ............... 190
I 6 . Completeness a n d Extendibility 197
6.1 Existence of Maximal Geodesic Segments . . . . . . . . . . . . . . . . . . . . 198
6.2 Geodesic Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
6.3 Metric Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
6.4 IdealBoundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
6.5 Local Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
6.6 Singularities ............................................. 225
7 . Stability of Completeness and Incompleteness 239
7.1 Stable Properties of Lor(M) and Con(M) ................... 241
7.2 The C1 Topology and Geodesic Systems . . . . . . . . . . . . . . . . . . . . . 247
7.3 Stability of Geodesic Incompleteness for Robertson-Walker Space-times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
7.4 Sufficient Conditions for Stability ........................... 263
8 . Maximal Geodesics and Causally Disconnected Space-times 271
8.1 Almost Maximal Curves and Maximal Geodesics ............. 273
8.2 Nonspacelike Geodesic Rays in Strongly Causal Space-times . . . 279
8.3 Causally Disconnected Space-times and Nonspacelike GwdesicLines ........................................... 282
CONTENTS xi
9 . T h e Lorentzian C u t Locus 295
9.1 The Timelike Cut Locus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298
9.2 The Null Cut Locus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
9.3 The Nonspacelike Cut Locus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
9.4 The Nonspacelike Cut Locus Revisited . . . . . . . . . . . . . . . . . . . . 318
10 . Morse Index Theory o n Lorentzian Manifolds 323
........................ 10.1 The Timelike Morse Index Theory 327
10.2 The Timelike Path Space of a Globally Hyperbolic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Space-time 354
10.3 The Null Morse Index Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365
11 . Some Results in Global Lorentzian Geometry 399
11.1 The Timelike Diameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401
11.2 Lorentzian Comparison Theorems . . . . . . . . . . . . . . . . . . . . . . . . 406
11.3 Lorentzian Hadamard-Cartan Theorems . . . . . . . . . . . . . . . . . . 411
12 . Singularities 425
12.1 JacobiTensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426
12.2 The Generic and Timelike Convergence Conditions . . . . . . . . . 433
12.3 Focal Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444
12.4 The Existence of Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467
12.5 SmoothBoundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472
13 . Gravitational P lane Wave Space-times 479
13.1 The Metric. Geodesics. and Curvature . . . . . . . . . . . . . . . . . . . . 480
13.2 Astigmatic Conjugacy and the Nonspacelike Cut Locus . . . . . 486
13.3 Astigmatic Conjugacy and the Achronal Boundary . . . . . . . . . 493
14 . T h e Splitting Problem in Global Lorentzian Geometry 501
14.1 The Busemann Function of a Timelike Geodesic Ray . . . . . . . . 507
14.2 Co-rays and the Busemann Function . . . . . . . . . . . . . . . . . . . . . . 519
14.3 The Level Sets of the Busemann Function . . . . . . . . . . . . . . . . . 538
14.4 The Proof of the Lorentzian Splitting Theorem ............. 549
14.5 Rigidity of Geodesic Incompleteness ...................... 563
xii CONTENTS !
Appendixes 567 A. Jacobi Fields and Toponogov's Theorem for Lorentzian
. . . . . . . . . . . . . . . . . . . . . . . . . . . Manifolds by Steven G. Harris 567
B. From the Jacobi, to a Riccati, t o the Raychaudhuri Equation: Jacobi Tensor Fields and the Exponential Map 1 Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573
LIST OF FIGURES
References
List of Symbols
Index
Figure Page Brief Description of Figure
chronological and causal future
basis of Alexandrov topology, I f ( p ) n I - ( q )
reverse triangle inequality for p << T << q
failure of causal continuity
nonspacelike curves may be imprisoned in compact sets
hierarchy of causality conditions
limit curve convergence but not C0 convergence
C0 convergence but not limit curve convergence
for the proof of Proposition 3.34
for the proof of Proposition 3.42
for the proof of Theorem 3.43
warped product structure of M x f H
for the proof of Theorem 3.68
yn -+ y , but lim L (7,) = 0 < L ( y )
d ( p , q ) = cc in Reissner-Nordstrijm space-time
pn + p, but d(p, q ) < lim inf d ( ~ n , q )
inner ball B+(p, E )
outer balls O+(P, E ) and 0- (p, E )
causally continuous with d ( p , q) not continuous
horismos E + ( p ) in Minkowski space-time
unit sphere K ( p , 1 )
horismos in W: = L~ with one point removed
conformal Wy = Ln with null, spacelike, timelike infinity
xiv I
LIST OF FIGURES !
Figure Page Brief Description of Figure
Penrose diagram for Minkowski space-time
Kruskal diagram for Schwarzschild space-time
de Sitter space-time
2-dimensional universal anti-de Sitter space-time
spacelike and null complete, but timelike incomplete
d(p , x,) -+ 0, but no accumulation point for {x,)
TIP'S and TIF's
local b-boundary extension
compact closure in a local extension
local extension of Minkowski space-time
for the proof of Theorem 6.23
Reissner-Nordstrom space-time with e2 = m2
for the proof of Proposition 8.18
example with s(v) = m, v, -t v, and s(v,) 5 4
simply connected but not future 1-connected
for the proof of Proposition 10.36
focal points in the Euclidean plane
focal points t o a spacelike submanifold
a focal point example in Wi = L3
Cauchy development D+(S)
a trapped set example in S1 x S1
a causal completion example
a causally disconnecting set example
causal disconnection for a Robertson-Walker space-time
the conjugate locus and astigmatic conjugacy
the null tail and astigmatic conjugacy
CHAPTER 1
INTRODUCTION: RIEMANNIAN THEMES IN LORENTZIAN GEOMETRY
In the 1970's, progress on causality theory, singularity theory, and black
holes in general relativity, described in the influential text of Hawking and
Ellis (1973), resulted in a resurgence of interest in global Lorentzian geometry.
Indeed, a better understanding of global Lorentzian geometry was required for
the development of singularity theory. For example, i t was necessary to know
that causally related points in globally hyperbolic subsets of spacetimes could
be joined by a nonspacelike geodesic segment maximizing the Lorentzian arc
length among all nonspacelike curves joining the two given points. In addi-
tion, much work done in the 1970's on foliating asymptotically flat Lorentzian
manifolds by families of maximal hypersurfaces has been motivated by general
relativity [cf. Choquet-Bruhat, Fischer, and Marsden (1979) for a partial list
of references].
All of these results naturally suggested that a systematic study of global
Lorentzian geometry should be made. The development of "modern" global
Riemannian geometry as described in any of the standard texts [cf. Bishop
and Crittenden (1964), Gromoll, Klingenberg, and Meyer (1975), Helgason
(1978), Hicks (1965)] supported the idea that a comprehensive treatment of
global Lorentzian geometry should be grounded in three fundamental topics:
geodesic and metric completeness, the Lorentzian distance function, and a
Morse index theory valid for nonspacelike geodesic segments in an arbitrary
Lorentzian manifold.
Geodesic completeness, or more accurately geodesic incompleteness, played
a crucial role in the development of singularity theory in general relativity and
has been thoroughly explored within this framework. However, the Lorentzian
distance function had not been as well investigated, although it had been of
2 1 INTRODUCTION
some use in the study of singularities [cf. Hawking (1967), Hawking and Ellis
(1973), Tipler (1977a), Beern and Ehrlich (1979a)l. Some of the properties of
the Lorentzian distance function needed in general relativity had been briefly
described in Hawking and Ellis (1973, pp. 215-217). Further results relat-
ing Lorentzian distance to causality and the global behavior of nonspacelike
geodesics had been given in Beem and Ehrlich (1979b). Hence, as part of the
first edition, a systematic study of the Lorentzian distance function was made.
Uhlenbeck (1975), Everson and Talbot (1976), and Woodhouse (1976) had
studied Morse index theory for globally hyperbolic space-times, and we had
sketched [cf. Beem and Ehrlich (1979c,d)] a Morse index theory for nonspace-
like geodesics in arbitrary space-times. But no complete treatment of this
theory for arbitrary space-times had been published prior to the first edition
of the current book.
The Lorentzian distance function has many similarities with the Riemann-
ian distance function but also many differences. Since the Lorentzian distance
function is still less well known, we now review the main properties of the Rie-
mannian distance function and then compare and contrast the corresponding
results for the Lorentzian distance function.
For the rest of this portion of the introduction, we will let (N, go) denote a
Riemannian manifold and (M, g) denote a Lorentzian manifold, respectively.
Thus N is a smooth paracompact manifold equipped with a positive definite
inner product golp : T,N x TpN -+ R on each tangent space T,N. In addition,
if X and Y are arbitrary smooth vector fields on N , the function N -+ R given
by p 4 go(X(p), Y(p)) is required to be a smooth function. The Riemannian
structure go : T N x T N 4 R then defines the Riemannian distance function
as follows. Let Q , , denote the set of piecewise smooth curves in N from p
to q. Given a curve c E flp,, with c : [O,1] + N, there is a finite partition
0 = t l < t z < . . . < tk = 1 such that c 1 It,, ti+l] is smooth for each i. The
Riemannian arc length of c with respect to go is defined as
RIEMANNIAN THEMES IN LORENTZIAN GEOMETRY 3
The Riemannian distance do(p, q) between p and q is then defined to be
For any Riemannian metric go for N , the function do : N x N + [0, co) has
the following properties:
(1) ~ o ( P , q) = do(q,p) for all P, q E N. (2) do(p, q) I do (P, r ) + do(r, q) for all P, q, r E N . (3) do(p, q) = 0 if and only if p = q.
More surprisingly,
(4) do : N x N -+ 10, co) is continuous, and the family of metric balls
for all p E N and E > 0 forms a basis for the given manifold topology.
Thus the metric topology and the given ~narlifold topology coincide. Further-
more, by a result of Whitehead (1932), given any p E N, there exists an R > 0
such that for any E with 0 < E < R, the metric ball B ( p , E) is geodesically
convex. Thus for any E with 0 < E < R, the set B(p, E) is diffeomorphic to the
n-disk, n = dim(N), and the set {q E N : do(p,q) = E ) is diffeomorphic to sn-1
Removing the origin from R2 equipped with the usual Euclidean metric and
setting p = (-1, O), q = (1, O), one calculates that do(p, q) = 2 but finds no
curve c E R,,, with Lo(c) = &(p,q) and also no smooth geodesic from p to
q. Thus the following questions arise naturally. Given a manifold N, find
conditions on a Riemannian metric go for N such that
(1) All geodesics in N may be extended to be defined on all of R.
(2) The pair (N, do) is a complete metric space in the sense that all Cauchy
sequences converge.
(3) Given any two points p,q E N, there is a smooth geodesic segment
c E Q,,, with Lo(c) = do(p, q).
A distance realizing geodesic segment as in (3) is called a minimal geodesic
segment. The word minimal is used here since the definition of Riemannian
4 1 INTRODUCTION
distance implies that Lo(y) 2 do(p, q) for all y E R,,,. More generally, one
may define an arbitrary piecewise smooth curve y E R,,, to be minimal if
Lo(y) = &(p, q). Using the variation theory of the arc length functional, it
may be shown that if y f a,,, is minimal, then y may be reparametrized to a
smooth geodesic segment.
The question of finding criteria on go such that (I), (2), or (3) holds was
resolved by H. Hopf and W. Rinow in their famous paper (1931). In modern
terminology the Hopf-Rinow Theorem asserts the following.
Theorem (Hopf-Rinow). For any Riemannian manifold (N, go) the fol-
lowing are equivalent:
(1) Metric completeness: (N, &) is a complete metric space.
(2) Geodesic completeness: For any v E T N , the geodesic c(t) in N with
c'(0) = v is defined for all positive and negative real numbers t E R.
(3) For some p E N , the exponential map exp, is defined on the entire
tangent space TpN to N at p.
(4) Finite compactness: Every subset K of N that is do bounded (i.e.,
sup{&(p, q) : p, q E K) < m) has compact closure.
Furthermore, if any one of (1) through (4) holds, then
(5) Minimal geodesic connectibility: Given any p, q E N , there exists a
smooth geodesic segment c from p to q with Lo(c) = do(p, q).
A Riemannian manifold (N, go) is said to be complete provided any one (and
hence all) of conditions (1) through (4) is satisfied. It should be stressed that
the Hopf-Rinow Theorem guarantees the equivalence of metric and geodesic
completeness and also that all fiemannian metrics for a compact smooth
manifold are complete. Unfortunately, none of these statements is valid for
arbitrary Lorentzian manifolds.
A remaining question for noncompact but paracompact manifolds is the
existence of complete Riemannian metrics. This was settled by Nomizu and
Ozeki's (1961) proof that given any Riemannian metric go for N, there is a
complete Riemannian metric for N globally conformal to go. Since any para-
compact connected smooth manifold N admits a Riemannian metric by a par-
RIEMANNIAN THEMES IN LORENTZIAN GEOMETRY 5
tition of unity argument, it follows that N also admits a complete Riemannian
metric.
We now turn our attention to the Lorentzian manifold (M, g). A Lorentzian
metric g for the smooth paracompact manifold M is the assignment of a nonde-
generate bilinear form glp : T,M x TpM -+ R with diagonal form (-, +, . . . , +)
to each tangent space. It is well known that if M is compact and x ( M ) # 0,
then M admits no Lorentzian metric. On the other hand, any noncompact
manifold admits a Lorentzian metric. Geroch (1968a) and Marathe (1972)
have also shown that a smooth Hausdorff manifold which admits a Lorentzian
metric is paracompact.
Nonzero tangent vectors are classified as timelike, spacelike, nonspacelike,
or null according to whether g(v,v) < 0, > 0, 5 0, or = 0, respectively.
[Some authors use the convention (+, -, . . . , -) for the Lorentzian metric,
and hence all of the inequality signs in the above definition are reversed for
them.] A Lorentzian manifold (M, g) is said to be time oriented if M admits
a continuous, nowhere vanishing, timelike vector field X. This vector field is
used to separate the nonspacelike vectors a t each point into two classes called
future directed and past directed. A space-time is then a Lorentzian manifold
(M,g) together with a choice of time orientation. We will usually work with
space-times below.
In order to define the Lorentzian distance function and discuss its properties,
we need to introduce some concepts from elementary causality theory. It is
standard to write p << q if there is a future directed piecewise smooth timelike
curve in M from p to q, and p < q if p = q or if there is a future directed
piecewise smooth nonspacelike curve in M from p to q. The chronological past
and future of p are then given respectively by I-(p) = {q E M : q << p) and
I+(p) = {q E M : p << q). The causal past and future of p are defined as
J-(p) = {q E M : q < p) and J+(p) = {q E M : p < q). The sets I-(p) and
I+(p) are always open in any space-time, but the sets J-(p) and J+(p) are
neither open nor closed in general (cf. Figure 1.1).
The causal stmcture of the space-time (M, g) may be defined as the collec-
tion of past and future sets at all points of M together with their properties.
6 1 INTRODUCTION
:..:..:...... ....;, / . . . . . . . . . . . . . . . . :.. :_. :_. :.. , . . . . . . . . . . .
- - +i, . . . . . . . . . . . . . . . . . . . . .?. . . . . . . . . .'.. :.. :.. :./ . . . . . . . . . . .
t remove this closed set
r
FIGURE 1.1. The chronological (respectively, causal) future of a
point consists of all points which can be reached from that point
by future directed timelike (respectively, nonspacelike) curves. In
this example, the causal future J + ( r ) of r is the closure of the
chronological future I + ( r ) of r. However, the set J+(q) is not the
closure of I+(q). In particular, the point w is in the closure of I+(q)
but is not in J+(q).
I t may be shown that two strongly causal Lorentzian metrics gl and g2 for
M determine the same past and future sets at all points if and only if the
two metrics are globally conformal [i.e., gl = Rg2 for some smooth function
0 : M --+ (0, a)]. Letting C(M,g) denote the set of Lorentzian metrics glob-
ally conformal to g, it follows that properties suitably defined using the past
and future sets hold simultaneously either for all metrics in C(M, g) or for no
metric in C(M,g). Thus all of the basic properties of elementary causality
RIEMANNIAN THEMES IN LORENTZIAN GEOMETRY 7
theory depend only on the conformal class C(M,g) and not on the choice of
Lorentzian metric representing C(M, g).
Perhaps the two most elementary properties to require of the conformal
structure C(M, g) are either that (M, g) be chronological or that (M, g) be
causal. A space-time (M,g) is said to be chronological if p $! I+(p) for all
p E M. This means that (M,g) contains no closed timelike curves. The
space-time (M, g) is said to be causal if there is no pair of distinct points
p, q E M with p < q < p. This is equivalent to requiring that (M, g) contain
no closed nonspacelike curves.
Already at this stage, a basic difference emerges between Lorentzian and
Riemannian geometry. On physical grounds, the space-times of general rela-
tivity are usually assumed to be chronological. But it is easy to show that if
M is compact, (M, g) contains a closed timelike curve. Thus the space-times
usually considered in general relativity are assumed to be noncompact.
In general relativity each point of a Lorentzian manifold corresponds to an
event. Thus the existence of a closed timelike curve raises the possibility that
a person might traverse some path and meet himself at an earlier age. Wlore
generally, closed nonspacelike curves generate paradoxes involving causality
and are thus said to "violate causality." Even if a space-time has no closed
nonspacelike curves, it may contain a point p such that there are future di-
rected nonspacelike curves leaving arbitrarily small neighborhoods of p and
then returning. This behavior is said to be a violation of strong causality at
p. Space-times with no such violation are strongly causal.
The strongly causal space-times form an important subclass of the causal
space-times. For this class of space-times the Alexandrov topology with basis
{I+(p) n I-(q) : p, q E M) for M and the given manifold topology are related
as follows [cf. Kronheimer and Penrose (1967), Penrose (1972)l.
Theorem. The following are equivalent:
(1) (MI g) is strongly causal.
(2) The Alexandrov topology induced on M agrees with the given manifold
topology.
(3) The Alexandrov topology is Hausdorff
1 INTRODUCTION
q
FIGURE 1.2. Sets of the form I+(p)nI-(q) with arbitrary p, q E M
form a basis of the Alexandrov topology. This topology is always
at least as coarse as the original topology on M. The Alexandrov
topology agrees with the original topology if and only if (M,g) is
strongly causal.
Ure are ready a t last to define the Lorentzian distance function
d = d(g) : M x M -+ [0, m]
of an arbitrary space-time. If c : [O, 11 -+ M is a piecewise smooth nonspacelike
curve differentiable except a t 0 = t l < t2 < ... < t k = 1, then the length
L(c) = L,(c) of c is given by the formula
If p (< q, there are timelike curves from p to q (very close to piecewise null
curves) of arbitrarily small length. Hence the infimum of Lorentzian arc length
of all piecewise smooth curves joining any two chronologically related points
RIEMANNIAN THEMES IN LORENTZIAN GEOMETRY 9
p (< q is zero. On the other hand, if p (< q and p and q lie in a geodesically
convex neighborhood U, the future directed timelike geodesic segment in U
from p to q has the largest Lorentzian arc length among all nonspacelike curves
in U from p to q. Thus the following definition for d(p, q) is natural: fixing
a point p E M, set d(p, q) = 0 if q J+(p), and otherwise calculate d(p, q)
for q E J+(p) as the supremum of Lorentzian arc length of all future directed
nonspacelike curves from p to q. Thus if q E J+(p) and y is any future
directed nonspacelike curve from p to q, we have L(y ) 5 d(p, q). Hence unlike
the Riemannian distance function, the Lorentzian distance function is not
a priori finite-valued. Indeed, a secalled totally vicious space-time may be
characterized in terms of its Lorentzian distance function by the property that
d ( p , q) = m for all p, q E M. Also, if (M, g) is nonchronological and p E I+(p),
it follows that d(p,p) = w.
The Reissner-Nordstrijm space-times, physically important examples of ex-
act solutions to the Einstein equations in general relativity, also contain pairs
of chronologically related distinct points p << q with d(p, q) = co.
By definition of Lorentzian distance, d(p, q) = 0 whenever q f M - J'(p),
We have even seen that d(p, p) = rn is possible. Thus for arbitrary Lorentzian
manifolds there is no analogue for property (3) of the Riemannian distance
function. Also, the Lorentzian distance function tends from its definition to
be a nonsymmetric distance. In particular, for any space-time it may be shown
that if 0 < d(p, q) < m, then d(q,p) = 0. But the Lorentzian distance function
does possess a useful analogue for property (2) of the Riemannian distance
function. Namely, d(p, q) 2 d(p, r) + d(r, q) for all p, q, T f IvI with p < r < q.
The reversal of inequality sign is to be expected since nonspacelike geodesics
in a Lorentzian manifold locally maximize rather than minimize arc length.
Since d(p,q) > 0 if and only if q E I+(p), and d(q,p) > 0 if and only if
q E I-(p), the distance function determines the chronology of (M, g). On
the other hand, conformally changing the metric changes distance but not the
chronology, so that the chronology does not determine the distance function.
Clearly, the distance function does not determine the sets J+(p) or J-(p) since
d(p, q) = 0 not only for q E J+(p) - I+(p) but also for q E M - J+(p).
1 INTRODUCTION
FIGURE 1.3. If r is in the causal future of p and q is in the causal
future of r, then the distance function satisfies the reverse triangle
inequality d(p, q) > d(p, r ) + d(r, q). The reverse triangle inequality
will not be valid in general for some point r' which is not causally
between p and q.
Property (4) of the Riemannian distance function is the continuity of this
function for all Riemannian metrics. For space-times, on the other hand, the
Lorentzian distance function may fail to be upper semicontinuous. Indeed, the
continuity of d : M x M + 10, m ] has the following consequence for the causal
structure of (M,g) [cf. Theorem 4.241. If (M,g) is a distinguishing space-
time and d is continuous, then (M, g) is causally continuous (cf. Chapter 3 for
definitions of these concepts). Hence it is necessary to accept the lack of con-
tinuity and lack of finiteness of the Lorentzian distance function for arbitrary
space-times. The Lorentzian distance function is a t least lower semicontinu-
ous where it is finite. This may be combined with the upper semicontinuity
in the C0 topology of the Lorentzian arc length functional for strongly causal
space-times to construct distance realizing nonspacelike geodesics in certain
classes of space-times (cf. Sections 8.1 and 8.2).
RIEMANNIAN THEMES IN LORENTZIAN GEOMETRY 11
With these remarks in mind, it is natural to ask if some class of spacetimes
for which the Lorentzian distance function is finite-valued and/or continuous
may be found. The globally hyperbolic spacetimes turn out to be such a
class. Here a space-time (M, g) is said to be globally hyperbolic if it is strongly
causal and satisfies the condition that J+(p)nJ-(q) is compact for allp,q E M.
It has been most useful in proving singularity theorems in general relativity
to know that if (M,g) is globally hyperbolic, then its Lorentzian distance
function is finite-valued and continuous. Oddly enough, the finiteness of the
distance function, rather than its continuity, characterizes globally hyperbolic
space-times in the following sense (cf. Theorem 4.30). We say that a space-
time (M, g) satisfies the finite distance condition provided that d(g)(p, q) < oo for all p,q E M. It may then be shown that the strongly causal Lorentzian
manifold (M, g) is globally hyperbolic if and only if (M, g') satisfies the finite
distance condition for all g' C(M, g).
Motivated by the definition of minimal curve in Riemannian geometry, we
make the following definition for space-times.
Definition. (Mazimal Curve) A future directed nonspacelike curve y
from p to q is said to be maximal if L(y) = d(p, q).
It may be shown (cf. Theorem 4.13), just as for minimal curves in Rie-
mannian spaces, that if y is a maximal curve from p to q, then -f may be
reparametrized to a nonspacelike geodesic segment. This result may be used
to construct geodesics in strongly causal spacetimes as limit curves of appro-
priate sequences of "almost maximal" nonspacelike curves (cf. Sections 8.1,
8.2).
In view of (5) of the Hopf-Rinow Theorem for Riemannian manifolds, it
is reasonable to look for a class of space-times satisfying the property that if
p < q, there is a maximal geodesic segment from p to q. Using the compactness
of J+(p) n J-(q), one can show that globally hyperbolic space-times always
contain maximal geodesics joining any two causally related points.
We are finally led to consider what can be said about Lorentzian analogues
for the rest of the Hopf-Rinow Theorem. Here, however, every conceivable
thing goes wrong. Thus much of the difficulty in Lorentzian geometry from
12 1 INTRODUCTION
the viewpoint of global Riemannian geometry or its richness from the viewpoint
of singularity theory in general relativity stems from the lack of a sufficiently
strong analogue of the Hopf-Rinow Theorem.
We now give a basic definition which corresponds to (2) of the Hopf-Rinow
Theorem.
Definition. (Timelike, Null, and Spacelike Geodesic Completeness) A
space-time (M,g) is said to be timelike (respectively, null, spacelike, non-
spacelike) complete if all timelike (respectively, null, spacelike, nonspacelike)
geodesics may be defined for all values -co < t < cm of an aRne parameter t.
A space-time which is nonspacelike incomplete thus has a timelike or null
geodesic which cannot be defined for all values of an affine parameter. Such
space-times are said to be singular in the theory of general relativity.
It is first important to note that global hyperbolicity does not imply any
of these forms of geodesic completeness. This may be seen by fixing points
p and q in Minkowski space with p << q and equipping M = I+(p) n I-(q)
with the Lorentzian metric it inherits as an open subset of Minkowski space.
This space-time M is globally hyperbolic. Since geodesics in M are just the
restriction of geodesics in Minkowski space to M , it follows that every geodesic
in M is incomplete.
It was once hoped that timelike completeness might imply null complete-
ness, etc. However, a series of examples has been given by Kundt, Geroch,
and Beem of globally hyperbolic space-times for which timelike geodesic com-
pleteness, null geodesic completeness, and spacelike geodesic completeness are
all logically inequivalent. Thus, there are globalIy hyperbolic spacetimes that
are s~acelike and timelike complete but null incomplete.
Metric completeness and geodesic completeness [(I) iff (2) of the Hopf-
Rinow Theorem] are unrelated for arbitrary Lorentzian manifolds. There are
also Lorentzian metrics which are timelike geodesically complete but also con-
tain points p and q with p << q such that no timelike geodesic from p to q
exists (cf. Figure 6.1).
On the brighter side, there is some relationship between (1) and (4) of the
Hopf-Rinow Theorem for globally hyperbolic space-times. Since d(p, q) = 0
RIEMANNIAN THEMES IN LORENTZIAN GEOMETRY 13
if q $ J+(p), convergence of arbitrary sequences in (M,g) with respect to
the Lorentzian distance function does not make sense. But timelike Cauchy
completeness may be defined (cf. Section 6.3). It can be shown for globally
hyperbolic space-times that timelike Cauchy completeness and a type of finite
compactness are equivalent.
In addition, results analogous to the Nomizu-Ozeki Theorem mentioned
above for Riemannian metrics have been obtained. For instance, given any
strongly causal space-time (M, g), there is a conformal factor fl : M -+ (0, m)
such that the space-time (M, fig) is timelike and null geodesically conlplete
(cf. Theorem 6.5). It is still unknown, however, whether this result can be
strengthened to include spacelike geodesic completeness as well.
It should now be clear that while there are certain similarities between
the Lorentzian and the Riemannian distance functions, especially for globally
hyperbolic space-times, there are also striking differences. In spite of these
differences, the Lorentzian distance function has many uses similar to those of
the Riemannian distance function.
In Chapter 8 the Lorentzian distance function is used in constructing max-
imal nonspacelike geodesics. These maximal geodesics play a key role in the
proof of singularity theorems (cf. Chapter 12). In Chapter 9 the Lorentzian
distance function is used to define and study the Lorentzian cut locus.
In Chapter 10 a Morse index theory is developed for both timelike and null
geodesics. A number of global results for Lorentzian manifolds are obtained
in Chapter 11 using the index theory and the Lorentzian distance function.
Also, results are presented concerning the relationship of geodesic connectibil-
ity to geodesic pseudoconvexity and geodesic disprisonrnent. In Chapter 13 a
nontrivial example is given of the cut locus structure and certain associated
achronal boundaries for the class of gravitational plane wave space-times from
general relativity. Finally, Chapter 14 treats the concept of rigidity of geo-
desic incompleteness and the Lorentzian Splitting Theorem for space-times
satisfying the timelike convergence condition which also contain a complete
Lorentzian distance-realizing timelike geodesic line. In this setting, the almost
maximal curves of Chapter 8 again play a role [cf. Galloway and Horta (1995)l.
CHAPTER 2
CONNECTIONS AND CURVATURE
Let (M, g) be an n-dimensional manifold M with a semi-Riemannian met-
ric g of arbitrary signature (-, . . . , -, +, . . . , +). Then there exists a unique
connection V on M which is both compatible with the metric tensor g and
torsion free. This connection, which is called the Levi-Czvita connection of
(M, g ) , satisfies the same formal relations whether or not (M, g) is positive
definite. Thus geodesics, curvature, Ricci curvature, scalar curvature, and sec-
tional curvature may all be defined for semi-Riemannian manifolds using the
same formulas as for positive definite Riemannian manifolds. The only diffi-
culty which arises is that the sectional curvature is not defined for degenerate
sections of the tangent space when (M,g) is not of constant curvature. In fact,
the sectional curvature is only bounded near all degenerate sections a t a point
in the case of constant sectional curvature [see Kulkarni (1979)l. This generic
"blow up" of the sectional curvature at degenerate sections corresponds to a
generic unboundedness of tidal accelerations [cf. Beem and Parker (1990)] for
objects moving arbitrarily close to the speed of light.
In the first section of this chapter there is an introduction to connections,
and in the second section semi-Riemannian manifolds are discussed. Riemann-
ian manifolds are positive definite semi-Riemannian manifolds and thus have
metrics of signature (+, +, . - . , +). Consequently, the metric induced on the
tangent space of a Riemannian manifold is Euclidean, The types of semi-
Riemannian manifolds of primary interest in this book are Lorentzian mani-
folds. These manifolds have metric tensors of signature (-, +, . . . , +). Thus,
the tangent spaces of a Lorentzian manifold have induced Minkowskian met-
rics. The tangent vectors a t p E M of length zero form the null cone at p. For
semi-Riemannian manifolds which are neither positive nor negative definite, a
16 2 CONNECTIONS AND CURVATURE
null cone is an (n - 1)-dimensional surface in the tangent space. In the third
section of this chapter we show that null cones determine the metric up to a
conformal factor for metrics which are not definite. In the fourth section we
consider sectional curvature. This curvature is related to tidal accelerations
using the Jacobi equation. The Jacobi equation analyzes the relative behavior
of "nearby" geodesics and will be of fundamental importance in later c h a p
ters. We introduce the generic condition in the fifth section. This condition
corresponds to a tidal acceleration assumption. The Einstein equations are
introduced in the last section. These are partial differential equations relating
the metric tensor and its &st two derivatives to the energy-momentum ten-
sor T. The Einstein equations thus link geometry in terms of the metric and
curvature to physics in terms of the distribution of mass and energy.
The manifold M will always be smooth, paracompact, and Hausdorff. The
tangent bundle will be denoted by TM, and the tangent space at p E M will
be denoted by TpM. If (U, x) is an arbitrary chart for M, then (x', x2,. . . , xn)
will denote local coordinates on M and 6/6x1, 6/6x2,. . . , 6/6xn will denote
the natural basis for the tangent space.
2.1 Connections
Let X(M) denote the set of all smooth vector fields defined on M , and let
$(M) denote the ring of all smooth real-valued functions on M. A connection
is a function
V : X(M) x X(M) + X(M)
with the properties that
(2.1) Vv(X + Y) = v v x + VvY,
(2.2) Vfv+hw(X) = f V v X + hVwX, and
(2.3) Vv(fX) = f v v x + Vtf ) X
for all f , h E z (M) and all X, Y, V, W E X(M).
The vector V X ( ~ ) Y = VxYIp at the point p E M depends only on the
connection V, the value X(p) = Xp of X at p, and the values of Y along any
2.1 CONNECTIONS 17
smooth curve which passes through p and has tangent X(p) at p [cf. Hicks
(1965, p. 57)]. To see this, let El, Ez,. . . , E n be smooth vector fields defined
near p which form a basis of the tangent space at each point in a neighborhood
of p. Then X(p) = C Xi(p) Ei (p) and Y = C Yi Ei. Hence
It follows that Xi(p), Yi(p), and X(p)(Yi) determine VxYI, completely if
the VE,(,)Ei's are known.
Given the connection V on M and a curve y : [a, b ] -+ M , we may define
parallel translation of vector fields along y. Here a vector field Y along y is a
smooth mapping Y : [a, b ] + TM such that Y(t) E T,(t)M for each t E [a, b ] .
For to E [a, b ] we may locally extend Y to a smooth vector field defined on a
neighborhood of $to). Then we may consider the vector field V,,(,]Y along
y. The preceding arguments show that this vector field along y is independent
of the local extension, and consequently V,,Y (= Y') is well defined. A vector
field Y along y which satisfies V,lY(t) = 0 for all t E [a, b ] is said to move by
parallel translation along y. A geodesic c : (a, b) -+ M is a smooth curve of M
such that the tangent vector c' moves by parallel translation along c. In other
words, c is a geodesic if
(2.4) Vc~c1 = 0. (geodesic equation)
A pregeodesic is a smooth curve c which may be reparametrized to be a geo-
desic. Any parameter for which c is a geodesic is called an afine parameter.
If s and t are two affine parameters for the same pregeodesic, then s = at + b
for some constants a , b f R. A pregeodesic is said to be complete if for some
affine parametrization (hence for all affine parametrizations) the domain of the
parametrization is all of R. The equation VC,c1 = 0 may be expressed as a
18 2 CONNECTIONS AND CURVATURE
system of linear differential equations. To this end, we let (U, ( x l , x2 , . . . , x n ) )
be local coordinates on M and let d / d x l , d /dx2 , . . . , d/dxn denote the natural
basis with respect to these coordinates. The connection coefficients rfj of V
with respect to ( x l , x2, . . . , xn) are defined by
" d
(2.5) v (5) = f - (connection coefficients) k= I
dxk
Using these coefficients we may write equation (2.4) as the system
d2xk " dxi dxj (2.6) + x- = 0. (geodesic equatzons in coordinates) dt i,j=l dt
The connection coefficients (Christoffel symbols of the second kind) may
also be used to give a local representation of the action of V. If the vector
fields X and Y have local representations as
n d n
d x = x ~ ' ( x ) ~ and Y = ~ Y ' ( ~ ) - ,
i=l i=l axz
then VxY has a local representation
The vector field V x Y is said to be the covariant derivative of Y with respect
to X . A semicolon is used to denote covariant differentiation with respect to a
natural basis vector. If X = d / d x J , then the components of VxY = Va/az ,Y
are denoted by Y k i j where
The Lie bracket of the ordered pair of vector fields X and Y is a vector field
[X, Y ] which acts on a smooth function f by [ X , Y ] ( f ) = X ( Y ( f)) - Y ( X ( f ) ) .
If X = C ~ " x ) ) d / d x ~ and Y = C Y i ( x ) d / d x i , then
(Lie bracket)
2.1 CONNECTIONS 19
The torsion tensor T of V is the function T : X ( M ) x X(M) + X(M) given
by
(2.8) T ( X , Y ) = VxY - V y X - [ X , Y] . (torsion tensor)
The mapping T is said to be f -bilinear since it is linear in both arguments
and also satisfies T ( f X , Y ) = T ( X , f Y ) = f T ( X , Y ) for smooth functions f .
The value T ( X , Y)I, depends only on the connection V and the values X ( p )
and Y ( p ) . Consequently, T determines a bilinear map T,M x T,M + T,M
a t each point p E M . Using the skew symmetry ( [ X , Y ] = -[Y, XI ) of the Lie
bracket, it is easily seen that T ( X , Y) = -T (Y , X ) , and hence T is also skew.
Since [d /dx i , d / d x j ] = 0 for all 1 5 i , j < n, it follows that
Consequently,
where
Tik j = rtj - I'$ (torsion components)
Clearly, the torsion tensor provides a measure of the nonsymmetry of the
connection coefficients. Hence, T = 0 if and only if these coefficients are
symmetric in their subscripts. A connection V with T = 0 is said to be
torsion free or symmetric.
The curvature R of V is a function which assigns to each pair X , Y E X ( M )
the f-linear map R ( X , Y ) : X ( M ) -+ X(M) given by
Curvature provides a measure of the nuncommutativity of Vx and Vy. I t
should be noted that some authors define the curvature as the negative of the
above definition. Consequently, they differ in sign for some of the definitions
of curvature quantities given below. The curvature R represents a tensor field.
20 2 CONNECTIONS AND CURVATURE
Hence, the map (X, Y, Z) -+ R(X, Y)Z from X(M) x X(M) x T(M) to X(M) is
f-trilinear, and the vector R(X, Y)Zlp depends only on X(p), Y(p), Z(p), and
V. If x, y, z E TpM, one may extend these vectors to corresponding smooth
vector fields X, Y, Z and define R(x, y)z = R(X, Y)Zlp.
If w E T,*M is a cotangent vector at p and x, y, z E TpM are tangent vectors
at p, then one defines
for X , Y, and Z smooth vector fields extending x, y, and z, respectively. The
curvature tensor R is a (1,3) tensor field which is given in local coordinates
by
where the curvature components R z j k m are given by
Notice that R(X, Y)Z = -R(Y, X)Z, R(w, X, Y, 2) = -R(w, Y, X, Z), and
Rijkm = - ~ i ,mk. F'urthermore, if X = CXid/dxi , Y = Yi6'/dx" Z =
C Zia/dxi, and w = C widxi, then
and
Consequently, one has R(dxi, d/dxk, a/dxm, a/dxj) = Rijkm.
2.2 Semi-Riemannian Manifolds
A semi-Riemannian metric g for a manifold M is a smooth symmetric tensor
field of type (0,2) on M which assigns to each point p E M a nondegenerate
2.2 SEMI-RIEMANNIAN MANIFOLDS 21
inner product glp : TpM x TpM -+ R of signature (-, . . . , -, +, . . . , +). Here
nondegenerate means that for each nontrivial vector v E T,M there is some
w E TpM such that gp(v, w) # 0. If g has components gij in local coordinates,
then the nondegeneracy assumption is equivalent to the condition that the
determinant of the matrix (gij) be nonzero.
Although we consider only smooth metrics, some authors have studied dis-
tributional semi-Riemannian metrics [cf. Parker (1979), Taub (1980)l. Also,
a number of authors have studied semi-Riemannian metrics for which degen-
eracy is allowed [cf. Bejancu and Duggal (1991), Katsuno (1980), Kossowski
(1987, 1989)l.
In local coordinates (U, (xl, x2,. . . , xn)) on M, the metric g is represented
by n
g 1 U = gij(r) dxi 8 dxi (metric tensor) i,j=l
with
,, - ,, (symmetric) and det(gij) # 0. (nondegenerate) 9 ' . - 9 . .
If g has s negative eigenvalues and r = n - s positive eigenvalues, then the
signature of g will be denoted by (s, r). For each fixed p E M , there exist lo-
cal coordinates (U, (xl, x2,. . . , xn)) such that gp = g I TpM can be represented
as the diagonal matrix diag{-, . - . , -, +, . . . , +). For each semi-FLiemannian
manifold (M, g) there is an associated semi-Riemannian manifold (M, -9) ob-
tained by replacing g with -g. Aside from some minor changes in sign, there is
no essential difference between (M, g) and (M, -g). Thus, results for spaces of
signature (s, r ) may always be translated into corresponding results for spaces
of signature (r, s) by appropriate sign changes and inequality reversals.
Two vectors in TpM are orthogonal if their inner product with respect to
gp is zero. Note that when g has eigenvalues with different signs, there will
be some nontrivial vectors which are orthogonal to themselves (i.e., satisfy
g(v,v) = 0). These are known as null vectors. A given vector is said to be a
unit vector if it has inner product with itself equal to either +1 or -1. Thus,
an orthonormal basis {el, ez, . . . ,en) of T,M satisfies Jg(e,, e,)] = 6',.
22 2 CONNECTIONS AND CURVATURE
Given a semi-Riemannian manifold (M,g), there is a unique connection V
on M such that
(2.16) Z(g(X, Y)) = g(VzX, Y) + g(X, V z Y ) (metric compatible)
and
(2.17) [X, Y] = VxY - V y X (torsion free)
for all X, Y, Z E X(M). This connection V is called the Levi-Czvita connection
of (M,g). As indicated above, (2.16) is the condition that the connection V
be compatible with the metric g, and (2.17) is the condition that V be torsion
free. Setting 2 = c' in (2.16), one finds that parallel translation of vector fields
along any smooth curve c of M preserves inner products. For semi-Riemannian
manifolds, the connection coefficients are given by
1 " dgia 89. . (2.18) rk . = - Igak (- - -' + %) '3 2
(connection coeficzents) a=l dxj dxa dx2
where gij represents the (2,O) tensor defined by
n
(2.19) C g i a 9 a j = bij for 1 5 i , j 5 n. a=l
The local representations gij and gij may be used to raise and lower indices.
For example, if the upper index of the curvature tensor is lowered, one obtains
the components of the Riemann-Christoffel tensor which is also known as the
couariant curvature tensor.
n
(2.20) &jkm = ):gat Rajkrn (couariant curvature components) a=l
Alternatively, one may define the Riemann-Christoffel tensor as the (0,4)
tensor such that
2.2 SEMI-RIEMANNIAN MANIFOLDS 23
Some standard curvature identities satisfied by the components of this tensor
are &jkm = Rkrnrj = -Rjikrn = -Rijmk and Rilkm + Rikmj + Rimjk = 0.
The trace of the curvature tensor is the Ricci curvature, a symmetric (0,2)
tensor. For each p E M , the Ricci curvature may be interpreted as a symmetric
bilinear map Ric, : T p M x T p M + R. To evaluate Ric(u, w), let e l , ea, . . . , en
be an orthonormal basis for T p M . Then
or equivalently,
n
(2.23) Ric(v,w) = x g ( e i , ei) x(ei , v: ei, w). r=l
One may express v and w in the natural basis as v = C v a d / d z b n d w =
C wid/dxi and then write
n
where n
(2.25) Ri, = C Raiaj. (Ricci curvature components) a=l
If one uses the Einstein summation convention of summing over repeated
indices, then equations (2.24) and (2.25) become Ric(v, w) = Rijviw3 and
Rij = Raiaj, respectively. The Ricci tensor is the ( 1 , l ) tensor field which
corresponds to the Ricci curvature. The components of the Ricci tensor may
be obtained by raising one index of the Ricci curvature. Either index may be
raised since the Ricci curvature is symmetric. Thus,
n n
(2.26) R> = ) g a ' ~ a 3 = ) g a a ~ j a . (Rzccz tensor components) a=l a=l
The trace of the Ricci curvature is the scalar curvature T. Historically, this
function has been denoted by the much used symbol R. Accordingly, n
(2.27) 7 = R = Raa. (scalar curuature) a=l
24 2 CONNECTIONS AND CURVATURE
Thus i f e l , e2, . . . ,en is an orthonormal basis of T p M , one has
n
(2.28) T = R = g(ei, ei) Ric(ei, ei). i=l
The gradient and Hessian are defined for semi-Riemannian manifolds just
as for Riemannian manifolds. I f f : M --+ R is a smooth function, then df is
a ( 0 , l ) tensor field (i.e., one-form) on M , and grad f is the (1,O) tensor field
(i.e., vector field) which corresponds to df. Thus,
(2.29) Y ( f ) = df ( Y ) = g(grad f , Y ) (gradient)
for an arbitrary vector field Y . In local coordinates (U, ( x l , x2 , . . . , xn)), the
vector field grad f is represented by
* . . a f a (2.30) grad f = gZ3 - -. (gradient using coordinates)
i,i=l axi 8x3
The Hessian Hf is defined to be the second covariant differential o f f:
~f = V ( V f ). (Hessian)
For a given f E 5 ( M ) , the Hessian Hf is a symmetric (0,2) tensor field which
is related to the gradient o f f through the formula
for arbitrary vector fields X and Y . The Laplacian Af = div (grad f ) is now
defined to be the divergence o f the gradient o f f.
A tangent vector v f T p M is classified as timelike, nonspacelike, null, or
spacelike i f g(v, v ) is negative, nonpositive, zero, or positive, respectively:
g(v, v ) < 0, (timelike)
g(v, v ) 1 0, (nonspacelike or causal)
g(v, v ) = 0, (null or lightlike)
g(v, v ) > 0. (spacelike)
2.3 NULL CONES AND SEMI-RIEMANNIAN METRICS 25
A Riemannian manifold ( M , g) is a semi-Riemannian manifold o f signature
(0, n ) [i.e., (+, . . . , + ) I . Thus, on each tangent space T p M o f a Riemannian
manifold the metric gp is positive definite. Consequently, the metric induced
on each tangent space is Euclidean, and all (nontrivial) vectors for Riemannian
manifolds are spacelike.
A Lorentzian manifold is a semi-Riemannian manifold ( M , g) o f signature
( 1 , n - 1) [i.e., (- ,+,. . . ,+)I. At each point p f M the induced metric on
the tangent space is Minkowskian. Each point o f a Lorentzian manifold has
timelike, null, and spacelike tangent vectors. A smooth curve is said to be
timelike, null, or spacelike i f its tangent vectors are always timelike, null, or
spacelike, respectively.
A timelike curve in a Lorentzian manifold corresponds to the path of an
observer moving at less than the speed of light. Null curves correspond to
moving at the speed of light, and spacelike curves correspond to the geometric
equivalent of moving faster than light. Although relativity predicts that physical
particles cannot move faster than light, spacelike curves are of clear geometric
interest.
A vector field X on M is timelike i f g (X , X ) < 0 at all points o f M . A
Lorentzian manifold with a given timelike vector field X is said to be time
oriented b y X . A space-time is a time oriented Lorentzian manifold. Not all
Lorentzian manifolds may be time oriented, but a Lorentzian manifold which
is not time orientable always admits a two-fold cover which is time orientable
(cf . Chapter 3).
2.3 Null Cones and Semi-Riemannian Metrics
One o f the folk theorems o f general relativity asserts that the space-time
metric is determined up to a conformal factor by the set o f null vectors. In
this section we examine more generally to what extent the null vectors o f a
nondefinite nondegenerate inner product on a vector space determine the given
inner product and obtain as a consequence in Theorem 2.3 an elementary proof
o f this folk theorem for spacetimes [cf. Ehrlich (1991)l.
26 2 CONNECTIONS AND CURVATURE
Lemma 2.1. Let V be a real n-dimensional vector space, n 2 2, and let
g and h be two nondefinite nondegenerate inner products on V of arbitrary
signature. Suppose that g and h satisfy the condition that for any v f V,
(2.31 ) g(v,v) = 0 iff h(v, v) = 0.
Th cn
(1) either g and h or g and -h have the same signature, and
(2) there exists X # 0 such that
(2.32 ) h(v, w) = Xg(v, w)
for all v,w E V.
Furthermore, i f g has signature ( s , ~ ) with T # s, then X > 0 if g and h have
the same signature and X < 0 if g and -h have the same signature.
Proof. Flrst conslder the case that r = s = 1. Let {el, e2) be an arbitrary
g-orthonormal basis for V such that g(el ,el) = -1 and g(e2, e2) = +1. Then
771 = el + e2 and 172 = el - e2 are both g-null vectors. Hence, setting h,, =
h(e,, e,), we obtain the system of equations
0 = h(711,771) = h l l + h22 + 2h12,
0 = h(~2,772) = h l l + h22 - 2h12,
from which we conclude hll = -hz2 and h12 = 0. Since g(e,,e,) # 0, we
also have hil, hz2 # 0. Thus if we set X = -hll, then h = Xg with X > 0 if
h(el ,el) < 0 and X < 0 if h(el,el) > 0.
Multiplying g by -1 if necessary, we consider the case that g has signature
(r,s) with r 2 1 and s 2 2. Let {el,. . . ,e,, e,+l,. . . , en ) be any g-orthonormal
basis for V with {el,. . . , e,} g-unit timelike and {e,+l,. . . ,en) g-unit space-
like. Again by (2.31), we have hJ3 = h(e,, e,) # 0 for all j. Fix any j , k > r + 1
with j # k. Then we have a one-parameter family of g-null vectors
(2.33) v(8) = el + cos(8) . e, + sin(8) . ek
2 3 NULL CONES AND SEMI-RIEMANNIAN METRICS 27
for any 8 f R. In vlew of (2.31), we obtain
(2.34) 0 = h(v(@), ~ ( 0 ) )
= hll + cos2(8) . h,, + sin2(6) . hkk
+ 2 cos(8) . hl, + 2sin(8) . hlk + sin(28) . hJk.
Taking 8 = 0 and 8 = .rr respectively in (2.34) yields
0 = hi1 + h,, + 2h1,,
and 0 = hll + h,, - 2hl, ,
from which we conclude hl, = 0 and h,, = -hll for all J 2 r + 1. With this
information, equation (2.34) reduces to the equation
0 = hl l - (cos2 8 + sin2 6) . hll + sln(26) hJk
or simply
0 = sin(28) - hJk.
Hence, hJk = 0 for all j, k 2 r + 1, j # k .
We now have for r = 1 that hpq = 0 if p # q , and h l l = -hz2 = -hS3 =
... - - -hnn. Hence if X = -hll > 0, then g and h both have signature
(-, +, . . . , +), whereas if X = -hli < 0, then h has signature (+, -, . . . , -). In either case, h = Xg with X = -hll as required.
If r 2 2, we have a little more work left. First, consideration of the one-
parameter family
with j, k 5 r and j # k yields hJk = h,, = 0 and hll = hz2 = .. . = h,, =
-h,,. It remains to show that hl, = 0 and h,, = 0 if p < r < q. But for this
we need simply consider the one-parameter null family
x(0) = cos(8) . el + sin(8) . ep + eq,
since the equation 0 = h(x(8), x(8)) reduces to
28 2 CONNECTIONS AND CURVATURE
in view o f our above results. Thus we have obtained hll = hz2 = . . . =
h,, = -h,+l,,+l = ... = -hnn and hJk = 0 i f j # k , from which the desired
conclusion follows as above.
As a corollary, we obtain
Corollary 2.2. Let V be a real n-dimensional vector space, n 2 3, and
assume V is equipped with inner products g and h , both o f Lorentzian signa-
ture (-, +, . . . , +). Suppose g and h determine the same null vectors. Then
h = Xg for some constant A > 0.
W i t h this last result applied to each tangent space, we obtain the desired
result.
Theorem 2.3. Let M be a smooth manifold of dimension n 2 3 with
metrics g and h, both o f Lorentzjan signature (-, +, . . . , +). Suppose for any
v E T M that g(v ,v) = 0 i f f h(v ,v) = 0. Then there exists a smooth function
R : M -+ (0, m) such that h = Rg.
A somewhat more general class o f objects than globally conformally r e
lated metrics has been considered in differential geometry and general rel-
ativity, namely, conformal transformations. Here a global diffeomorphism
F : ( M , g ) -+ ( N , h ) between two semi-Riemannian manifolds ( M I g) and
( N , h) is said to be a global conformal transformation i f there exists a smooth
function R : M -t (0 , m) such that
(2.35) h(F*v, F,w) = R(p) g(v, w)
for all v , w f T p M and p E M . In the Riemannian case, such maps would be
angle preserving. Evidently, condition (2.35) implies that ( M , g ) and ( N , h )
have the same signature.
Conversely, given two semi-Ftiemannian manifolds ( M , g) and ( N , h ) , we
may apply Lemma 2.1 t o g and F'h in order t o obtain that the condition
(2.36) g(v, v ) = 0 i f f h(F.v, F,v) = 0 for all v f T M
serves to ensure that F is a global conformal transformation o f ( M , g ) onto
either ( N , h ) or ( N , -h). Especially for space-times, one has the following
well-known extension of Theorem 2.3.
2.4 SECTIONAL CURVATURE 29
Corollary 2.4. Let ( M I g) and ( N , h ) be smooth manifolds of dimension
n 2 3 having Lorentzian signature (-, +, . . . , +). Suppose that f : M -+ N is
a diffeomorphism which satisfies condition (2.36). Then f is a global conformal
transformation o f ( M , g) onto ( N , h).
One could paraphrase this last result as follows: a diffeomorphism o f space-
times which preserves null vectors is a conformal transformation. Hence Corol-
lary 2.4 is a differentiable version o f a much deeper a priori topological result
o f Hawking, King, and McCarthy (1976).
2.4 Sectional Curvature
Let ( M , g) be a semi-Riemannian manifold. A two-dimensional linear sub-
space E of TpM is called a plane section. The plane section E is said to be
nondegenerate if for each nontrivial vector v 6 E there is some vector u E E
with g (u ,v ) # 0. This is equivalent to the requirement that gp I E be a nonde
generate inner product on E. I f v and w form a basis o f the plane section E ,
then g(v, v)g(w, w ) - [g(v, w)j2 is a nonzero quantity i f and only i f E is non-
degenerate. This quantity represents the square o f the semi-Euclidean area o f
the parallelogram determined by v and w. The plane E is said to be timelike
i f the signature o f gp I E is (1 , I ) , i.e., (-, +). It is spacelike i f the signature is
(0,2), i.e., (+, +). For Lorentzian manifolds, degenerate planes are called either null or lightlike
and always have signature (0, +). A null plane in T p M is a plane tangent to
the null cone in T p M . Thus, in the Lorentzian case a degenerate plane contains
exactly one generator o f the null cone.
Using the square o f the semi-Euclidean area o f the parallelogram determined
by the basis vectors {v , w ) , one has the following classification for plane sec-
tions o f Lorentzian manifolds:
g(v, v)g(w, w ) - Ig(v, w)I2 < 0, (tzmelike plane)
g(v, v)g(w, w ) - [g(v, w)I2 = 0 , (degenerate plane)
g(v, v)g(w, W ) - lg(u, w)I2 > 0. (spacelike plane)
T h e sectional curvature o f the nondegenerate plane section E with basis
30 2 CONNECTIONS AND CURVATURE
{v, w) where v = Cvid /dx i and w = C wid/dxi is then given by
- - R w , v, w, V) (sectional curvature) g(v, v)g(w1 w) - [g(v, w)I2
For positive definite manifolds the Ricci curvature evaluated at a unit vector
is sometimes thought of as more or less an average sectional curvature weighted
by a factor of (n- I). More precisely, let w be a unit vector at p in the Riemann-
ian manifold (M, g), extend to an orthonormal basis {el, ez, . . . , en-l, en = w),
and let Ei = span{ei, w) for 1 5 i 5 n - 1. Equations (2.22) and (2.37) yield
. -
Ric(w, w) = C g ( ~ ( e i , W)W, ei) = C K(p, E,).
It is instructive to contrast the above with the Ricci curvature evaluated a t
a unit timelike vector in a Lorentzian manifold. In this case the interpretation
is in terms of the negative of the timelike sectional curvature. For let u be a
unit timelike vector at a point p of the Lorentzian manifold (M,g). Extend to
an orthonormal basis {el, e2,. . . , en-1, en = u) and let Ei = span{ei, u) for
I 5 i < n - I . Equations (2.22) and (2.37) yield
Thus, if u is a timelike vector with Ric(u, u) > 0, then in some sense the
"average" sectional curvature for planes in the pencil of u is negative.
For Riemannian manifolds one has a number important "pinching" theo-
rems [cf. Cheeger and Ebin (1975)l. However, similar results fail for Lorentzian
manifolds. In particular, if the sectional curvatures of timelike planes are
bounded both above and below, and if dim(M) > 3, then (M, g) has constant
curvature [cf. Harris (1982a), Dajczer and Nomizu (l980a)l. Nevertheless, fam-
ilies of Lorentzian manifolds conformal to ones of constant curvature may be
2 4 SECTIONAL CURVATURE 31
constructed which have all timelike sectional curvatures bounded in one direc-
tion [cf. Harris (1979)j. However, if dim(M) > 3 and if the sectional curvatures
of all nondegenerate planes are bounded either from above or from below, then
the sectional curvature is constant [cf. Kulkarni (1979)l. Sectional curvature
has been further investigated by Dajczer and Nomizu (1980b), Nomizu (1983),
Beem and Parker (1984), and Cordero and Parker (1995a,c).
A semi-Riemannian manifold (M, g) which has the same sectional curvature
on all (nondegenerate) sections is said to have constant curvature. If (M, g) has
constant curvature c, then R(X, Y)Z = c [g(Y, Z ) X - g(X, Z)Y] [cf. Graves
and Nomizu (1978)l. Thorpe (1969) showed that the sectional curvature can
only be continuously extended to degenerate planes in the case of constant
curvature. Spaces of constant sectional curvature have been investigated in
connection with the space form problem [cf. Wolf (1961, 1974)l.
The curvature tensor, Ricci curvature, scalar curvature, and sectional cur-
vature may all be calculated in local coordinates using the metric tensor com-
ponents and the first two partial derivatives of these con~por~ents. Thus, the
metric tensor determines the curvatures. In contrast, curvature does not nec-
essarily determine the metric. Nevertheless, for most Lorentzian manifolds
the metric will be determined either completely or up to a constant conformal
factor given sufficient information concerning curvature in local coordinates
[cf. Hall (1983, 1984), Hall and Kay (1988), Ihrig (1975), Quevedo (1992)j.
Conjugate points along geodesics may be defined using Jacobi fields. These
objects are studied in Chapter 10 in connection with the development of the
Morse index theory for timelike and null geodesics. If c : (a, b) -+ M is a
geodesic, then the smooth vector field J : (a, b) -+ T M along c is a Jacobi field
if it satisfies the Jacobi equation,
(2.38) J" + R(J, cJ)cJ = 0, (Jacobz equatzon)
where J" = V,,(V,lJ). In an intuitive sense, one thinks of a Jacobi field
as representing the relative displacement of "nearby" geodesics [cf. Hawking
and Ellis (1973), Hicks (1965), or Misner, Thorne and Wheeler (1973)l. In
particular, let c be a unit speed timelike geodesic. Then c represents the path
of a "freely falling" particle moving a t less than the speed of light. Taking J
32 2 CONNECTIONS AND CURVATURE
as a Jacobi field which is orthogonal to the tangent vector c', one interprets J
as a vector from the original particle to another particle moving on a nearby
timelike geodesic, and one interprets J" as the relative (or tidal) acceleration of
the second particle as measured by the first. If g(J, J ) # 0, then the definition
of sectional curvature together with g(cl, c') = -1 and g(J, c') = 0 yield
At points where J does not vanish, the vector J / d m is a unit vector in
the direction of J. Using J" = -R(J, cl)c', one finds that the radial component
of the tidal acceleration is given by
This equation shows that for "close" particles the radial component of the tidal
acceleration varies directly with the separation distance IJI and with the sec-
tional curvature K(cl, J ) of the plane containing both c' and J. Thus, using
our sign conventions, a timelike plane with positive sectional curvature cor-
responds to freely falling particles accelerating away from each other, and
negative sectional curvature of a timelike plane corresponds to particles accel-
erating toward each other. Since Ric(cl, c') > 0 corresponds to the timelike
planes containing c' having negative average sectional curvature, it follows that
Ric(cl, c') > 0 corresponds to average attractive (i.e., focusing) tidal forces. I t
should be kept in mind that some authors use different sign conventions and
may have sectional curvature equal to the negative of ours.
For a constant value of I JI the maximum tidal acceleration will be radial
[cf. Beem and Parker (1990, p. 820)j. Thus at any fied value to, an observer
traversing the timelike geodesic c will have zero tidal accelerations if and only
if all planes E containing cl(to) have zero sectional curvature.
2.5 THE GENERIC CONDITION
2.5 The Generic Condition
The sectional curvature can be used to study the generic condition, which
will be of importance in the singularity theorems to be considered in C h a p
ter 12. If W = Wad/dxa is a tangent vector, then the values W a
are the contravariant components. Using the metric g one has values Wb =
Cr=l gabWa which are the covariant components. Thus, xr=l Wb dxb is the
cotangent vector corresponding to the original W . The generic condition is
said to be satisfied for a vector W at p E M if
n
(2.40) wC wdwfa R ~ ~ ~ ~ [ ~ wfl # 0. (generic condition) c.d=l
~f condition (2.40) fails to hold (i.e., if C:,,=, WCWdWI,Rblcdl,Wfl = o), then
W will be called nongeneric.
The generic condition is said to hold for a geodesic c : (a, b) -+ M if at some
point c(t0) the tangent vector to the geodesic is generic, i.e., one has (2.40)
satisfied with W = cl(to). Notice from continuity that if (2.40) holds for some
W = cl(to), then it will hold for W = cl(t) whenever t is sufficiently close to
to.
We will show in Proposition 2.6 that for a vector W which is nonnull (i.e.,
g(W, W ) # O), the nongeneric condition is equivalent to requiring that all
plane sections containing W have zero sectional curvature. For null vectors,
the nongeneric condition is slightly more complicated.
Lemma 2.5. Let Rabcd represent the components of the curvature tensor
with respect to an oi.thonormal basis {vl , vz, . . . , v,) of T,M. The vector
W = vn satisfies the generic condition (2.40) iff there exist b and e with
1 < b, e < n - 1 such that Rbnne # 0.
Proof. The components of W are given by W 1 = . . . = Wn-' = Wl =
. . . = Wn-1 = 0, W" = 1, and W, = +l or -1. Consequently,
34 2 CONNECTIONS AND CURVATURE
- wa Rbnn j We + Wb Rann j We)
- bna Rbnn j bne + bnb Rann j S n e ) .
It is easily seen that this expression is nonzero i f and only i f Rbnne # 0 for
s o m e b , e w i t h l ~ b , e ~ n - 1 .
Proposition 2.6. If W E T p M is a nonnull vector, then W fails to be
generic (i.e., is nongeneric) i f f for each nondegenerate plane section E contain-
ing W the sectional curvature K ( p , E ) vanishes.
Proof. W e may assume without loss o f generality that W is a unit vector.
Set W = vn and extend to an orthonormal basis { v l , v2, . . . , v,) o f T p M . Let
Rabcd represent the components o f the curvature tensor with respect t o this
basis.
From Lemma 2.5, i f W fails to satisfy the generic condition, then Rbnne = 0
for all 1 5 b, e 5 n - 1. This and the skew symmetry of Rabcd in both the
first pair o f indices and the second pair o f indices yield Rbnne = 0 for all
1 5 b, e 5 n. Hence Rnbne = -Rbnne = 0 for all such b and e. I t follows that
i f U is another tangent vector at p, one has z:,b,c,e=l RabceWaUbWCUe =
~ ~ , e = l R,~,,U~U" = 0. Fkom (2.37), we conclude that i f E is the plane
spanned by { W , U), and i f E is nondegenerate, then K(p, E ) = 0.
Conversely, assume all nondegenerate planes containing W have zero sec-
tional curvature. Then (2.37) easily shows that all terms o f the form Rbnnb
must be zero. Assume b # e. Notice that E = span{vb,vb + 2ve) cannot - - be degenerate. Using (2.37) we obtain R(vn, vb, v,, vb) = R(vn, v,, v,, v,) = - - R(vn, vb + 2ve, vn, vb + 2ve) = 0. The multilinearity of R and standard curva- - ture identities then yield 0 = R(vn, vb + 2ve, v,, vb + 2ve) = 4 @v,, vb, v,, we) .
This shows f i b n e = 0 and hence Rbnne = 0. Using Lemma 2.5, it follows that
W fails to satisfy the generic condition as desired. U
2.5 THE GENERIC CONDITION 35
Proposition 2.6 implies that the only way for a timelike geodesic c to fail to
satisfy the generic condition is for the corresponding observer to fail to ever
experience any tidal accelerations.
In Chapter 12 we will make use o f an alternative formulation o f the generic
condition using the curvature tensor. Let c be a unit speed timelike geo-
desic, p = c(to), and W = cl(to). The set of vectors orthogonal to W is an
( n - 1)-dimensional linear subspace W' = V L ( c ( t o ) ) lying in T p M , and the
metric induced on this linear subspace is positive definite. Thus vL(c ( to ) ) is
a spacelike hyperplane in T p M . I f y E T p M , then g (W, R(y, W ) W ) = 0 which
implies that R ( y , W ) W lies in VL(c( to)) . It follows that the curvature tensor
R induces a linear map from VL(c ( to ) ) to VL(c( to)) :
Since g I VL(c( to ) ) is positive definite, this curvature map is nontrivial i f and
only i f there arc vectors yl, y2 f V L ( c ( t o ) ) with g ( y n , R ( y l , W ) W ) # 0. The
next result shows this map is nontrivial i f and only i f W is generic.
Proposition 2.7. If W = cl(to) is a timelike vector in the Lorentzian
manifold ( M , g) , then the following three conditions are equivalent:
(1 ) The timelike vector W is generic.
( 2 ) At least one plane containing W has nonzero sectional curvature.
(3) R( . , W ) W is not the trivial map.
Proof. Clearly, Proposition 2.6 shows the first two conditions are equiva-
lent. Let Rabcd represent the components o f the covariant curvature tensor with
respect to an orthonormal basis {v l ,v2 , . . .vn-l,vn = W ) of T p M . I f W sat-
isfies the generic condition, then Lemma 2.5 implies that -Rbnen = Rbnne # 0
for some 1 5 b, e 5 n - 1. Consequently, g(vb, R(ve , vn)vn) # 0 which shows
R(v,, vn)vn is a nontrivial vector. Thus, when W is generic the map R( . , W ) W
is not the trivial map.
Conversely, assume that R ( . , W ) W is not trivial. Let yl , yz be vectors in
W L = spanjvl, v2, . . . , vn-l) with g(y2, R(y l , vn)vn) # 0. This last inequality
together with the multilinearity o f 2(~, Z , X , Y ) = g(W, R ( X , Y ) Z ) yield b
36 2 CONNECTIONS AND CURVATURE
and e with 1 5 b, e < n - 1 such that Rbnne # 0. Thus, Lemma 2.5 shows that
W is generic as desired.
The next proposition shows that a sufficient condition for the nonnull vector
W to be generic is that Ric(W, W) # 0.
Proposition 2.8. If W E TpM is a nonnull vector with Ric(W, W) # 0,
then W is generic.
Proof. We may assume without loss of generality that W is a unit vector
and extend to an orthonormal basis {vl, v2,. . . , vn-1, vn = W) of TpM. Then,
which shows that Ric(W, W) # 0 implies Rnin # 0 for some 1 5 i 5 n - 1.
The result now follows from Lemma 2.5. 0
In particular, this last proposition shows that if c : (a, b) --+ M is a timelike
geodesic with Ric(c1, c') > 0 for some t = t o , then c is generic.
In order to investigate the generic condition for (nontrivial) null vectors,
we first define a pseudo-orthonoma1 basis Icf. Hawking and Ellis (1973)l. Let
{vl, 7.12,. . . , vn-2) be n - 2 unit spacelike vectors in TpM that are mutually
orthogonal, and let W, N be two null vectors which satisfy g(W, N ) = -1
and which are both orthogonal to the first n - 2 vectors. Then a pseudo-
orthonormal basis of TpM is formed by {vl, va, . . . , vn-2, N, W). In this basis
the metric tensor at p has the form
g. . V - - 6.. ,3 and g in- l=g in=O f o r l < i , j S n - 2 ,
with g = -1 and gnn = gn-ln-l = 0.
It is not difficult to show that any nontrivial null vector W can be extended
to a pseudo-orthonormal basis. One first takes any timelike plane E containing
2.5 THE GENERIC CONDITION 37
W and finds an orthonormal basis {en-', en) of E with W = (en-l +en)/&,
en-1 spacelike, and en timelike. Then one extends to an orthonormal basis
{el, e2,. . . ,en) and sets vi = ei for 1 < i 5 n - 2. Finally, one assigns
vn-1 = N = (-en-l +en)/&, and vn = W.
The next lemma provides the null version of Lemma 2.5 using a pseudo-
orthonormal basis. In the lemma below, the values of b and e only go to n - 2
as opposed to n - 1 in Lemma 2.5.
Lemma 2.9. Let Rabcd represent the components of the curvature tensor
with respect to a pseudo-orthonormal basis { q , . . . , vn-2, vn-1 = N, vn = W)
ofTpM. The null vector W = vn satisfies thegeneric condition (2.40) iff there
exist b and e with 1 5 b, e 5 n - 2 such that Rbnne # 0.
Proof. The components of W are given by W' = . .. = Wn-' = Wl =
. . = Wn-2 = Wn = 0, Wn = 1, and Wn-l = -1. Consequently,
- wa Rbnnf We + Wb Rannf We)
It is easily seen that this expression is zero for all a , b, e, f whenever one or
more of a , b, e, f equal n. If it is nonzero for some a, b, e, f , then exactly one of
a , b must be n - 1, and also exactly one of e, f must be n - 1. It follows that
x:d=l WCWdWIaRblcdIeWfl # 0 iff Rbnne # 0 for some 1 5 b, e 5 n - 2.
If n = 2, then Lemma 2.9 shows all null vectors fail to be generic since there
are no values of b and e with 1 5 b, e I n - 2.
Corollary 2.10. If dim(M) = 2, then all null vectors are nongeneric
If W is a nontrivial null vector in TpM, then the orthogonal space W-' =
N(W) = {v E TpM / g(W, v) = 0) is an (n - 1)-dimensional linear subspace
of the tangent space and contains the null vector W. The set N(W) is a
hyperplane which is tangent to the null cone at p along one generator. The i
38 2 CONNECTIONS AND CURVATURE
signature of g I N(W) is degenerate of order one and positive of order n - 2.
Notice that g(W, R(v, W)W) = 0 implies that R(v, W)W lies in N(W).
The one-dimensional linear subspace determined by W will be denoted by
[W]. Let G(W) = N(W)/[W] be the (n - 2)-dimensional quotient space and
T : N(W) -+ G(W) be the natural projection map. Since R(W, W) = 0 and
the multilinearity of the curvature tensor yield R(v + aW, W)W = R(v, W)W
for all v, each element of a given coset of G(W) is mapped to the same element
of N(W) by R( . , W)W. It follows that one has a linear map
defined by (G, W) W = x (R(v, W) W) where v is any vector in 7r-l(8). The
degenerate metric g 1 N(W) projects to a positive definite metric 3 on G(W)
since vl,vz E N(W) must satisfy g(v1 + aW,v2 + bW) = g(vl,v2) for all real
a, b. Thus,
(2.44) - g(G1, G2) = g(v1, v2) (quotient metric on G(W))
where V1 = x(v1) and 5 2 = x(v2).
The next result is the null version of Proposition 2.7. We do not give a
statement corresponding to condition (2) of Proposition 2.7. In general, a null
vector may lie in planes with nonzero sectional curvature and yet fail to be
generic.
Proposition 2.11. Let (M, g) be a Lorentzian manifold, and let W E TpM
be a nontrivial null vector. Then the following two conditions are equivalent:
(1) The null vector W is generic.
(2) R ( . , W)W is not the trivial map.
Proof. Let Rabcd represent the components of the covariant curvature tensor
with respect to a pseudo-orthonormal basis {vl,. . . ,v,-2,vn-l,vn = W) of
T,M, N(W) = WL, and G(W) = N(W)/[W] as above. Notice that N(W) =
span(vl, ~ 2 , . . . , vn-2, W).
Assuming W is generic, Lemma 2.9 yields b and e with Rbnne # 0 for 1 5 b, e 5 n - 2. Consequently, g ( ~ ( v b ) , Z(T(V,), W)W) = g(vb, R(ve, W)W) =
Rbnen = -Rbnne # 0 which shows that R ( . , W)W is not trivial.
2.5 THE GENERIC CONDITION 39
Conversely, assume R ( . , W)W is not the trivial map. Since 3 is positive
definite, there must exist E and 8 in G(W) with 3 ( i i , R(v, W)W ) # 0.
Choose u f n-'(E) and v E a-'(5). Then multilinearity, the fact that
3 ( E , R(u, W)W ) = g(u, R(v, W)W) # 0, and R(W, W)W = 0 together im-
ply there must be b and e, 1 5 b,e < n - 2, such that g(vb, R(ve, W)W) # 0.
Hence, Rbnen = -Rbnne # 0, and W is generic by Lemma 2.9. 0
The next proposition shows that Ric(W, W) # 0 is a sufficient condition for
a vector W to be generic. This was already proven for timelike and spacelike
vectors in Proposition 2.8.
Proposition 2.12. Let (MI g) be a Lorentzian manifold, and let W E TpM
be a tangent vector. If Ric(W, W) # 0, then W is generic.
Proof. We may assume without loss of generality that W is a null vector
because of Proposition 2.8. It is always possible to construct both an orthonor-
ma1 basis {el, e2, . . . ,en) and a pseudo-orthonormal basis
with v, = e, for 15 i 5 n - 2,
and en timelike. Let Rabcd be the components of the Riemann-Christoffel
curvature tensor with respect to the pseudo-orthonormal basis and Rabcd be
the components with respect to the orthonormal basis. Then,
40
Hence,
2 CONNECTIONS AND CURVATURE
which shows that Ric(W, W ) # 0 implies %,in # 0 for some 1 5 i 5 n - 2.
The result now follows from Lemma 2.9. 0
The next corollary follows from Corollary 2.10 and Proposition 2.12
Corollary 2.13. Let ( M , g ) be a Lorentzian manifold with d im(M) = 2.
I f W is null, then Ric(W, W ) = 0.
The generic condition holds generically in each tangent space o f any point
where there is some component o f the curvature tensor not equal to zero. More
precisely, one can establish the following result [cf. Beem and Harris (1993a,
p. 950)].
Proposition 2.14. Let ( M , g) be a Lorentzian manifold of dimension four
and let p E M . If T p M has five nonnull and nongeneric vectors with four
o f them linearly independent and with the fifth not in any plane determined
by any two o f the original four, then the curvature tensor vanishes at p. In
particular, i f any component o f the curvature tensor fails to be zero at p, then
one cannot find five nonnull nongeneric vectors at p in general position.
The situation for nongeneric null vectors is somewhat different. Having all
null vectors nongeneric does not necessarily imply zero curvature at a point.
For dimension three and higher, all null vectors at a point will be nongeneric
i f and only i f there is constant sectional curvature at the point. In dimension
four, one can have a "cubic" o f nongeneric null vectors at a point and yet fail
2.5 THE GENERIC CONDITION 41
t o have constant sectional curvature at that point. Using a cubic definition
for "generically situated" [cf. Beem and Harris (1993b, pp. 969-972)], one may
establish the following result.
Proposition 2.15. Let ( M , g) be a Lorentzian manifold o f dimension four,
and let p E M. I f T,M has 11 null nongeneric vectors generically situated,
then all sectional curvatures at p are equal. In particular, i f ( M , g) does not
have constant sectional curvature at p, then the generic null directions at p
form an open dense subset o f the two-sphere o f all null directions at p.
This last proposition yields that "generically" a given null direction at a
point p satisfies the generic condition.
Assume that ( M , g) is a four-dimensional Lorentzian manifold, and recall
that the Jacobi equation is J" + R(J, cl)c' = 0. Let c be a unit speed timelike
vector, and assume { E l , E2, E3, E4) is an orthonormal basis along c which
moves by parallel translation along c and satisfies Eq = cl(t) . I f J = c : ~ ~ JzEl
is a Jacobi field, the Jacobi equation can then be written as
3
= - C ~ ' 4 k 4 J~ (Jacobi equation) k=l
using ~ ~ 4 . 4 4 = 0. Since { E l , E2, Esl E4) is an orthonormal basis with c l ( t ) =
E4, the vectors { E l , E2, E3) are unit spacelike vectors, and the Jacobi equation
becomes
W e now illustrate the unboundedness of tidal accelerations for observers
with velocity vectors close to the direction o f a null vector W which sat-
isfies the generic condition [cf. Beem and Parker (1990)l. To this end, as-
sume that (U, Xl,X2,X3,~4) are local coordinates near c(t0) = p. Assume
also that the natural basis for the x,-coordinates at p is the orthonormal
set { E l , E2, E3, Ed). Denote the components o f the curvature tensor for this
42 2 CONNECTIONS AND CURVATURE
basis by Rabcd. Regard 8 as temporarily fixed, and define new coordinates
(YI, ~ 2 , ~ 3 , ~ 4 ) near P by YI = 21, y2 = 22, y3 = 5 3 cosh(8) - 2 4 sinh(B), and
y4 = -23 sinh(8) + 2 4 cosh(8). Let the natural basis a t p for the y-coordinates - - - -
be denoted by {El , E p , E3, E4), and let the curvature tensor components at p - - - -
with respect to this basis be denoted by Rabcd. The new basis {E l , E2, E3, E 4 )
is also orthonormal, and if one lets 8 --+ +m, the direction of & converges to
the null direction determined by W = E3 + E4. This change of coordinates
corresponds in T,M t o what is called a "pure boost." The new curvature
components fZabcd are related to the original components Rabcd by
Using XI = yl, x2 = y2, 2 3 = y3 cosh(0) + y4sinh(8), and x 4 = y~sinh(8) + y4 cosh(8), one finds
An observer traversing the timelike geodesic which has tangent vector - - -
a t p has a rest space at p given by F: = span{El, E2, E3). TO investigate
tidal accelerations for this observer, one may consider the Jacobi equation -
d2J'/dt2 = - ~ i , ~ Rt4k4Jk using vectors in this rest space of the form J =
c:=, J"% with IJI = 1. Notice that the tidal accelerations will be bounded
as 8 4 +m if and only if the components fZz4k4 are bounded for large 8, and
2.5 THE GENERIC CONDITION 43
this will hold if and only if the following equations hold:
The vector W is nongeneric if and only if (2.53), (2.54), and (2.55) all hold,
and it is nondestructive if (2.51)-(2.55) all hold [cf. Beem and Parker (1990),
Beem and Harris (1993a,b)]. Tidal forces and radiation for a falling body in
Schwarzschild space-time have been studied by Mashhoon (1977). Also, tidal
impulses have been investigated by Mashhoon and McClune (1993). Further
results on nondestructive directions have been obtained by Hall and Hossack
(1993).
Remark 2.16. In order to get a physical interpretation of (2.51)-(2.55),
consider a "freely falling" steel ball with center having velocity vector E4 at
p. Assume the ball has rest mass mo and radius = a. Tidal accelerations for
points on the surface of the ball will correspond to IJI = a. Recall that the
formula for the special relativistic increase in mass (or energy) is given by
where y = 1 / d m = cosh(8), v is the speed measured with respect to
the original rest frame, and c is the speed of light. Of course, v -+ c corresponds
to 8 -+ +m. In other words, the classical special relativistic magnification
factor of mass is y = cosh(8). If one of (2.53), (2.54), or (2.55) fails to hold,
then W = E3 + E4 is generic, and the increase in tidal accelerations for large 8
is approximately proportional to -y2 which is the square of the increase in the
mass. For particles approaching the speed of light in a direction corresponding
to a generic null direction, eventually the increase in tidal accelerations will
become a bigger difficulty than the increase in mass. If (2.53)-(2.55) all hold,
44 2 CONNECTIONS AND CURVATURE
but at least one of (2.51) or (2.52) fails to hold, then the increase in tidal
accelerations for large B is approximately proportional to y. In this case, the
null direction corresponding to W is (tidally) destructive but is not generic.
2.6 T h e Einstein Equations
In this section we give a brief description of the Einstein equations. A
heuristic derivation of these equations may be found in Frankel (1979, Chap-
ter 3). Since these equations apply to manifolds of dimension four, we restrict
our attention in this section to this dimension. The Einstein equations re-
late purely geometric quantities to the energy-momentum tensor T, which is a
physical quantity. They may thus be used to state energy conditions in terms
of T. In the case of a perfect fluid, the energy-momentum tensor also takes
a simple form. This is important in general relativity because the matter of
the universe is assumed to behave like a perfect fluid in the standard cosmo-
logical models. The physical motivation for studying Lorentzian manifolds is
the assumption that a gravitational field may be effectively modeled by some
Lorentzian metric g defined on a suitable four-dimensional manifold M. Since
every manifold which admits a Lorentzian metric clearly admits uncountably
many such metrics, it is necessary to decide both which manifold and which
Lorentzian metric on that manifold should be used to model a given grav-
itational problem. The Einstein equations relate the metric tensor g, Ricci
curvature Ric, and scalar curvature T (= R) to the energy-momentum tensor
T. The tensor T is to be determined from physical considerations dealing with
the distribution of matter and energy [cf. Hawking and Ellis (1973, Chapter
3), Misner, Thorne, and Wheeler (1973, Chapter 5)]. The Einstein equations
may be written invariantly as
1 (2.57) Ric - - R g + A g = 87rT (Einstein equations)
2
where A is a constant known as the cosmological constant. The constant factor
of 87r is present for scaling purposes. In local coordinates, one has
1 (2.58) Rij - -Rgij + hgi j = 87rTij (Einstein equations in coordinates)
2
2.6 THE EINSTEIN EQUATIONS 45
where 1 < i, j 5 4. The Ricci curvature and scalar curvature involve the first
and second partial derivatives of the components gij of the metric g but do not
involve any higher derivatives. Hence, the Einstein equations represent (non-
linear) partial differential equations in the metric and its first two derivatives.
These sixteen equations reduce to ten because all of the tensors in equation
(2.58) are symmetric. There is a further reduction to six equations [cf. Misner,
Thorne, and Wheeler (1973, p. 409)] using the curvature identity
which yields four conservation laws given by
4
C ~ ' j ; j = O. (conservation laws) j=1
Here, "; j" denotes covariant differentiation in the xj direction (i.e., 'i7a/dzl).
The Einstein equations do not determine the metric on M without sufficient
boundary conditions. For example, let M = { ( t , r ) It E R and r > 2m) x S2.
Then M is topologically lR2 x S2. Let A = 0 and T = 0, and set dR2 =
do2 + ~ i n ~ ( e ) d $ ~ . Then M with this A and T admits both the flat metric
ds2 = -dt2 + dr2 + r2dR2 as well as the Schwarzschild metric
(2.59) ds2 = - dr2 + r2dR2 (Schwarzschild)
as solutions to the Einstein equations. Each of these rnetrics is asymptotically
flat, and each is Ricci JEat (i.e., Ric = 0). However, the Schwarzschild metric
has a nonzero curvature tensor, and hence the two metrics cannot be isometric.
Nevertheless, a counting argument shows that, in general, one expects the Ein-
stein equations to determine the metric up to diffeomorphism [cf. Hawking and
Ellis (1973, p. 74)j. First, notice that the metric tensor g has 16 components
which, by symmetry, reduce to ten independent components. Furthermore,
four of these ten components can be accounted for by the dimension of M
which allows four degrees of freedom. Thus the metric tensor is thought of
46 2 CONNECTIONS AND CURVATURE
as having six independent components after symmetry and diffeomorphism
freedom are taken into account. Consequently, the Einstein equations yield
six independent equations to determine six essential components of the metric
tensor.
More rigorous approaches to the problem of existence and uniqueness of
solutions to the Einstein equations using Cauchy surfaces with initial data may
be found in a number of articles and books such as Chrusciel (1991), Hawking
and Ellis (1973, Chapter 7), Marsden, Ebin, and Fischer (1972, pp. 233-264),
and Choquet-Bruhat and Geroch (1969).
The Einstein equations may be used to relate the timelike convergence con-
dition (Ric(v, v) 2 0 for all timelike, and hence also all null vectors v) to the
energy-momentum tensor. In order to evaluate the scalar curvature R in terms
of T at p E M , let {el, e 2 , . . . , en ) be an orthonormal basis of T,M, and use
equation (2.57) to obtain
Using the fact that the scalar curvature R is the trace of the Ricci curvature,
this last equation becomes
Hence,
(2.60) R = - 8 ~ tr(T) + 411.
The Einstein equations become
1 Ric - - 2 ( - 8 ~ tr(T) + 4A)g + Ag = 8x2".
Thus,
2.6 THE EINSTEIN EQUATIONS 47
This last equation shows that the condition Ric(v,v) 2 0 is equivalent to
the inequality T(v, v) 2 [tr(T)/2 - A/8n] g(v, v). If follows that when A = 0
and dim(M) = 4, the condition
is equivalent to the condition
[cf. Hawking and Ellis (1973, p. 95)]. Note that (2.60) and (2.61) show that if
A = 0, then T = 0 (i.e., vacuum) is equivalent to Ric = 0 (i.e., Ricci flat).
The Einstein equations are fundamental in the construction of cosmological
models. Consider a fluid which moves through space. This motion generates
timelike flow lines in space-time. Let v be the unit speed timelike vector field
which is everywhere tangent to the flow lines of the fluid. The fluid is said to be
a perfect fluid if it has an energy density p , pressure p, and energy-momentum
tensor T such that
(2.62) T = (p + p) w 8 u + p g, (perfect fiuid)
which is
Tij = (p+p)vivj +pg i j
in local coordinates. Here w = z v i d x i is the one-form corresponding to the
vector field v = via/dxi. It follows from the above form of T that a perfect
fluid is an isotropic fluid which is free of shear and viscosity. Let (M,g)
be a manifold for which T has the above perfect fluid form. If the vectors
{el, e:!, e3, e4) form an orthonormal basis for T,M, then the trace of T may be
calculated as follows:
4
(2.63) tr(T) = g(ei, el) T(e,, e,) i=l
= -(P+P) $ 4 ~
= 3p - p.
48 2 CONNECTIONS AND CURVATURE
Using equation (2.61), it follows that the timelike convergence condition for a
perfect fluid is equivalent to
for all timelike (and null) w. For Lorentzian manifolds, it is easy to verify
that the inner product of a timelike vector and another timelike (or nontrivial
null) vector is nonzero. Thus, we may assume without loss of generality that
g(v, w) # 0. Using equation (2.62) we obtain
which simplifies to
Since g(w, w) 5 0 and g(v, w) # 0, equation (2.64) shows that a negative
cosmological constant has the effect of making the timelike convergence condi-
tion more plausible and that a positive cosmological constant has the opposite
effect. Einstein originally introduced the cosmological constant because the
Einstein equations with A = 0 predict a universe which is either expanding
or contracting, and in the early part of this century it was believed that the
universe was essentially static. After the discovery that the universe was ex-
panding, the original motivation for the cosmological constant was removed;
however, removing A from the theory has been more difficult. While astro-
nomical experiments have failed to detect a A different from zero, one may
always argue that A is so small that the experiments have not been sufficiently
sensitive.
Discussions of the experimental evidence for general relativity may be found
in a number of books such as Misner, Thorne, and Wheeler (1973) and Will
(1981).
CHAPTER 3
LORENTZIAN MANIFOLDS AND CAUSALITY
Sections 3.1 and 3.2 give a brief review of elementary causality theory basic
to this monograph as well as to general relativity. Then Section 3.3 describes an
important relationship between the limit curve topology and the C0 topology
for sequences of nonspacelike curves in strongly causal space-times. Namely,
if y : [a, b ] --+ M is a future directed nonspacelike limit curve of a sequence
{y,) of future directed nonspacelike curves, then a subsequence converges to y
in the C0 topology. This result is useful for constructing maximal geodesics in
strongly causal spacetimes using the Lorentzian distance function (cf. Chapter
8 and Chapter 12, Section 4).
In Section 3.4 we study the causal structure of two-dimensional Lorentzian
manifolds. In particular, we show that if (M, g) is a space-time homeomorphic
to R2, then (M,g) is stably causal.
Section 3.5 gives a brief discussion of the theory of Lorentzian submani-
folds and the second fundamental form needed for our discussion of singularity
theory in Chapter 12.
An important splitting theorem of Geroch (1970a) guarantees that a glob-
ally hyperbolic space-time may be written as a topological (although not nec-
essarily metric) product R x S where S is a Cauchy hypersurface. This result
suggests that product space-times of the form (R x M, -dt2 @ g) with (M, g)
a Riemannian manifold should be studied. While this class of space-times in-
cludes Minkowski space and the Einstein static universe, it fails to include the
physically important exterior Schwarzschild and Robertson-Walker solutions
to Einstein's equations.
In Sections 3.6 and 3.7 we study a more general class of product space-times,
the so-called warped products, which are spacetimes MI x f M2 with metrics
50 3 LORENTZIAN MANIFOLDS AND CAUSALITY
of the form gl @I fg2. This class of metrics, studied for Riemannian manifolds
by Bishop and O'Neill (1969) and later for semi-Riemannian manifolds by
O'Neill (1983), includes products, the exterior Schwarzschild space-times, and
Robertson-Walker space-times. The following result, which may be regarded
as a "metric converse" to Geroch's splitting theorem, is typical of the results
of Section 3.6. Let (R x M, -dt2 @ g) be a Lorentzian product manifold with
(M, g) an arbitrary Riemannian manifold. Then the following are equivalent:
(1) (M, g) is a complete Riemannian manifold.
(2) (R x M, -dt2 @ g) is globally hyperbolic.
(3) (R x M, -dt2 @I g) is geodesically complete.
3.1 Lorentzian Manifolds a n d Convex Normal Neighborhoods
Let M be a smooth connected paracompact Hausdorff manifold, and let
TM denote the tangent bundle of M with n : T M -+ M the usual bundle map
taking each tangent vector to its base point. Recall that a Lorentzzan metric g
for M is a smooth symmetric tensor field of type (0,2) on M such that for each
p f M, the tensor glp : TpM x TpM -+ R is a nondegenerate inner product
of signature (1, n - 1) [i.e., (-, +, . . . , +)I. All noncompact manifolds admit
Lorentzian metrics. However, a compact manifold admits a Lorentzian metric
if and only if its Euler characteristic vanishes [cf. Steenrod (1951, p. 207)].
The space of all Lorentzian metrics for M will be denoted by Lor(M).
A continuous vector field X on M is timelike if g(X(p),X(p)) < 0 for all
points p E M . In general, a Lorentzian manifold does not necessarily have
globally defined timelike vector fields. If (M,g) does admit a timelike vector
field X E X(M), then (M,g) is said to be time oriented by X . The timelike
vector field X divides all nonspacelike tangent vectors into two separate classes,
called future and past directed. A nonspacelike tangent vector v E T,M is
said to be future [respectively, past] directed if g(X(p), v) < 0 [respectively,
g(X(p),v) > 01. A Lorentzian manifold (M,g) is said to be time onentable
if (M,g) admits a time orientation by some timelike vector field X. In this
case, (M,g) admits two distinct time orientations defined by X and -X,
respectively. A time oriented Lorentzian manifold is called a space-time.
3.1 LORENTZIAN MANIFOLDS, CONVEX NORMAL NEIGHBORHOODS 51
More precisely, we have the following definition.
Definition 3.1. (Space-time) A space-time (M,g) is a connected Cm Hausdorff manifold of dimension two or greater which has a countable basis,
a Lorentzian metric g of signature (-, +?. . . , +), and a time orientation.
We now show how to construct a time oriented two-sheeted Lorentzian
covering manifold n : ( E , i j ) -+ (M,g) for any Lorentzian manifold (M, g)
which is not time orientable.
To this end, first let (M,g) be an arbitrary Lorentzian manifold. Fix a base point PO E M . Give a time orientation to TpoM by choosing a timelike
tangent vector vo E TpoM and defining a nonspacelike w E TpoM to be future
[respectively, past] directed if g(v0, w) < 0 [respectively, g(v0, w) > 01. Now let
q be any point of M. Piecewise smooth curves y : [O, 11 -+ M with y(0) = po
and $1) = q may be divided into two equivalence classes as follows. Given
~ 1 ~ 7 2 : [0,11 -+ M with ~ l ( 0 ) = ~ ( 0 ) = PO and yl(1) = y2(1) = 91 let Vl
(respectively, T/z) be the unique parallel field along yl (respectively, yz) with
Vl(0) = V2(0) = vo. Wesay that yl and 7 2 areequivalent ifg(Vl(l), V2(1)) < 0.
If yl and 7 2 are homotopic curves from po to q, then yl and 7 2 are equivalent.
But equivalent curves are not necessarily homotopic.
Given y : [O, 11 -+ M with y(0) = po, let [y] denote the equivalence class
of y. Let 2 consist of all such equivalence classes of piecewise smooth curves
y : [O, 11 -+ M with $0) = PO. Define n : z -+ M by ~ ( [ y ] ) = y(1). If (M, g) - is time orientable, then M = M . Otherwise, T : % -+ M is a two-sheeted
covering [cf. Markus (1955, p. 412)].
Suppose now that the Lorentzian manifold (M, g) is not time orientable. I t
is standard from covering space theory to give the set ii;j a topology and dif-
ferentiable structure such that n : M + M is a two-sheeted covering manifold. - Define a Lorentzian metric g for M by ij = n*g, i.e., ;(v, w) = g ( ~ , v , n,w).
Then the map 7r : z -+ M is a local isometry.
In order to show that (%,g) is time orientable, it is useful to establish a
preliminary lemma. Fix a base point Fo E T - ' ( ~ ~ ) for M. Let Go E T~,M be - the unique timelike tangent vector in TFo M with r,Go = VO.
52 3 LORENTZIAN MANIFOLDS AND CAUSALITY
Lemma 3.2. Let q E be arbitrary and let yl,;j;2 : [0, I] -+ % be two
piecewise smooth curves with Yl(0) = 72(O) = Po and ;j;l(l) = 72(1) = g. If
fi, 1/2 are the parallel vector fields along y1 and y2, respectively, with (0) = - V2(0) = 6, then g(v1(1), %(I)) < 0.
- Proof. Let y l = .n o yl and 72 = 7r o 72. Since .n : ( M , $ ) ---t (M, g) is
a local isometry, the vector fields Vl = .n,(vl) and 6 = 7riT,(v2) are parallel - fields along 71 and 7 2 , respectively, with Vl(0) = h(O) = n,vo = vo. Also,
g (K( l ) , Wl)) = g(x*G(l) ,x*%(l)) = g(G(l )> %(I)).
Suppose now that $(Ql(l), V2(1)) y! 0. Since ?1(1) and ?2(1) are timelike
tangent vectors, it follows that 5(?1(1), V2(1)) > 0. Thus g(Vl(l), h ( 1 ) ) > 0
at q = .n(@). By definition of the equivalence relation on piecewise smooth
curves from po to q, we have [yl] # [y2]. F'rom the construction of M , we know
that T1(l) = [yl] and T2(l) = Im]. Thus %,(I) # 72(1), in contradiction.
Theorem 3.3. Suppose that (M,g) is not time orientable. Then the two-
sheeted Lorentzian covering manifold (%, $) of (M, g) constructed above is
time orientable and hence is a space-time.
Proof. Let & and Go be as above. Given any T E %, let o : [O, 11 -+ % be - a smooth curve with a(0) = Po, a(1) = $ Let V be the unique parallel vector
field along a with ?(o) = Go. Set F + ( q ) = (timelike w E T& : ~ ( v ( l ) , w) < 0). By Lemma 3.2, the definition of F+(a ) is independent of the choice of
u. Hence if+ F+(T) consistently assigns a future cone to each tangent space
T~%, q E G. Now let h be an auxiliary positive definite Riemannian metric for %. We
may define a continuous nowhere zero timelike vector field X on by choosing
X(5) to be the vector in F+(q) which is the unique h-unit vector in F + ( q )
having a negative eigenvalue of g with respect to h. That is, we may find a
continuous function X : M + (-a, 0) and a continuous timelike vector field
X on % satisfying X ( q ) E F+(q"), h(X(T), X(c)) = 1, and $(X(g), v) =
X(y) h(X(q), v) for all v f T& and 5 E %. U
Implicit in the proof of Theorem 3.3 is an alternative definition for the time
orientability of a Lorentzian manifold (M, g). Namely, ( M g) is time orientable
3.1 LORENTZIAN MANIFOLDS, CONVEX NORhlAL NEIGHBORHOODS 53
if, fixing any base point po E M and timelike tangent vector vo E Tp,M, the
following condition is satisfied for all q f M. Let yl,yz : [0,1] --+ M be any
two smooth curves from p to q. If V, is the unique parallel vector field along
yi with V,(O) = vo for i = 1,2, then g(V1(l), V2(1)) < 0. This condition means
that parallel translation of the future cone determined by vo at po to any
other point q of M is independent of the choice of path from p to q. Hence
a consistent choice of future timelike vectors for each tangent space may be
made by parallel translation from po.
Recall that a smooth curve in (M,g) is said to be timelike (respectively,
nonspacelike, null, spacelike) if its tangent vector is always timelike (respec-
tively, nonspacelike, null, spacelike). As in the Riemannian case, a geodesic
c : (a , b) -+ M is a smooth curve whose tangent vector moves by parallel dis-
placement, i.e., V,jcl(t) = 0 for all t E (a, b). The tangent vector field cl(t) of
a geodesic c satisfies g(cl(t), cl(t)) = constant for all t 6 (a , b) since
Consequently, a geodesic which is timelike (respectively, null, spacelike) for
some value of its parameter is timelike (respectively, null, spacelike) for all
values of its parameter.
The exponential map expp : TpM + M is defined for Lorentzian manifolds
just as for Riemannian manifolds. Given v E TpM, let cu(t) denote the unique
geodesic in M with cv(0) = p and cul(0) = v. Then the exponential exp,(v)
of v is given by expp(v) = cv( l ) provided c,(l) is defined.
Let vl, v2, . . . , vn be any basis for the tangent space TpM. For sufficiently
small (21, 22,. . . , x,) E Rn, the map
xlvl + 22212 + . . . + X n V n --' ~ X P ~ ( X ~ U ~ + 2 2 ~ 2 + . . . + xnvn)
is a diffeomorphism of a neighborhood of the origin of T,M onto a neigh-
borhood U(p) of p in M . Thus, assigning coordinates (xl,x2,. . . ,xn) to the
point expp(xlvl + 22712 + . . . + xnvn) in U(p) defines a coordinate chart for M
called normal coordinates based at p for U(p). The set U(p) is said to be a
(simple) convex neighborhood of p if any two points in U(p) can be joined by
54 3 LORENTZIAN MANIFOLDS AND CAUSALITY
a unique geodesic segment of (M,g) lying entirely in U(p). Whitehead (1932)
has shown that any semi-Riemannian (hence Lorentzian) manifold has convex
neighborhoods about each point [cf. Hicks (1965: pp. 133-136)]. In fact, it may
even be assumed that for each q E U(p), there are normal coordinates based
a t q containing U(p). We call such a neighborhood U(p) a convex normal
neighborhood [cf. Hawking and Ellis (1973, p. 34)].
The next proposition is essential to the study of the local behavior of causal-
ity. A proof is given in Hawking and Ellis (1973, pp. 103-105). It is important
to carefully note the phrase "contained in U" in the statement of this result.
Proposit ion 3.4. Let U be a convex normal neighborhood of q. Then
the points of U which can be reached by timelike {respectively, nonspacelike)
curves contained in U are those of the form expq(u), u E TqM, such that
g(v, v) < 0 {respectively, g(v, v) 5 0).
3.2 Causality Theory of Space-times
In a space-time (M, g) a (nowhere vanishing) nonspacelike vector field along
a curve cannot continuously change from being future directed to being past
directed. It follows that a smooth timelike, null, or nonspacelike curve in
(M, g) is either always future directed or always past directed.
We will use the standard notation p << q if there is a smooth future directed
timelike curve from p to q, and p < q if either p = q or there is a smooth future
directed nonspacelike curve from p to q. Furthermore, p < q will mean p 6 q
and P # q-
A continuous curve y : (a, b) --+ M is said to be a future directed non-
spacelike curve if for each to E (a, b) there is an E > 0 and a convex normal
neighborhood U(y(t0)) of y(t0) with y(t0-E, to+€) C U(y(t0)) such that given
any tl , tz with to - E < t l < t2 < to + E, there is a smooth future directed non-
spacelike curve in (U(y(to)), g I U(y(t0))) from y(t1) to y(t2). It is necessary
to use the convex normal neighborhood U(y(to)) in this definition for the fol-
lowing reason. There exist space-times for which p << q for all (p, q) E M x M.
But in these space-times, any continuous curve y satisfies both y(tl) << y(ta)
and y(t2) << y(tl) for all tl and t2 in the domain of y.
3.2 CAUSALITY THEORY OF SPACE-TIMES 55
For a given p E M, the chronological future Ii(p), chronological past I-(p),
causal future Ji(p), and causal past J-(p) of p are defined as follows:
I f (p) = { q f M : p << q) , (chronological future)
I-(p) = { q E M : q << p), (chronological past)
~ + ( p ) = { q E M : p 6 q), (causal future)
J-(p) = { q E M : q < p). (causal past)
For general subsets S 2 M, the sets I i (S) , I - (S) , J+(S) , and J - (S) are
defined analogously: for example, I+(S) = { q E A4 / s << q for some s E S ) .
The relations << and Q are clearly transitive. Moreover,
p << q and q < r implies p << r,
and
p 6 q and q << r implies p << r
[cf. Penrose (1972, p. 14)]. If there is a future directed timelike curve from p
to q, there is a neighborhood U of q such that any point of U can be reached
by a future directed timelike curve. Consequently, it follows that
Lemma 3.5. If p is any point of the space-time (M,g), then I+(p) and
I-(p) are open sets of M.
An example has been given in Chapter 1, Figure 1.1, to show that the sets
J+(p) and J-(p) are neither open nor closed in general.
Two especially important classes of subsets of a space-time are future sets
and past sets. In this section we will restrict our attention to open future and
past sets, which may be defined as follows.
Definition 3.6. (Future and Past Sets) The (open) subset F (respec-
tively, P) of the space-time (M,g) is said to be a .future (respectively, past)
set if F = I+(F) (respectively, P = I-(P)).
These sets will be used in studying the causality of the gravitational plane
wave space-times in Chapter 13. Future and past sets have often proven useful
56 3 LORENTZIAN MANIFOLDS AND CAUSALITY
in causality proofs [cf. Hawking and Sachs (1974), Dieckmann (1987)l. Rather
complete discussions of this topic may be found in Penrose (1972, Section 3)
and in Hawking and Ellis (1973).
The simplest examples are formed by taking F = I+(S) or P = IP(S) ,
where S is an arbitrary subset of M. (Indeed, these sets may also be defined
in this manner.) They also share the following property with the chronological
past and future sets I-(p) and I+(p) of an arbitrary point p in M:
x in F and x << y implies y in F ; (future set)
x in P and y << x implies y in P. (past set).
The next two results give simple characterizations of the closures and bound-
aries of sets which are either future or past.
Proposition 3.7.
(1) If F is a future set, then F = {x E M : I+(x) c F) .
(2) If P is a past set, then P = {x E M : I-(2) C P}.
Proof. As usual, it suffices to establish (1). First, suppose I+(x) 2 F. We
may take {q,) 5 Z+(x) with lim,,,q, = x. Since {q,) 2 F, this yields
x E 7. Conversely, let x E 7 be given. Select any z E I+(x). Then x E I- (z)
which is open, and since x E 7, there exists y E I - (z )nF . But then y E F and
y << z implies z f F since F is a future set. Thus I+(x) C_ F as required. 0
Corollary 3.8. Let F (respectively, P) be a future (respectively, past) set.
Then
(1) B(F) = {x 6 M : x @ F and Z+(x) E F) , and
(2) d(P) = {x E M : x 4 P and I-(2) E P).
In the particular cases where F = I+(p) or P = I-(p), the following useful
characterizations of
-- -- I- (p) = J-(p) and I+(p) = J+ (p)
3.2 CAUSALITY THEORY OF SPACE-TIMES 57
are obtained for arbitrary space-times:
- I+(p) = {x E M : I+(%) C I+(p)), (closure)
= {x E M : I-(2) I-(p)),
d(I+(p)) = {x E M : x 4f I+(p) and I+(x) C I+(p)), (boundary)
and d(I-(p)) = {x E M : x 4 I-(p) and I-(x) C I-(p)).
Past and future sets have played an important role in singularity theory
in general relativity as treated in Penrose (1972) or Hawking and Ellis (1973)
via the allied concept of achronal boundaries. A subset B of M is said to be
an achronal boundary if B = d(F) for some future set F . (Of course, it is
necessary to prove that a ( F ) is achronal, i.e., that no two points of d(F) may
be joined by a future timelike curve, for this definition to make sense.) In
particular, for any p in M the set
is a simple example of an achronal boundary. Achronal boundaries have the
following important regularity properties.
Theorem 3.9. The (nonempty) boundary d ( F ) of a future set is a closed
achronal (Lipschitz) topological hypersurface.
Discussions of this result are given in Penrose (1972, pp. 21-23) and in
Hawking and Ellis (1973, p. 187) from the viewpoint of applications in sin-
gularity theory. A rather complete treatment of a number of mathematical
aspects may be found in O'Neill (1983, pp. 413-415).
I t may happen that p E I+(p). If so, there is a closed timelike curve through
p, and the space-time is said to have a causality violation. For example, on
the cylinder M = S1 x W with the Lorentzian metric ds2 = -do2 + dt2, the
circles t = constant are closed timelike curves. In this space-time, I+(p) = M
for all p E M. A number of causality conditions have been defined in general
relativity in recent years because of the problems associated with examples of
causality violations.
58 3 LORENTZIAN MANIFOLDS AND CAUSALITY
Space-times which do not contain any closed timelike curves [i.e., p # I+(p)
for all p f M ] are said to be chronological. A space-time with no closed
nonspacelike curves is said to be causal. Equivalently, a causal space-time
contains no pair of distinct points p and q with p < q < p. The cylinder M =
S 1 x R with the Lorentzian metric ds2 = do d t is an example of a chronological
space-time that fails to be causal. The only closed nonspacelike curves in this
example are the circles t = constant, which with proper parametrization are
null geodesics.
The chronological condition is the weakest causality condition which will be
introduced. The next proposition guarantees that no compact space-time is
either causal or chronological.
Proposition 3.10. Any compact space-time (M, g ) contains a closed time-
like curve and thus fails to be chronological.
Proof. Since the sets of the form I + ( p ) are open, it may be seen that ( I f ( p ) :
p f M ) forms an open cover of M . By compactness, we may extract a finite
subcover ( I f ( P I ) , I f (p2) , . . . , I f ( p k ) ) . Now pl E I+(p ip) ) for some i ( 1 ) with
1 5 i ( 1 ) 5 k. Similarly, p,(l) E I+(pq2)) for some index i ( 2 ) with 1 5 i ( 2 ) 5 k .
Continuing inductively, we obtain an infinite sequence . . . << p , ( ~ ) << pi(2) << pi(l) << pl. Since k is finite, there are only a finite number of distinct P ~ ( ~ ) ' s .
Thus there are repetitions on the list, and from the transitivity of <<, it follows
that pi(,) f I'-(pi(,)) for some index pi(,). Thus (M,g) contains a closed
timelike curve through pi(n).
Tipler (1979) has proved that certain classes of compact space-times contain
closed timelike geodesics, not just closed timelike curves. Since the proof
uses the Lorentzian distance function as a tool, discussion of Tipler's result is
postponed until Section 4.1, Theorem 4.15.
A spacetime is said to be dzstinguishing if for all points p and q in M,
either I+(p) = I+(q) or I - ( p ) = I - ( q ) implies p = q. In a distinguishing
space-time, distinct points have distinct chronological futures and chronolog-
ical pasts. Thus, points are distinguished both by their chronological futures
and pasts.
3.2 CAUSALITY THEORY OF SPACE-TIMES 59
A distinguishing space-time is said to be causally continuous if the set-
valued functions I + and I - are outer continuous. Since I+ and I - are always
inner continuous [cf. Hawking and Sachs (1974, p. 291)] , the causally con-
tinuous spacetimes are those distinguishing space-times for which both the
chronological future and past of a point vary continuously with the point.
Here I+ is said to be inner continuous a t p E M if for each compact set
K I+(p) , there exists a neighborhood U ( p ) of p such that K c I+(q) for
each q f U ( p ) . The set-valued function I+ is outer continuous a t p E M if -
for each compact set K E M - I+(p) , there exists some neighborhood U ( p ) -
of p such that K c M - I+(q) for each q E U ( p ) . Inner and outer continuity
of I - may be defined dually. An example of a space-time for which I - fails
t o be outer continuous is given in Figure 3.1. The concept of causal continu-
ity was introduced by Hawking and Sachs (1974). For these space-times the
causal structure may be extended to the causal boundary [cf. Budic and Sachs
(1974)l. Fbrthermore, a metrizable topology may be defined on the causal
completion of a causally continuous space-time [cf. Beem (1977)l.
An open set U in a space-time is said to be causally convex if no nonspace-
like curve intersects U in a disconnected set. Given p E M , the space-time
( M , g ) is said to be strongly causal at p if p has arbitrarily small causally con-
vex neighborhoods. Thus, p has arbitrarily small neighborhoods such that no
nonspacelike curve that leaves one of these neighborhoods ever returns. The
space-time (M, g) is strongly causal if it is strongly causal a t every point. It
may be shown that the set of points of an arbitrary space-time (M, g ) a t which
(M, g) is strongly causal is an open subset of M fcf. Penrose (1972, p. 30)]. I t
is not hard to show that strongly causal spacetimes are distinguishing.
Strongly causal space-times may be characterized in terms of the Alexan-
drov topology for M. The Alexandmv topology on an arbitrary space-time
(M,g) is the topology given M by taking as a basis all sets of the form
I+(p) n I - ( q ) with p, q E M (cf. Figure 1.2). The given manifold topology on
M is always at least as fine as the Alexandrov topology since I+(p) n I - ( q )
is an open set in the given topology by Lemma 3.5. The following result has
been obtained by Kronheimer and Penrose [cf. Penrose (1972, p. 34)] .
60 3 LORENTZIAN MANIFOLDS AND CAUSALITY
delete 1 /
FIGURE 3.1. A space-time which is not causally continuous is
shown. The map p -+ I-(p) fails to be outer continuous at the -
point q. The compact set K is contained in M - I - ( q ) , yet each
neighborhood U(q) of q contains some point T such that K is not -
contained in M - I-(r).
Proposition 3.11. The Alexandrov topology for (M, g) agrees with the
given manifold topology iff (M, g) is strongly causal.
Proof. Assume first that (M,g) is strongly causal. Then each p E M has
some convex normal neighborhood U(p) such that no nonspacelike curve in-
tersects U(p) more than once. The set U(p) is a convex normal neighborhood
of each of its points, and hence Proposition 3.4 implies that for each q E U(p),
the chronological future (respectively, past) of q in (U(p), g IU(,)) consists of
3.2 CAUSALITY THEORY OF SPACE-TIMES 61
all points which can be reached by geodesic segments in U of the form exp,(tu)
for 0 5 t 5 1, where v is a future (respectively, past) directed timelike vector at
q. This demonstrates that the Alexandrov topology on (U(p), g l U ( p ) ) agrees
with the given manifold topology on U(p). Using the fact that no nonspacelike
curve of (M, g) intersects U(p) more than once, it follows that the Alexandrov
topology agrees with the given manifold topology.
Now assume that strong causality fails to hold at p E M. Then there is a
convex normal neighborhood V(p) of p such that if W(p) is any neighborhood
of p with W(p) V(p), a nonspacelike curve starts in W(p), leaves V(p),
and returns to W(p). It follows that all neighborhoods of p in the Alexandrov
topology contain points outside of V(p). Thus, the Alexandrov topology differs
from the given manifold topology. O
In order to study causality breakdowns and geodesic incompleteness in gen-
eral relativity, it is helpful to formulate the concept of inextendibility for non-
spacelike curves. This may be done as follows. Let y : [a, b) -+ M be a curve
in M. The point p E M is said to be the endpoint of y corresponding to t = b
if
lim y (t) = p. t - b -
If y : [a, b) -4 M is a future (respectively, past) directed nonspacelike curve
with endpoint p corresponding to t = b, the point p is called a future (re-
spectively, past) endpoint of y. A nonspacelike curve is said to be future
znextendible (or future endless) if it has no future endpoint. Dually, a past
inextendible nonspacelike curve is one that has no past endpoint.
Convention 3.12. A nonspacelike curve y : (a, b) -+ M is said to be
znextendible (or endless) if it is both future and past inextendible.
Causal space-times exist that contain inextendible nonspacelike curves hav-
ing compact closure. An example given by Carter is displayed in Figure 3.2
Icf. Hawking and Ellis (1973, p. 195)]. An inextendible nonspacelike curve
which has compact closure and hence is contained in a compact set is said to
be imprisoned. Thus, Carter's example shows that imprisonment can occur in
causal space-times.
62 3 LORENTZIAN MANIFOLDS AND CAUSALITY
identify
___f
z identify-
FIGURE 3.2. A causal space-time (M, g) is shown which has im-
prisoned nonspacelike curves that are inextendible. Let a be an
irrational number, and let M = W x S' x S1 = {(t, y, z) E IR3 :
(t, y, z) -- (t, y, z + l ) and (t, y, z) -- (t, y+1, z+a)). The Lorentzian
metric is given by ds2 = (cosht - 1)'(dy2 - dt2) - dt dy + dz2.
Let y : [a, b) -+ M be a future directed nonspacelike curve. Then y is said
to be future imprisoned in the compact set K if there is some to < b such
that y(t) E K for all to < t < b. The curve y is said to be partially future
imprisoned in the compact set K if there exists an infinite sequence t, f b
with y(t,) E K for each n.
If (M,g) is strongly causal and K is a compact subset of M , then K may
be covered with a finite number of convex normal neighborhoods {Ui) such
that no nonspacelike curve which leaves some U, ever returns to that Ui. This
observation leads to the following result.
3.2 CAUSALITY THEORY OF SPACE-TIMES 63
Proposition 3.13. If (M, g) is strongly causal, then no inextendible non-
spacelike curve can be partially future (or past) imprisoned in any compact
set.
We now discuss another important class of space-times in general relativity,
stably causal space-times. For this purpose as well as for later use, it is helpful
to define the fine CT topologies on Lor(M).
Recall that Lor(M) denotes the space of all Lorentzian metrics on M. The
fine CT topologies on Lor(M) may be defined by using a fixed countable cov-
ering B = {Bi) of M by coordinate neighborhoods with the property that the
closure of each Bi lies in a coordinate chart of M and such that each compact
subset of M intersects only finitely many of the Bi's. This last requirement
is the condition that the covering be locally finite. Let 6 : M -+ (0 ,m) be
a continuous function. Then gl, g2 E Lor(M) are said to be 6 : M -+ (0, m )
close in the CT topology, written Igl - g21r < 6, if for each p E M all of the
corresponding coefficients and derivatives up to order r of the two metric ten-
sors gl and g2 are 6(p) close at p when calculated in the fixed coordinates of
all Bi 6 I3 which contain p. The sets {gl E Lor(M) : Igl - gzlr < 6 ) , with
g2 E Lor(M) arbitrary and 6 : M 4 (0 ,m) an arbitrary continuous function,
form a basis for the fine CT topology on Lor(M). This topology may be shown
to be independent of the choice of coordinate cover B.
The C' topologies for r = 0,1,2 may be given the following interpretations.
Remark 3.14. (1) Two Lorentzian metrics for M which are close in the
fine C0 topology have light cones which are close.
(2) Two Lorentzian metrics for M which are close in the fine C' topology have
geodesic systems which are close (cf. Section 7.2).
(3) Two Lorentzian metrics for M which are close in the fine C 2 topology have
curvature tensors which are close.
A space-time (M, g) is said to be stably causal if there is a fine CO neighbor-
hood U(g) of g in Lor(M) such that each gl E U(g) is causal. Thus a stably
causal space-time remains causal under small Co perturbations.
64 3 LORENTZIAN MANIFOLDS AND CAUSALITY
Stably causal space-times may be characterized in terms of a partial or-
dering < for Lor(M) defined using light cones to compare Lorentzian metrics.
Explicitly, if A is a subset of M , one defines gl < A g2 if for each p E A and
v E TpM with v # 0, gl(v,v) 5 0 implies g2(v,v) 5 0. One also defines
gl < A 92 if for each p E A and v E T,M with v # 0, gl(v,v) 5 0 implies
g2(v,v) < 0. We will write gl < g2 (respectively, gl < gz) for gl <M g2 (re-
spectively, gl <M g ~ ) . Thus gl < g2 means that a t every point of M the light
cone of gl is smaller than the light cone of g2, or g2 has wider light cones than
gl. I t may be shown that (M,g) is stably causal if and only if there exists
some causal gl E Lor(M) with g < gl.
A C0 function f : M -+ W is a global time function if f is strictly increasing
along each future directed nonspacelike curve. A space-time (MI g) admits
a global time function if and only if it is stably causal [cf. Hawking (1968),
Seifert (1977)l. However, there is generally no natural choice of a time function
for a stably causal space-time.
Let f : M -+ W be a smooth function such that the gradient V f is always
timelike. If y : (a, b) -+ M is a future directed nonspacelike curve with nonvan-
ishing tangent vector yl(t), then g(Of (y(t)), yl(t)) = yl(t)(f) must either be
always positive or always negative. Thus f must be either strictly increasing
or strictly decreasing along y. It follows that f must be strictly increasing or
decreasing along all future directed nonspacelike curves. Hence f or - f is a
smooth global time function for M. Furthermore, V f must be orthogonal to
each of the level surfaces f -'(c) = {p E M : f (p) = c), c f W, of f . These
level surfaces are hypersurfaces orthogonal to a timelike vector field and are
spacelike, i.e., g restricted to each of these hypersurfaces is a positive definite
metric. Since the gradient of f is nonvanishing and df is an exact 1-form, it
follows that M is foliated by the level surfaces {f-'(c) : c E W). Each non-
spacelike curve y of M can intersect a given level surface a t most once since f
must be strictly increasing or decreasing along y.
One of the most important causality conditions which we will discuss in
this section is global hyperbolicity. Globally hyperbolic space-times have the
important property, frequently invoked during specific geodesic constructions,
3.2 CAUSALITY THEORY OF SPACE-TIMES 65
that any pair of causally related points may be joined by a nonspacelike geo-
desic segment of maximal length.
Definition 3.15. (Globally Hyperbolic) A strongly causal space-time (M,g) is said to be globally hyperbolic if for each pair of points p, q E M ,
the set Jf (p) n J-(q) is compact.
A distinguishing space-time (M,g) is causally simple if J+(p) and J-(p)
are closed subsets of M for all p E M. It then follows that
Proposi t ion 3.16. A globally hyperbolic space-time is causally simple.
- Proof. Suppose q E J+(p) - J+(P) for some p E M. Choose any r E I+(q).
- Since I - ( r ) is open and q E J+(p), it may be seen that r E I+(p) by taking
a subsequence {q,) 2 J+(p) with q, -+ q and using the fact that p 6 q, and
qn << r imply p << r. Consequently, q E Jt(p) n J - ( r ) - ( J+(p)n J- ( r ) ) . But
this is impossible since J+(p) fl J - ( r ) is compact hence closed. 0
Globally hyperbolic space-times may be characterized using Cauchy sur-
faces. A Cauchy surface S is a subset of M which every inextendible non-
spacelike curve intersects exactly once. (Some authors only require that every
inextendible timelzke curve intersect S exactly once.) It may be shown that
a space-time is globally hyperbolic if and only if it admits a Cauchy surface
[cf. Hawking and Ellis (1973, pp. 211-212)). Furthermore, Geroch (1970a) has
established the following important structure theorem for globally hyperbolic
space-times [cf. Sachs and Wu (1977b, p. 1155)l.
T h e o r e m 3.17. If (M, g) is a globally hyperbolic space-time of dimension
n, then M is homeomorphic to W x S where S is an (n - 1)-dimensional
topological submanifold of M, and for each t , { t ) x S is a Cauchy surface.
The proof of this theorem uses a function f : M -+ W given by f(p) =
m(J+(p))/m(J-(P)) where m is a measure on M with m ( M ) = 1. The level
sets off may be seen to be Cauchy surfaces as desired, but f is not necessarily
smooth.
A time function f : M --+ W will be said to be a Cauchy time function if
each level set f -'(c), c E W, is a Cauchy surface for M. In studying globally
66 3 LORENTZIAN MANIFOLDS AND CAUSALITY
hyperbolic space-times, it is helpful to use Cauchy time functions rather than
arbitrary time functions.
In a complete Riemannian manifold, any two points may be joined by a
geodesic of minimal length. Avez (1963) and Seifert (1967) have obtained a
Lorentzian analogue of this result for globally hyperbolic space-times.
Theorem 3.18. Let (M, g) be globally hyperbolic and p < q. Then there
is a nonspacelike geodesic from p to q whose length is greater than or equal to
that of any other future directed nonspacelike curve from p to q.
It should be emphasized that the geodesic in Theorem 3.18 is not necessarily
unique. This result will also be discussed from the viewpoint of the Lorentzian
distance function in Section 6.1.
It is natural to consider how continuity properties of a given candidate for
a time function on a given space-time are influenced by the causal structure
of the space-time. We have just indicated how Geroch (1970a) used past and
future volume functions to obtain the globally hyperbolic topological splitting
theorem (Theorem 3.17) and how stably causal space-times may be character-
ized in terms of the existence of a (continuous) global time function [cf. Hawk-
ing (1968), Seifert (1977), Hawking and Sachs (1974)l. It is often stated that
an auxiliary "additive measure H on M which assigns positive volume H[U] to
each open set U and assigns finite volume HIM] to M . . ." may be employed
in this context. For such a measure, it has been asserted that a distinguishing
space-time (M,g) is reflecting (cf. Definition 3.20), hence causally continuous,
if and only if the past and future volume functions t- and t+ associated to
any such measure H on M are continuous global time functions.
J. Dieckmann (1987, 1988) has noted that the above assertion fails to be
valid unless certain further regularity properties are imposed on the measure
H . In particular, one needs to restrict attention to measures that are not
positive on the boundaries of chronological futures and pasts (cf. Definition
3.19). The need for this restriction is best understood by means of an example.
Let (M, g) be four-dimensional Minkowski space-time, and let m be any Borel
measure for M with m(M) finite and with m(d(I+(p))) = m(b(l-(p))) = 0
for all p in M. Such a measure may be constructed using a partition of unity
3.2 CAUSALITY THEORY OF SPACE-TIMES 67
and local volume forms as follows [cf. Hawking and Ellis (1973, p. 199), Geroch
(1970a, p. 446), Dieckmann (1988, p. 860)]. Let w be the usual volume form
for all of M , and let {p,) be a partition of unity subordinate to a covering
{Un) with each Un being a simple region and with
Then let m be the Borel measure associated (by integration) to the 4-form
It is easily seen that m ( d ( ~ + ( ~ ) ) ) = m(d(1-(p))) = 0 for any p in M .
Now define a measure H for M as follows. Put 0 = (0,0,0,0) and
Then even though (M,g) is certainly causally continuous, the past volume
function t-(p) = H[I-(p)] fails to be continuous at 0. The difficulty is that H
assigns measure one to the point 0. Thus, H assigns positive measure to light
cones containing this point.
This counterexample led Dieckmann (1987, 1988) to investigate more pre-
cisely, for a subclass of probability measures for a given space-time, how the
past and future volume functions are related to space-time causality. We will
summarize certain of these results. We begin with an arbitrary spacetime
(M,g) and, following Dieckmann, abstract the crucial properties of the Borel
measure m whose construction was sketched above.
Definition 3.19. (Admissible Measure) A Borel measure m on (M, g) is said to be admissible provided that
(3.1) m(U) > 0 for all nonempty open sets U,
(3.2) m(M) < +oo, and
(3.3) m(d(If (p))) = m(d(I-(p))) = O for all p in M .
Clearly, the above measure H fails to satisfy condition (3.3) and thus fails
to be admissible.
68 3 LORENTZIAN MANIFOLDS AND CAUSALITY
The past (respectively, future) volume functions (associated to the measure
m) are given respectively by
and
for all p in M.
The crucial importance of property (3.3) is that it implies
and
m (m) = 41- (PI)
for all p in M .
If the given space-time happens to be totally vicious, so that I-(p) =
I+(p) = M for all p in M, then t- and t+ take on the constant values m(M)
and -m(M), respectively. More generally, in the presence of certain causality
violations the past and future volume functions are only weakly increasing
along future causal curves and may also fail to be continuous. Hence the
volume functions do not, in general, define generalized time functions in the
sense of Definition 3.23 below without imposing some causality conditions on
the space-time in question.
For convenience, we will employ the notational convention used in O'Neill
(1983) that p < q if there exists a (nontrivial) future directed nonspacelike
curve from p to q. Thus, p < q if p < q and p # q. The usual transitivity
relationships (i.e., r << p, p < q, and q << s together imply r << q and p (< s)
yield that the volume functions t- and tf are (not necessarily continuous)
"semi-time functions" in the sense that for any p, q in M
(3.8) p < q implies t- (p) 5 t- (q) and t + (PI 5 t+(q).
3.2 CAUSALITY THEORY O F SPACE-TIMES 69
It is also interesting that the usual continuity properties of a measure imply
that t- (respectively, t+) is lower semicontinuous (respectively, upper semi-
continuous).
We now introduce a causality condition important in Hawking and Sachs
(1974) and in Dieckmann (1987, 1988).
Definition 3.20. (Reflecting) A space-time (M,g) is said to be past
reflecting (respectively, future reflecting) a t q in M if for all p in M
(3.9) I+(P) 2 I+(q) implies I-(p) C I-(q),
respectively,
(3.10) I-(p) _> I-(q) implies ~ ' ( p ) I + ( ~ )
and is said to be reflecting at q if it satisfies both conditions. The space-time
is said to be reflecting if it is reflecting at all points.
It is well known that for space-times which fail to be reflecting, the past and
future volume functions may fail to be continuous [cf. Figure 1.2 in Hawking
and Sachs (1974, p. 289)]. Even more strikingly, Dieckmann obtained the
following more precise relationship between continuity and reflexivity. In this
result, properties (3.6) and (3.7) for the admissible measure are crucial.
Proposit ion 3.21. Let (M, g) be a (not necessarily distinguishing) space-
time, and let t- and t+ be the past and future volume functions associated to
an admissible Bore1 measure. Then
(1) t- is continuous at q iff (M, g) is past reffecting at q, and
(2) t+ is continuous a t q iff (M, g) is future reflecting a t q.
Moreover, from his proof of a version of Proposition 3.21, Dieckmann is able
to conclude that the set of points a t which the volume function t- (respectively,
t+) fails to be continuous is a union of null geodesics without past (respectively,
future) endpoints. Especially, a space-time may not fail to be reflecting a t
isolated points, as had already been noted in Vyas and Akolia (1986).
Thus, reflexivity settles the question of the continuity of the volume func-
tions independent of choice of admissible measure. The "time function" aspect
70 3 LORENTZIAN MANIFOLDS AND CAUSALITY
o f being strictly increasing along future directed nonspacelike curves employs
the following lemma [cf. Dieckmann (1987, p. 47)).
L e m m a 3.22. Let ( M , g ) be an arbitrary {not necessarily distinguishing)
space-time, and let t - be the past volume function associated to an admissible
Borel measure. Suppose that p, q in M satisfy p < q and I - (p ) # I - (9) . Then
Lemma 3.22 has the immediate consequence that ( M , g) is chronological i f
and only i f some past (or future) volume function is strictly increasing along
all future timelike curves. More importantly, with Lemma 3.22 in hand the
following proposition may now be obtained with the help o f future set tech-
niques. For clarity, let us first make precise the notion o f "generalized time
function" as employed in the present context.
Definit ion 3.23. (Generalized Time Function) A (not necessarily con-
tinuous) function t : ( M , g ) -+ W is said to be a generalized time fisnction i f ,
for all p, q in M ,
(3.12) p < q implies t (p) < t(q).
Thus t is strictly increasing along all future nonspacelike curves but is not
required to be continuous.
Proposit ion 3.24. Let t - (respectively, t+) be the past (respectively, fu-
ture) volume function on ( M , g ) associated to an admissible Borel measure.
Then
(1 ) ( M , g ) is past distinguishing i f f t- is a generalized time function; and
(2) ( M , g ) is future distinguishing i f f t+ is a generalized time function.
Now it has been established that "causal continuity" (i.e., ( M , g ) is distin-
guishing and I+ and I - are outer continuous) is equivalent to "reflecting" and
"distinguishing" [cf. Hawking and Sachs (1974, p. 292)). Hence, Propositions
3.21 and 3.24 imply the following characterization o f causal continuity in terms
o f volume functions.
3.2 CAUSALITY THEORY O F SPACE-TIMES
T h e o r e m 3.25. The following are equivalent:
(1 ) The space-time ( M , g ) is causally continuous.
(2 ) For any (and hence all) admissible Borel measures, the associated vol-
ume functions t - and t+ are both continuous time functions.
The condition o f "causal continuity" implies the "stable causality" condi-
tion but not conversely. Stable causality has the characterization that ( M , g)
admits some continuous global time function (which will not, in general, be a
volume function associated to some admissible Borel measure).
W e now state the characterization of global hyperbolicity in terms of volume
functions [implicit in Geroch (1970a)l which is given in Dieckmann (1987).
T h e o r e m 3.26. Let t - (respectively, t+) be the past (respectively, future)
volume functions associated to an admissible Borel measure m for the space-
time ( M , g). Define t : ( M , g) -+ R by
for p in M . Then the following are equivalent:
(1 ) The space-time ( M , g) js globally hyperbolic.
(2 ) The past and future volume functions t - , t+ are continuous time
functions, and for any inextendible future directed nonspacelike curve
y : (a , b) -+ ( M , g ) we have lim,,bt+(~(u)) = l imu4at-(y(u)) = 0.
(3 ) The function t given by (3.13) is a continuous time function, and for
any inextendible future directed nonspacelike curve y, range(to y ) = W. (4 ) For all a in R, the set t - ' ({a)) is a Cauchy surface.
The next proposition gives sufficient conditions for a space-time to be causal
in terms o f the volume functions [cf. Dieckmann (1987)l.
Proposition 3.27. Let t be a past or future volume function associated to
an admissible Borel measure for the given space-time ( M , g ) . Suppose further
that
(1 ) p << q implies t ( p ) < t (q) for all p, q in M (which implies chronology),
and
72 3 LORENTZIAN MANIFOLDS AND CAUSALITY
(2) for each complete null geodesic ,f3 : R -+ (M, g), there exist u l , uz in W such that t(P(u1)) < t(P(u2)).
Then (M, g) is causal.
In Vyas and Joshi (1983), a discussion is given of how causal functions sim-
ilar to (3.5) may be related to the ideal boundary points and singularities of
a space-time. An interesting list of 71 assertions on causality to be proved
or disproved (together with answers) has been given in Geroch and Horowitz
(1979, pp. 289-293). We now give a diagram (Figure 3.3) indicating the rela-
tions between the causality conditions discussed above [cf. Hawking and Sachs
(1974, p. 295), Carter (1971a)l.
3.3 Limit Curves and the C O Topology on Curves
Two different forms of convergence for a sequence of nonspacelike curves
{y,) have been useful in Lorentzian geometry and general relativity [cf. Pen-
rose (1972), Hawking and Ellis (1973)l. The first type of convergence uses
the concept of a limit curve of a sequence of curves, while the second type
uses the C0 topology on curves. For arbitrary space-times, neither of these
types of convergence is stronger than the other. However, we will show that
for strongly causal space-times, these two forms of convergence are closely
related. This relationship will be useful in constructing maximal geodesics in
strongly causal space-times (cf. Sections 8.1 and 8.2).
Definition 3.28. (Limit Curve) A curve y is a limit curve ofthe sequence
(7,) if there is a subsequence (y,) such that for all p in the image of y, each
neighborhood of p intersects all but a finite number of curves of the subsequence
{y,). The subsequence (7,) is said to distinguish the limit curve y.
In general, a sequence of curves (7,) may have no limit curves or may
have many limit curves. This is true even if the curves (y,) are nonspace-
like. Furthermore, even in causal space-times a limit curve of a sequence of
nonspacelike limit curves is not necessarily nonspacelike. For example, the
curve ~ ( u ) = (0,0, u) in Carter's example (cf. Figure 3.2) is not nonspacelike
although it is a limit curve of any sequence {y,) of inextendible null geodesics
3.3 LIMIT CURVES AND THE C0 TOPOLOGY ON CURVES 73
globally hyperbolic
causally simple I 4
causally continuous
C I stably causal 1
1 strongly causal
C I distinguishing
4 I causal 1
I chronological 1 FIGURE 3.3. This diagram illustrates the strengths of the causal-
ity conditions used in this book. Global hyperbolicity is the most
restrictive causality assumption that we will use.
contained in the set t = 0. Recently, the phrase "cluster curve" has been sug-
gested as a more mathematically precise term for the convergence in Definition
3.28.
In contrast, we have the following result for strongly causal space-times.
Lemma 3.29. Let (M, g) be a strongly causal space-time. If y is a limit
curve of the sequence {y,) of nonspacelike curves, then y is nonspacelike.
76 3 LORENTZIAN MANIFOLDS AND CAUSALITY
where hij are the components of h with respect to the local coordinates
XI , . . . , 2,. Since Ixi'l 5 K1, the length Lo(y I [tl, ta]) satisfies
where H is the supremum of lhijj on the compact set for 1 5 i, j 5 n. Thus,
any nonspacelike curve from the level set f -l(tl) to the level set f -'(t2) which
lies in U has length bounded by n ~ ~ / ~ K1 Itl - t21. Furthermore, covering
(M, g) by a locally finite cover of sets with the properties of U and (V, x) above,
it follows that any nonspacelike curve of (M, g) defined on a compact interval of
R must have finite length with respect to h. Thus every nonspacelike curve of
(M, g) may be given a parametrization which is an arc length parametrization
with respect to h. Also, an inextendible curve y which has an arc length
parametrization with respect to h must be defined on all of W because do is
complete (cf. Lemma 3.65).
We now state a version of Arzela's Theorem which may be established using
standard techniques [cf. Munkres (1975, Section 7.5)].
Theorem 3.30. Let X be a locally compact Hausdorff space with a count-
able basis, and let (M, h) be a complete Riemannian manifold with distance
function do. Assume that the sequence {f,) of functions fn : X -+ M is
equicontinuous and that for each s o 6 X the set Un{fn(xo)) is bounded with
respect to 4. Then there exist a continuous function f : X + M and a sub-
sequence of {f,) which converges to f uniformly on each compact subset of
X.
Using Arzela's Theorem, we may now obtain the next proposition, given in
Hawking and Ellis (1973, p. 185), which guarantees the existence of limit curves
for a sequence (7,) of nonspacelike curves having points of accumulation.
Proposition 3.31. Let {y,) be a sequence of (future) inextendible non-
spacelike curves in (M, g). If p is an accumulation point of the sequence {y,),
then there is a nonspacelike limit curve y of the sequence {yn) such that p E y
and y is (future) inextendible.
Proof. We will give the proof only for inextendible curves since the proof
for future inextendible curves is similar.
3.3 LIMIT CURVES AND THE C0 TOPOLOGY ON CURVES 77
Let h be an auxiliary complete Riemannian metric for M with distance
function 4 as above, and give each y,, an arc length parametrization with
respect to h. Then the domain of each y, i s R as each curve is assumed to
be inextendible. Shifting parametrizations if necessary, we may then choose
a subsequence {y,) of (7,) such that ym(0) + p as m + co since p is an
accumulation point of the sequence y,. Using the fact that each y, has an
arc length parametrization with respect to h, we obtain
for each m and tl,t2 E R. Thus the curves {y,) form an equicontinu-
ous family. Furthermore, since y,(O) + p, there exists an N such that
do(y,(O),p) < 1 whenever m > N. This implies that for each fixed to E R,
the curve ym 1 [-to, to] of the subsequence lies in the compact set {q E M :
do(p, q) 5 to + 1) whenever m > N. Hence the family {y,) satisfies the hy-
potheses of Theorem 3.30, and we thus obtain a (continuous) curve y : R -+ M
and a subsequence {yk) of the subsequence {y,) such that {yk) converges to
y uniformly on each compact subset of R. Clearly, yk(0) + p = y(0). The
convergence of {yk) to y also yields the inequality do(y(tl), y(t2)) 5 Itl - t21
for all t l , t2 E W. It remains to show that y is nonspacelike and inextendible.
To show that y is nonspacelike, fix t l E R and let U be a convex normal
neighborhood of (M,g) containing y(t1). Choose 6 > 0 such that the set
{q E M : do(y(tl),q) < 6 ) is contained in U . If tl < t2 < t l + 6, then (3.16)
and the uniform convergence on compact subsets yields that for all large k, the
set ~ k [ t l > t2] lies in U . Using yk(t1) ~ ( t l ) , ~ k ( t 2 ) --' ~ ( t 2 ) 1 ~ k ( t 1 ) <U ~ k ( t 2 ) for all large k, and the fact that U is a convex normal neighborhood, we obtain
that y(t1) y(t2). Thus y ] [tl,t2] is a future directed nonspacelike curve
in U [cf. Hawking and Ellis (1973, Proposition 4.5.1)]. It follows that y is a
future directed nonspacelike curve in (M, g).
It remains to show that y is inextendible. We will give the proof only of the
future inextendibility since the past inextendibility may be proven similarly.
To this end, assume that y is not future inextendible. Then y(t) -+ qo f M
as t -+ m. Let U' be a convex normal neighborhood of qo such that U' is a
compact set contained in a chart (V, x) of M with local coordinates (xl, . . . , x,)
78 3 LORENTZIAN MANIFOLDS AND CAUSALITY
such that f = x1 : U' -+ W is a time function for U'. An inequality of the
form of (3.15) shows that if y I [tl, m ) C U', then no nonspacelike curve in U'
from the level set f -I( f ( ~ ( t l ) ) ) to the level set f -'( f (qo)) can have arc length
with respect to h greater than some number 6' > 0. On the other hand, for
sufficiently large k we must have yk[tl+ l , t l +6' +2] 5 f -l([f (y(tl)), f (qo)]).
Since the length LO(yk[tl + l , t l + 6' + 21) = 6' + 1 for all k, this yields a
contradiction. (3
Even if all of the inextendible nonspacelike curves of the sequence {y,)
are parametrized by arc length with respect to a complete Riemannian met-
ric h, the limit curve y obtained in the proof of Proposition 3.31 need not
be parametrized by arc length. This is a consequence of the fact that the
Riemannian length functional, while lower semicontinuous, is not upper semi-
continuous in the topology of uniform convergence on compact subsets. Even
though the curve y constructed in the proof of Proposition 3.31 need not be
parametrized by arc length, the curve y will still be defined on all of R pro-
vided each y, is inextendible. Furthermore, if (M,g) is strongly causal, the
Hopf-Rinow Theorem and Proposition 3.13 imply that do(y(O), y(t)) --t co as
I t -+ co. Here, do denotes the complete Riemannian distance function in-
duced on M by h as above. An alternative treatment of the technicalities of
Proposition 3.31, closer in spirit to that given in Hawking and Ellis, may be
found in O'Neill (1983, p. 404) in the section on "quasi-limits."
In the globally hyperbolic case, a stronger version of Proposition 3.31 may
be obtained.
Corollary 3.32. Let (M, g) be globally hyperbolic. Suppose that {p,) and
{q,) are sequences in M converging to p and q in M respectively, with p < q,
p # 4, and p, < q, for each n. Let y, be a future directed nonspacelike curve
from p, to q, for each n. Then there exists a future directed nonspacelike
limit curve y of the sequence (7,) which joins p to q.
Proof. Let h be an auxiliary complete Riemannian metric on M with length
functional Lo. Choose a finite cover of the compact set J+(p) n J-(q) by con-
vex normal neighborhoods Ul, U2, . . . , Uk, each of which has compact closure
and such that no nonspacelike curve which leaves any U, ever returns to that
3.3 LIMIT CURVES AND THE C0 TOPOLOGY ON CURVES 79
Ui. As in the proof of Proposition 3.31, there exists a number N, for each i such
that each nonspacelike curve y : [a, b ] -+ Ui has length less than N, with re-
spect to h [cf. equation (3.15)]. Thus if U = UIU.. . u U ~ and N = Nl +. . -+Nk,
every nonspacelike curve y : [a, b] + U must satisfy Lo(?) 5 N.
Extend each given nonspacelike curve y, to a future inextendible nonspace-
like curve, also denoted by y,. We may assume that each y, : [0, co) -+ M has
been parametrized by arc length with respect to h [cf. equation (3.15)]. Thus
if U = Ul U...UUk and Proposition 3.31 is applied to {y,) with accumulation
point p of {y,(O) = p,), then there exist a future inextendible nonspacelike
limit curve y : [0, m ) -+ M with $0) = p and a subsequence {y,) of {y,)
such that y, -+ y uniformly on compact subsets of [0, a). Using y,(t,) = q,
for 0 < t, 5 N and q, -+ q, we conclude that y passes through q for some
parameter value T which satisfies 0 < T I N. It follows that y 1 10, T] is a
nonspacelike limit curve of {y, 1 10, t,]) which joins p to q. 0
We now consider convergence in the C0 topology [cf. Penrose (1972, p. 49)].
Definition 3.33. (Convergence of Curves in C0 Topology) Let y and all curves of the sequence {y,) be defined on the closed interval [a, b]. The
sequence {y,) is said to converge to y in the C0 topology on curves if ?,(a) -+
y(a), yn(b) --+ y(b), and given any open set V containing y, there is an integer
N such that y, C V for all n 2 N.
Any space-time contains a sequence {y,) that has a limit curve y, yet {y,)
does not converge to y in the C0 topology. For, let a, P : [ O , 1 ] -+ M be any
two future directed timelike curves with a([O, I]) n P([O, I]) = 0. Set
Then {y,) does not converge to either a or P in the C0 topology. However, the
subsequence (72,) (respectively, {y2,-1)) of {-/n) converges to a (respectively,
p) in the C0 topology. A space-time which is not strongly causal may also
contain a sequence {y,) of nonspacelike curves which has a nonspacelike limit
curve y, yet no subsequence {y,) of {y,) converges to 7 in the C0 topology on
3 LORENTZIAN MANIFOLDS AND CAUSALITY
remove -- FIGURE 3.4. A causal space-time (M,g), in which a sequence
{y,} o f nonspacelike curves may have a limit curve 7 yet fail t o
have a subsequence which converges to y in the C0 topology on
curves, may be formed from a subset o f Minkowski space as shown.
curves. This is illustrated in Figure 3.4 [cf. Hawking and Ellis (1973, p. 193)
for a discussion of the causal properties o f this example].
Conversely, a sequence o f nonspacelike curves (7,) may converge in the C0
topology to some nonspacelike curve y but fail t o have y as a limit curve. This
may be seen on the cylinder M = S1 x W with the Lorentzian metric ds2 =
d8dt. Let y, be the segment on the generator 8 = 0 given by yn(t) = (0 , t )
for 0 5 t 5 1 and for all n. I f y is the piecewise smooth nonspacelike curve
3.3 LIMIT CURVES AND THE C0 TOPOLOGY ON CURVES 81
obtained by going around the circle on the null geodesic t = 0 and then up
the generator B = 0 from t = 0 to t = 1 , then (7,) converges t o y in the C0
topology, but y is not a limit curve o f {y,) (c f . Figure 3.5).
In strongly causal space-times, however, these two types o f convergence are
almost equivalent for sequences of nonspacelike curves [cf. Beem and Ehrlich
(1979a, p. 164)]. A more precise statement is given by the following result.
Proposition 3.34. Let ( M , g ) be a strongly causal space-time. Suppose
that (7,) is a sequence o f nonspacelike curves defined on [a, b ] such that
yn(a) -+ p and yn(b) --+ q. A nonspacelike curve y : [a, b] -+ M with y(a) = p
and y(b) = q is a limit curve o f {y,} i f f there is a subsequence {y,} o f (7,)
which converges to y in the C0 topology on curves.
Proof. (+) W e may assume without loss o f generality that y and {y,) are
all future directed curves. Let V be any open set with y V . Cover the
compact image of y with convex normal neighborhoods W l , W 2 , . . . , W k such
that each Wi V and no nonspacelike curve which leaves Wi ever returns to
Wi. There exists a subdivision a = to < tl < . . . < t j = b o f la, b ] such that
for all 0 5 i 5 j - 1, each pair y(t,), y(t,+l) lies in some Wh. Here h = h ( i )
and 1 < h( i ) 5 k for all i. Let {y,} be a subsequence that distinguishes
y as a limit curve. For each m , let p(0, m ) = y,(a) and p ( j , m ) = y,(b).
Furthermore, for each fixed i with 0 < i < j , choose p(i, m) E ym such that
{ p ( i , m ) } converges to $ti). Since y(ti+l) lies in the causal future o f y(t,) and
M is strongly causal, the point p(i + 1, m) lies in the causal future o f p(i, m)
for all m larger than some N l . Also, there is some N2 such that p(i, m) and
p ( i + l , m ) l i e i n W h ( i ) f o r a l l O ~ i ~ j - l a n d m ? Nz. L e t N = m a x { N l , N z ) .
The portion o f ym joining p ( i ,m) to p(i + 1, m) must lie entirely in Wh(;) for
m 2 N because no nonspacelike curve can leave W h and return. It follows
that ym 5 W l U . . . U W k V for all m > N as required.
(e) Let {y,} be a subsequence o f {y,} converging to y in the C 0 topology on
curves. Define A = { to E [a, b] : each point o f y I [a, to] is a limit point o f the
given subsequence }. W e wish to show that A = [a, b] . Clearly, ym(a) -+ y(a)
implies a E A. I f r = sup{to : to E A ) , then for each a 5 t < 7 the point
y ( t ) is a limit point o f the subsequence (7,). To show r E A we assume
3 LORENTZIAN MANIFOLDS AND CAUSALITY
FIGURE 3.5. In chronological space-times, a sequence of non-
spacelike curves {y,) may converge to the nonspacelike curve y in
the C0 topology on curves, and yet y may fail to be a limit curve
of any subsequence (7,) of {y,). The curves y, are segments on
the line 0 = 0 from t = 0 to t = 1. The curve y goes around the
cylinder once, then traverses y,.
T > a and let i t k ) be a sequence with t k -+ T-. Each neighborhood U(y(7))
of y ( r ) is also a neighborhood of y(tk) for sufficiently large k and hence must
intersect all but a finite number of curves of the subsequence {y,). Thus y ( r )
is a limit point of {y,), and A must be a closed subinterval of [a, b]. Assume
that 7 < b. Using the strong causality of (M,g), we may find a convex normal
neighborhood V of y ( r ) such that no nonspacelike curve of (M, g) which leaves
V ever returns. Letting V be sufficiently small we may assume that (V, gj,) is
globally hyperbolic and that f : V --+ W is a Cauchy time function for (V, gIv)
with f (V) = R and f ($7)) = 0. We may also assume y(b) # V. Then each
inextendible nonspacelike curve of (M, g) which has a nonempty intersection
3.4 TWO-DIMENSIONAL SPACE-TIMES 83
with V must intersect each Cauchy surface f-'(s) exactly once. Fix s with
0 < s < m, and define x(s) = y n fP'(s). This intersection exists because
y ( r ) E V and y(b) $ V. Since y ( r ) and y(b) are limit points of {y,), the
curves y, must have a nonempty intersection with f-'(s) for all sufficiently
large m. Set x,(s) = y, n f-'(s) for all such m. In order to verify that
x,(s) 4 x(s), we observe that for each neighborhood W of y the points
x,(s) must lie in W fl f -'(s) for all large m (cf. Figure 3.6). This shows
that each x(s) is a limit point of the subsequence {y,). Consequently, the
set A contains numbers greater than T, in contradiction to the definition of
T. We conclude A = [a, b] which shows y is a limit curve of the subsequence
{yrn). 0
Let y : [a, b] + (M,g) be a nonspacelike curve in a strongly causal space-
time (M,g). Choose a compact subset K of M such that y 2 Int(K). Let the
nonspacelike curves which are contained in K be given the C0 topology. It
is known [cf. Penrose (1972, p. 54)] that the Lorentzian arc length functional
L(y) [cf. Chapter 4, equation (4.l)] is upper semicontinuous with respect to
the C0 topology on curves fcf. Busemann (1967, p. lo)]. This is the analogue
of the well-known result that the Riemannian arc length functional is lower
semicontinuous.
R e m a r k 3.35. Let (M,g) be strongly causal, and let y be a given non-
spacelike curve in (M, g). If the sequence {y,) of nonspacelike curves converges
to y in the C0 topology on curves, then
L(y) L lim sup L(y,).
3.4 Two-Dimensional Space-times
In this section we consider the topological and causal structures of two-
dimensional Lorentzian manifolds. Using the pair of null vector fields generated
by the tangent vectors to the two null geodesics passing through each point
of M, we show that the universal covering manifold of any two-dimensional
Lorentzian manifold is homeomorphic to R2. We then show that any two-
dimensional Lorentzian manifold homeomorphic to R2 is stably causal. In
84 3 LORENTZIAN MANIFOLDS AND CAUSALITY
FIGURE 3.6. In the proof of Proposition 3.34 the globally hyper-
bolic neighborhood V of y(7) has a Cauchy time function f : V --+ R
with f ($7)) = 0. For all large m the curves ym must intersect
the Cauchy surface f-'(s) at a single point xm(s). If W is any
neighborhood of y, then xm(s) E W n fF1(s) for all large m. We
may choose W such that W n f -'(s) is as small a neighborhood of
Z(S) = y n f -'(s) in f -l(s) as we wish. Thus xm(s) -+ x(s), and
x(s) must be a limit curve of the subsequence {y,).
particular, every simply connected two-dimensional Lorentzian manifold is
causal. Thus no Lorentzian metric for R2 has any closed nonspacelike curves.
It should also be noted that two- (but not higher) dimensional Lorentzian man-
ifolds have the property that (M, -g) is also Lorentzian. This is sometimes
3.4 TWO-DIMENSIONAL SPACGTIMES 85
useful in obtaining results about all geodesics in (M,g) from results valid in
higher dimensions only for nonspacelike geodesics.
Let (M, g) be an arbitrary two-dimensional Lorentzian manifold, and fix a
point p E M. Choose a convex normal neighborhood U(p) based at p, and
consider the following method of assigning local coordinates to points in U(p)
sufficiently close to p. Let the two null geodesics yl and 72 through p be
given parametrizations yl : (-el, €1) -+ U(p) and 72 : (-e2, €2) -' U(p) with
yl(0) = yz (0) = p. For each point q E U(p) sufficiently close to p, the two null
geodesics through q will intersect yl and 7 2 in U(p) at unique points yl(to)
and y2(so) respectively. Assign coordinates (to, so) to q. In these coordinates
the null geodesics near p are contained in sets of the form t = to or s = so.
We have established
Lemma 3.36. Let (M, g) be a two-dimensional Lorentzian manifold. Then
each p E M has local coordinates x = (XI, 22) with x(p) = 0 such that each null
geodesic in this neighborhood is contained in a set of the form x1 = constant
or $2 = constant.
Suppose X is a future directed timelike vector field on M. Then at each p E
M, there are two uniquely defined future directed null vectors nl , n2 E T,M
such that X(p) = nl + n2. Clearly, a sufficiently small neighborhood U(p) of p
may be found such that nl and n2 may be extended to continuous null vector
fields XI, X2 defined on U(p) with X(q) = Xl(q) + X2(q) for all q E U(p). If
M is simply connected, we now show X1 and X2 can be extended to all of M.
Proposition 3.37. Let (M,g) be a simply connected Lorentzian manifold
of dimension two. Then two smooth nonvanishing null vector fields X1 and
X2 may be defined on M such that X1 and X2 are linearly independent a t
each point of M.
Proof. Since M is simply connected, (M,g) is time orientable. Thus we
may choose a smooth future directed timelike vector field X on M.
Fix a base point po E M, and let X(po) = nl + 7x2 as above. Given any
other point q E M, let y : [0,1] -4 M be a curve from po to q. There is exactly
one way to define continuous null vector fields X1 and X2 along y such that
86 3 LORENTZIAN MANIFOLDS AND CAUSALITY
Xl(0) = n l , Xz(0) = n2, and X(y(t)) = Xl(t) + Xz(t) for all t E [O,l]. If
77 : [O, 11 -+ M is any other curve from po to q, then y and 7 are homotopic since
M is assumed to be simply connected. Hence if Yl and Y2 were null vector
fields along q with Yl(0) = n l and E(0 ) = n2, we would have Yl(1) = X1(l)
and Yz(1) = Xz(1) by standard homotopy arguments. Thus this construction
produces a pair of continuous vector fields XI and X2 on M which are linearly
independent at each point. C1
Corollary 3.38. Let (M, g) be any two-dimensional Lorentzian manifold.
Then the universal Lorentzian covering manifold (z,~) of (M, g) is homeo-
morphic to R2.
Proof. Since % is simply connected and tw~dimensional, is homeornor-
phic to EC2 or S2. But since the Euler characteristic of S2 is nonzero, S2 does
not admit any nowhere zero continuous vector fields.
Recall that an integral curve for a smooth vector field X on M is a smooth
curve y such that yl(t) = X(y(t)) for all t in the domain of y [cf. Kobayashi
and Nomizu (1963, p. 12)]. The following result is well known [cf. Hartman
(1964, p. 156)].
Proposi t ion 3.39. Let X be a smooth nonvanishing vector field on R?,
and let y : (a, b) -, R2 be a maximal integral curve of X . Then y(t) does not
remain in any compact subset of R2 as t + a+ [or t + b-).
Now assume that ( M , g ) is a Lorentzian manifold homeomorphic to R2. Let XI , X2 be the null vector fields on M given by Proposition 3.37. Clearly,
each null geodesic of (M,g) may be reparametrized to an integral curve of
X1 or X2. Equivalently, the integral curves of XI and X2 are said to be null
pregeodesics. Suppose y : (a, b) + M is an inextendible null geodesic which
may be reparametrized to an integral curve of XI. If y(t1) = y(t2) for some
t l # t2, then since both yl(tl) and yli(tz) are scalar multiples of Xl(y(tl)), it
follows from geodesic uniqueness that y is a smooth closed geodesic. However,
this is impossible by Proposition 3.39. Thus Proposition 3.39 has the following
corollary.
3.4 TWO-DIMENSIONAL SPACGTIMES 87
Corollary 3.40. If (M, g) is a Lorentzian manifold homeomorphic to R2,
then (M, g) contains no closed null geodesics. Moreover, every inextendible
null geodesic y : (a, b) + M is injective and hence contains no loops.
A family F of inextendible null geodesics is said to cover a manifold M
simply if each point p f M lies on exactly one null geodesic of F. Suppose
(M, g ) is a Lorentzian manifold homeomorphic to W2. Then the integral curves
of the null vector field X1 (respectively, Xz) given in Proposition 3.37 may be
reparametrized to define a family Fl (respectively, F2) of geodesics on M . Each
family F, covers M since X,(p) # 0 for i = 1, 2 and all p E M . Furthermore,
since exactly one integral curve of X, passes through any p E M , each family
F, covers M simply. Consequently, Proposition 3.37 implies [cf. Beem and
Woo (1969, p. 51)]
Proposi t ion 3.41. Let (M, g) be a Lorentzian manifold homeomorphic to
R2. Then the inextendible null geodesics of (M,g) may be partitioned into
two families F1 and F2 such that each of these families covers M simply.
Let y : (a, b) -+ M be an inextendible timelike curve, and let c : (cu,P) -+
M be an inextendible null geodesic. Obviously, in arbitrary two-dimensional
Lorentzian manifolds, y and c may intersect more than once. However, if M
is homeomorphic to W2, y and c intersect in a t most one point [cf. Beem and
Woo (1969, p. 52), Smith (1960b)l.
Proposi t ion 3.42. Let (M, g) be a Lorentzian manifold homeomorphic to
R2. Then each timelike curve intersects a given null geodesic at most once.
Proof. Let co be an inextendible future directed null geodesic in M , which
we may assume belongs to the family Fl defined by the null vector field XI as
above. Suppose that a is a future directed timelike curve in M which intersects
co twice (possibly a t the same point). We may then find a, b f R with a < b
such that u(a), u(b) lie on co and u(t) 4 for a < t < b. Since u is timelike, u
is locally one-to-one. Hence if u / [a, b] is not one-to-one, u contains a t worst
closed timelike loops. Using one of these loops. it is possible to find a , /3 f R
with a < a < ,B < b and a second null geodesic cl E Fl such that u 1 [a, /3] is
one-to-one, u (a ) and u(p) lie on cl, and u(t) 4 cl for cu < t < p.
88 3 LORENTZIAN MANIFOLDS AND CAUSALITY
geodesic
FIGURE 3.7. In a Lorentzian manifold homeomorphic to R2, the
timelike curve y is assumed to cross the null geodesic Q at y(a)
and y(b). The null geodesic cl enters W at y(tl) and first leaves at
y(t;) where t i > t l .
We will show below that if y : [a, b] -, M is an injective future directed
timelike curve, there is no c E FI such that ~ ( a ) and y(b) lie on c but y(t) $ c for a < t < b. If the original timelike curve a 1 [a, b ] is injective, this
argument applied to a 1 [a, b] yields the desired contradiction. If a I [a, b ] is
not injective but intersects at a(a) and a(b), then this argument applied to
cl and a 1 [a, p] yields the desired contradiction.
3.4 TWO-DIMENSIONAL SPACE-TIMES 89
Thus the theorem will be established if we show that it is impossible to find
an injective future directed timelike curve y : [a, b] --+ M with y(a) and y(b)
on q and y(t) $ GJ for a < t < b. Traversing y from y(a) to y(b) and then the
portion of q from ?(b) to y(a) yields a closed Jordan curve which encloses a set
W with 'ii;i compact (cf. Figure 3.7). Let U be a convex normal neighborhood
based on y(a). Choose t l with a < t l < b and y(t1) E U. Let cl be the null
geodesic in Fl passing through y(tl). Since cl may be reparametrized to be
an integral curve of XI and cl enters W at y(tl) , it follows by Proposition
3.39 that cl leaves W a t some point y(ti) with t i > tl. As q intersects y at
y (b), we must have t i < b. In particular, itl, ti] C (a, b). Hence we have found
a closed interval [tl,t;] 2 (a,b) such that y([tl,t;]) 2 and y intcrsccts the
null geodesic cl E Fl at y(tl) and ?(t:) (cf. Figure 3.7).
We may now form a second closed Jordan curve by traversing y from tl to
t i followed by the portion of cl from y (t i) to y (tl). Repeating the argument
of the preceding paragraph, we obtain a closed interval [tz, t i] C (tl,t',) such
that the timelike curve y 1 [tz, t i] intersects a null geodesic c2 in the family
Fl at y(t2) and y(ti) and such that y(t2) is contained in a convex normal
neighborhood of y(tl). Inductively, we can construct a nested sequence of
intervals [tk+1, C (tk,tjc) such that ~ ( t k + ~ ) lies in a convex normal neigh-
borhood of y(tk) and y I [ t k + l r t ~ + l ] intersects a null geodesic ck+l € Fl at
Y(tk+l) and ~ ( t : + ~ ) . Moreover, the intervals [tk, t i ] may be chosen such that
n E l [ t k , t i ] = {to) for some to E (a,b). We thus have constructed two se-
quences tk 1 to and t: 1 to such that the timelike curve y intersects a null
geodesic in Fl at both y(tk) and y(tjc) for each k 2 1. But this is impossible
by Proposition 3.4. Hence the geodesic y : [a, b ] -+ M intersects at most
once.
Theorem 3.43. Let (M, g) be a Lorentzian manifold homeomorphic to R2.
Then (M, g) is stably causal.
Proof. Recall that g E Lor(M) is stably causal if there is a fine C0 neigh-
borhood U of g in Lor(M) such that all metrics in U are causal. Since strongly
causal implies causal, it will thus follow that all metrics in Lor(M) are stably
causal if all metrics in Lor(M) are strongly causal. Thus to prove the theorem,
90 3 LORENTZIAN MANIFOLDS AND CAUSALITY
it is enough to show that if g is any Lorentzian metric for M , then (M, g) is
strongly causal.
Thus suppose that g is a Lorentzian metric for A4 such that (M, g) is not
strongly causal. Then there is some p E M such that strong causality fails at
p. Let (U, x) be a chart about p, guaranteed by Lemma 3.36, such that the
null geodesics in U lie on the sets x l = constant and x2 = constant. Since
strong causality fails a t p, there are arbitrarily small neighborhoods V of p
with V C U and timelike curves which begin at p, leave V , and then return to
V. By Proposition 3.42, there are no closed timelike curves through p. Thus
if y is a future directed timelike curve with y(0) = p which leaves V and then
returns, we have y ( t ) # p for all t > 0. Since the null geodesics in U through p
are given by x1 = 0 and by 2 2 = 0 in the local coordinates x = (XI , x2) for U ,
it follows that y may be deformed to intersect one of the null geodesics through
p upon returning to V (cf. Figure 3.8). Hence y intersects a null geodesic in
Fl or F2 twice, contradicting Proposition 3.42.
Corollary 3.44. No Lorentzian metric for R2 contains any closed non-
spacelike curves.
A different proof of the result that any simply connected Lorentzian two-
manifold is strongly causal may be found in O'Neill (1983). For n > 3,
Lorentzian metrics which are not chronological and hence not strongly causal
may be constructed on Rn.
For every two-dimensional Lorentzian manifold (M,g), there is an associ-
ated Lorentzian manifold (MI -g). The timelike curves of (MI -g) are the
spacelike curves of (M, g) and vice versa. Using M = R2 and applying Corol-
lary 3.44 to (M, -g) we obtain
Corollary 3.45. No Lorentzian metric for R2 contains any closed spacelike
curves.
If (M, g) is two-dimensional and both (MI g) and (M, -g) are stably causal,
then using techniques given in Beem (1976a), one may show there is some
smooth conformal factor 52 : M --, (0 ,m) such that the manifold (M, Rg) is
geodesically complete. This yields the following corollary.
3.4 TWO-DIMENSIONAL SPACE-TIMES
FIGURE 3.8. (M,g) is a two-dimensional space-time such that
strong causality fails a t p. There is a future directed timelike curve
y which starts a t p, later returns close to p, and crosses one of the
null geodesics through p.
Corollary 3.46. Let (M,g) be a Lorentzian manifold homeomorphic to
R2. Then there is a smooth conformal factor 0 : M -+ (0,m) such that
(M, 52g) is geodesically complete.
There are examples of twudimensional space-times such that no global con-
formal change makes them nonspacelike geodesically complete (cf. Section 6.2).
Thus, Corollary 3.46 cannot be extended to all two-d~mensional space-times
by covering space arguments. The recent monograph by Weinstetn (1993)
contains many further interesting results on Lorentzian surfaces inspired in
part by Kulkarni's (1985) study of the conformal boundary for such a surface
[cf. Smyth and Weinstein (1994)l.
-
92 3 LORENTZIAN MANIFOLDS AND CAUSALITY
3.5 The Second Fundamental Form
Let N be a smooth submanifold of the Lorentzian manifold (M,g) . If
i : N + M denotes the inclusion map, then we may regard TpN as being a
subspace of TpM by identifying i,,(TpN) with TpN. Let go = z*g denote the
pullback of the Lorentzian metric g for M to a symmetric tensor field on N .
Under the identification of T p N and i ep(TPN) , we may also identify go at p
and g ( T,N x TpN for all p f N . This identification will be used throughout
this section.
Definition 3.47. (Nondegenerate Submanifold) The submanifold N of
( M , g ) is said to be nondegenemte if for each p E N and nonzero v E T p N ,
there exists some w f TpN with g(v, w) f 0. If, in addition, g 1 TpN x TpN is
positive definite for each p E N , then N is said to be a spacelike submanzfold.
If g I TpN x TpN is a Lorentzian metric for each p E N , then N is said to be
a timelike submanifold.
For the rest of this section, we will suppose that N is a nondegenerate
submanifold. Thus for each p E N , there is a well-defined subspace T k N of
T p M given by
T ~ N = { v ETpM : g(v,w) = 0 for all w E T p N )
which has the property that T k N n T p N = (0). Consequently, there is a well-
defined orthogonal projection map P : TpM + TpN. The connection V on
( M , g) may be projected to a connection V 0 on N by defining V g Y = P ( V x Y )
for vector fields X , Y tangent to N . It is easily verified that V 0 is the unique
torsion free connection on ( N , go) satisfying
X ( ~ ~ ( y l z ) ) = ~ O ( V % Y ? ~ ) + g ~ ( ~ l v % z )
for all vector fields X, Y, Z on N . The second fundamental form, which mea-
sures the difference between V and V O , may be defined just as for Riemannian
submanifolds [cf. Hermann (1968, p. 319), Bolts (1977, p. 25, pp. 51-52)].
Definition 3.48. (Sewnd Fundamental Form) Let N be a nondegener-
ate submanifold of (M,g) . Given n E T ~ N , define the second fundamental
3.5 THE SECOND FUNDAMENTAL FORM 93
form Sn : TpN x TpN -, W in the direction n as follows. Given x , y E T p N ,
extend to local vector fields X, Y tangent to N , and put
Define the second fundamental form S : T i N x T p N x TpN -+ W by
Given n E T i N , the second fundamental form operator L, : TpN + TpN is
defined by g(L,(x), y) = Sn(x, y) for all x, y E T p N .
It may be checked that this definition of S,(x, y) is independent of the
choice of extensions X , Y for x , y E TpN and also that S, : TpN x TpN -+ W is a symmetric bilinear map. Furthermore, S : T i n r x TpN x TpN -, W is
trilinear for each p E N .
Lemma 3.49. Let N be a nondegenerate submanifold of (M,g) . The
second fundamental form S = 0 on N iff V x Y = V g Y for all vector fields X
and Y tangent to N .
Proof. Obviously, Definition 3.48 implies that if V x Y = V$Y for all vector
fields tangent to N , then S = 0.
Now suppose S = 0. Let p E N be an arbitrary point. We then have
g ( V x Y I p - V $ Y l p , n ) = 0 for all n E T i N and vector fields X, Y tangent to
N . Since g T p N x T p N is nondegenerate, g 1 T i N x T i N is also nondegen-
erate. Thus V x Y I p and V $ Y l p have the same projection onto T k N . Since
TpM = TpN @ T k N , we have O X Y I P = V $ Y l p , as required. 0
The second fundamental form may be used to characterize totally geodesic
nondegenerate submanifolds of ( M , g). A submanifold N of ( M , g) is said to be
geodesic at p E N if each geodesic y of ( M , g) with y(0) = p and ~ ' ( 0 ) E TpN
is contained in N in some neighborhood of p. The submanifold N is said to be
totally geodesic if it is geodesic at each of its points. The following proposition
is the Lorentzian analogue of a well-known Riemannian result Icf. Hermann
(1968, p. 338), Cheeger and Ebin (1975, p. 23)].
94 3 LORENTZIAN MANIFOLDS AND CAUSALITY
Proposit ion 3.50. Let N be a nondegenerate submanifold of (M, g). Then
N is totally geodesic iff the second fundamental form S satisfies S = 0 on N .
Proof. Given that S = 0 on N , Lemma 3.49 implies that VxY = V$Y
for all vector fields X , Y tangent to N. Let c : (-6, c) --+ M be a geodesic in
(M,g) with cl(0) = v E TpN for some p E N. Also let y : (-45) --+ N be
the geodesic in (N,go) with yl(0) = v. Since Vy,yl = V:,yl = 0, the curve y
is also a geodesic in (M,g). Set 11 = min{c,6). From the uniqueness of the
geodesic in (M, g) with the given initial direction v, we have c(t) = y(t) for all
t E (-7, 7). Hence y 1 ( - q , ~ ) N as required.
Conversely, suppose N is totally geodesic in (M, g). Let p E N be arbitrary.
Given n E T i N and x E TpN, let c : J -, N be the geodesic (in both M
and N ) with c'(0) = x. Extend cl(t) to a vector field X tangent to N near p.
We then have S(n, x, x) = g(V,yXlp, n) = g(V,lcl(0), n) = g(0, n) = 0. By
polarization, it follows that S(n, x, y) = 0 for all x, y E TpN. Hence S = 0 on
N. 0
As will be seen in Chapter 12, the second fundamental form plays an im-
portant role in singularity theory in general relativity.
3.6 Warped Produc t s
If (M, g) and (H, h) are two Riemannian manifolds, there is a natural prod-
uct metric go defined on the product manifold M x H such that ( M x H, go) is
again a Riemannian manifold. Bishop and 07Neill(1969) studied a larger class
of Riemannian manifolds, including products, which they called warped prod-
ucts. If (M,g) and (H, h) are two Riemannian manifolds and f : M -+ (0, oo)
is any smooth function, the product manifold M x H equipped with the met-
ric g @ f h is said to be a warped product and f : M + (0, m) is called the
warping functzon. Following Bishop and O'Neill, we will denote the Riemann-
ian manifold ( M x H I g @I fh) by M x f H. Bishop and O'Neill (1969, p. 23)
showed that M x f H is a complete Riemannian manifold if and only if both
(M,g) and (H, h) are complete Riemannian manifolds. Utilizing this result,
they were able to construct a wide variety of complete Riemannian manifolds
of everywhere negative sectional curvature using warped products.
3.6 WARPED PRODUCTS 95
In this section, we will use warped product metrics to construct Lorentzian
manifolds and will then study the causal structure and completeness proper-
ties of this class of Lorentzian manifolds. The theory for Lorentzian mani-
folds differs from the Riemannian theory somewhat, since the product of two
Lorentzian manifolds (M, g) and (HI h) has signature (-, -, T, . . . , +) and
hence is not Lorentzian. Nevertheless, warped product Lorentzian metrics
may be constructed from products of Lorentzian and Riemannian manifolds.
In particular, this product construction may be used to produce examples of
bi-invariant Lorentzian metrics for Lie groups (cf. Section 5.5). A treatment of
warped products of semi-Riemannian (not necessarily Lorentzian) manifolds,
including a calculation of their Riemannian and Ricci curvature tensors, is
given in O'Neill (1983).
Throughout this section, we will let T : M x H -+ M and 7 : M x H -,
H denote the projection maps given by ~ ( m , h) = m and ~ ( m , h) = h for
( m , h ) ~ M X H.
Definition 3.51. (Lorentzian Warped Product) Let (M,g) be an n-
dimensional manifold (n 2 1) with a signature of (-, +, . . . , +), and let (H, h)
be a Riemannian manifold. Let f : M -+ (0, cm) be a smooth function. The
Lorentzian warped product M x f H is the manifold % = M x H equipped
with the Lorentzian metric ?j defined for v, w E T~~ R by
Definition 3.52. (Lorentzian Product) A warped product M x f H with
f = 1 is said to be a Lorentzian product and will be denoted by M x H .
R e m a r k 3.53. One may also obtain Lorentzian manifolds by considering
warped products H x f M , where (H, h) is a Riemannian manifold, (M, g) is a
Lorentzian manifold, and f : H --+ (0, a) is a smooth function. The universal
covering manifold of anti-de Sitter space (cf. Section 5.3) is an example of a
space-time important in general relativity which may be written as a warped
product of the form H x f M with H Riemannian and M Lorentzian but not
as a warped product of the form M x f H of Definition 3.51. We will only treat
warped products of the form M x f H in this section.
96 3 LORENTZIAN MANIFOLDS AND CAUSALITY f 3.6 WARPED PRODUCTS 97
We begin our study of the causal properties of warped products with the From the hierarchy of causality conditions given in Figure 3.3, we then
following lemma. obtain
Lemma 3.54. The warped product M x f H of ( M I g) and ( H , h) may be Corollary 3.56. Let ( H I h ) be an arbitrary Riemannian manifold, and let time oriented iff either ( M , g) is time oriented (if dim M 2 2) or ( M , g) is a M = (a , b) with -cm 5 a < b 5 +a begiven the negative definite metric -dt2. one-dimensional manifold with a negative definite metric. For any smooth function f : M -t (0, a), the warped product ( M x f H, 3 ) is
Proof. Suppose that M x f H is time orientable. If dimM = 1. then chronological, causal, distinguishing, and strongly causal.
( M , g ) has a negative definite metric by Definition 3.51. Now consider the
case dim M > 2. Since M x f H is time orientable, there exists a continuous
timelike vector field X for M x f H. Since f > 0 and h is positive definite, we
then have g(r,X, r , X ) 5 g ( X , X ) < 0. Thus the vector field n,X provides a
time orientation for ( M , g).
Conversely, suppose first that dim M 2 2 and ( M , g) is time oriented by the
timelike vector field V. Then V may be lifted to a timelike vector field 7 on
M x H which satisfies r,v = V and 77,V = 0. Explicitly, fixing j j = (m, b) E
M x H , there is a natural isomorphism
Thus we may define V at j j by setting V(p) = ( V ( m ) , Ob) using this isomor-
phism to identify TF (M x H ) and TmM x TbH. It is immediate from Definition - -
3.51 that 3 ( V , V ) = g(V, V ) < 0. Hence 7 time orients M x f H as required.
Now consider the case dim M = 1. It is then known that M is diffeomorphic
to S 1 or R. In either case, let T be a smooth vector field on M with g(T, T ) =
-1. Defining T(P) = (T(r(jj))lO,(ii)) as above, we have 71*T = 0, SO that T time orients B. Note also in the case that M = S1 the integral curves of T in - M are closed timelike curves. Thus M is not chronological. 0
Lemma 3.55. Let ( H I h) be an arbitrary Riemannian manifold, and let
M = (a, b) with -cc 5 a < b 5 +cm be given the negative definite metric -dt2.
For any smooth function f : M --, (0, a), the warped product ( M x f H , g ) is
stably causal.
Proof. The projection map n : M x H --+ M W serves as a time func-
tion. El
In the proof of Lemma 3.54 above, we have seen that if M = S1, then the
warped product (S1 x f H , g ) fails to be chronological and hence fails to be
causal, distinguishing, or strongly causal.
We now list some elementary properties of warped products that follow
directly from Definition 3.51. A homothetic map F : ( M I , g l ) -+ (M2, 92) is a
diffeomorphism such that F'(g2) = cgl for some constant c. We remark that
some authors only require homothetic maps to be smooth and not necessarily
one-to-one.
Remark 3.57. Let M x f H be a Lorentzian warped product.
(1) For each b 6 H , the restriction r/v-l(b) : 77-'(b) --+ M is an isometry
of q-l(b) onto M .
(2 ) For each m f MI the restriction qlT-l(m) : T-'(m) --+ H is a homo-
thetic map of r - ' ( m ) with homothetic factor I / f (m).
(3) If v E T ( M x H ) , then g(n,v,r,v) 5 g(v, v ) . Thus n , : T p ( M x H ) -+
TT(p)M maps nonspacelike vectors to nonspacelike vectors, and r maps
nonspacelike curves of M x f H to nonspacelike curves of M .
(4) Since l g ( n , v , ~ ~ v ) / > lg(v,v)/ if w 6 T(M x H ) is nonspacelike, the
map r is length nondecreasing on nonspacelike curves (cf. Section 4.1,
formula (4.1) for the definition of Lorentzian arc length).
(5) For each (m,b) E M x H , the submanifolds r-'(m) and q-vb) of
M x f H are nondegenerate in the sense of Definition 3.47.
(6) If q5 : H -+ H is an isometry, then the map @ = 1 x q5 : M x f H -+
M x f H given by @(m, b) = (m,q5(b)) is an isometry of M x f H.
(7) If $ : M -+ M is an isometry of M such that f o ?C, = f , then the map
Q =$J x 1 : M x f H -+ M x f H given by Q(m,b) = (?C,(m),b) is an
98 3 LORENTZIAN MANIFOLDS AND CAUSALITY
isometry of M x f H. Thus if X is a Killing vector field on M (i.e.,
Lxg = 0) with X ( f ) = 0, then the natural lift X of X to M x f H
given by x ( p ) = (X(n(p)), Oq(*)) is a Killing vector field on M x f H.
Lemma 3.58. Let M x f H be a Lorentzian warped product. Then for
each b E H , the leaf v-l(b) is totally geodesic.
Proof. Since the map n : M x f H -+ M is length nondecreasing on non-
spacelike curves and since nonspacelike geodesics are locally length maximiz-
ing, it follows that any nonspacelike geodesic of 77-'(b) (in the metric induced
by the inclusion q-'(b) 5 M x f H ) is a geodesic in the ambient manifold
M x f H. Thus the second fundamental form vanishes on all nonspacelike vec-
tors in T(q-'(b)). Since any tangent vector in T(q-'(b)) may be written as a
linear combination of nonspacelike vectors in T(q-'(b)), it follows that the sec-
ond fundamental form vanishes identically. Hence, 77-'(b) is totally geodesic
by Proposition 3.50. 0
In view of Corollary 3.56, we may now restrict our attention to studying
the fundamental causal properties of time oriented Lorentzian warped products
( M x H,i j ) with dim M 2 2.
Lemma 3.59. Let p = (pl, pz) and q = (ql, q2) be two points in M x f H
with p << q (respectively, p < q) in (M x f H , g ) . Then pl << ql (respectively,
Pl < q1) in (M,g).
Proof. If y is a future directed timelike (respectively, nonspacelike) curve
in M x f H from p to q, then n o y is a future directed timelike (respectively,
nonspacelike) curve in M from pl to ql.
While n : M x f H -+ M takes nonspacelike curves to nonspacelike curves,
.rr does not preserve null curves. Indeed, it follows from Definition 3.51 that if
y is any smooth null curve with q,y(t) # 0 for all t , then g(z,y(t), T, y(t)) < 0 for all t.
For points p and q in the same leaf q-l(b) of M x f H, Lemma 3.59 may be
strengthened as follows.
3.6 WARPED PRODUCTS 99
[isometric to M]
[homothetic to H]
FIGURE 3.9. Let (m, b) be a point of the warped product M x f H.
Then the projection map n restricted to q-l(b) is an isometry onto
M, and the projection map 77 restricted to ?rp'(m) is a homothetic
map onto H .
Lemma 3.60. If p = (pl, b) and q = (91, b) are points in the same leaf
~ - l ( b ) of M X I H , then p q (respectively, p Q q) in (M x f H , g ) iffpl << ql
(respectively, pl ,< ql) in (M, g).
Proof. By Lemma 3.59, it only remains to show that if pl << ql (respectively,
pl < ql) in (M,g), then p << q (respectively, p ,< q) in (M x f H,ij) . But if
yl : [O, 11 --, M is a future directed timelike (respectively, nonspacelike) curve
in M from pl to ql, then y ( t ) = (yl(t), b), 0 5 t 5 1, is a future directed
timelike (respectively, nonspacelike) curve in M x f H from p to q.
100 3 LORENTZIAN MANIFOLDS AND CAUSALITY
Lemma 3.60 implies that each leaf 77-'(b), b E H , has the same chronology
and causality as (M,g). In particular, Lemmas 3.59 and 3.60 imply that
( M x f H , g ) has a closed timelike (respectively, nonspacelike) curve iff (M,g)
has a closed timelike (respectively, nonspacelike) curve. Hence
Proposition 3.61. Let (M, g) be a space-time, m d let (H, h) be a Rie-
mannian manifold. Then the Lorentzian warped product (M x j H , g ) is
chronological (respectively, causal) iff (M, g) is chronological (respectively,
causal).
A similar result holds for strong causality.
Proposition 3.62. Let (M, g) be a space-time, and let (H, h) be a Rie-
mannian manifold. Then the Lorentzian warped product (M x j H,ij) is
strongly causal iff (MI g) is strongly causal.
Proof. We first show that if the space-time (M, g) is not strongly causal a t
pl , then ( M xf H , g ) is not strongly causal a t p = (pl, b) for any b E H. Since
(M, g) is not strongly causal at pl, there is an open neighborhood Ul of pl in
M and a sequence {yk : [O,1] -+ M ) of future directed nonspacelike curves
with yk(0) -+ pl and yk(1) -+ pl as k -+ m , but yk(1/2) $! U1 for all k. Define
uk : I0,1] -+ M x H by ak(t) = (yk(t), b ) Let Vl be any open neighborhood
of b in H, and set U = Ul x Vl in M x H. Then U is an open neighborhood of
p = (pl, b) in M x f H , and {ak) is a sequence of nonspacelike future directed
curves in M x f H with ak(0) -+ p and ak( l ) -+ p as k -+ m , but ak(1/2) $ U
for all k. Thus ( M x f H , g ) is not strongly causal a t p.
Conversely, suppose that strong causality fails a t the point p = (pl, q l ) of
( M x j H,ij). Let (xl,. . . ,x,) be local coordinates on M near pl such that g
has the form diag(-1, + I , . . . ,+I) at pl, and let (x ,+~, . . . , on) be local coor-
dinates on H near ql such that f h has the form diag(f1,. . . , +1) a t ql. Then
(51,. . . , x,, x,+l,. . . , x,) are local coordinates for M x f H near p. Further-
more, Fl = x1 and Fz = xl o .ir are (locally defined) time functions for M
near pl and for M x f H near p, respectively. The failure of strong causal-
ity a t p implies the existence of a sequence yk : I O , l ] -+ M x f H of future
directed nonspacelike curves with yk(0) -+ p and yk( l ) -+ p as k -+ m, but
3.6 W A R P E D PRODUCTS 101
Fz(yk(1/2)) 2 E > 0 for all k and some parametrization of ~ k . Choose a
neighborhood W of pl in M such that W is covered by the local coordinates
(XI, . . . , xi) above and such that sup(F1 (r) : r E W ) 5 ~ / 2 . The curves
.rr 0 yk are then future directed nonspacelike curves in ,nil with .ir o ~ ~ ( 0 ) -+ pl ,
.ir 0 yk(l) -+ pl, and .ir o yk(1/2) $! W . Hence W and {T o yk) show that strong
causality fails a t pl in (M, g) required. I?
In Proposition 3.64 we prove the equivalence of stable causality for (M, g)
and ( M x j H, ij) for dim M 2 2. From this proposition and the last two
propositions, it follows that the basic causal properties of (M x f H,i j ) are
determined by those of (M, g).
Remark 3.63. If g < gl on M , then there is a smooth conformal factor
R : M -+ (0, m) such that Rgl(v, 21) < g(71,~) for all nontrivial vect.ors which
are nonspacelike with respect to g.
Proposition 3.64. Let (M, g) be a space-time and (H, h) a Riemannian
manifold. Then the Lorentzian warped product (M x f H , ij ) is stably causal
iff (M, g) is stably causal.
Proof. In this proof we will use the identification Tp(M x H ) 2 Tp, M x TbH
for a l l p = (pl,b) E M x H .
Assuming that ( M x f H, 3 ) is stably causal, there exists ij, E Lor(M x H)
such that ij < gl and ?jl is causal. If b is a fixed point of H , then we may
assume without loss of generality that ?jlI,-,(,) is nondegenerate since ?jI,-,(,)
is nondegenerate. Setting = ijll,-l(b, and using T/,-,(~) to identify 77-'(b)
with M , we obtain a metric gl E Lor(M) such that .irl,-l(b) is an isometry
of (7-'(b),:~) onto (M,gl). Notice that since ( M x H,G1) is causal, the
space-time (q-l(b),&) is causal, and hence (M,gl) is also causal. To show
g < g1 on M , we choose a nonzero vector vl E Tpl M such that g(v1, vl) 5 0.
If Ob denotes the zero vector in TbH, then g(v,v) = g(vl,vl) 5 0. where
v = (vl, Ob) E Tpl M x TbH. Since ij < gl, we obtain ij,(v, v) = gl(vl, v l ) < 0.
Hence g < gl, and (M, g) is stably causal.
Conversely. we now assume that ( M g) is stably causal. Let gl E Lor(M)
be a causal metric with g < gl. By Remark 3.63 we may also assume that
102 3 LORENTZIAN MANIFOLDS AND CAUSALITY
g1 ( v l , v l ) < g(v1, v l ) for all vectors vl # 0 which are nonspacelike with respect
to g. Since Proposition 3.61 implies that g1 = gl @ f h is a causal metric on
M x H , it suffices to show that 3 < 3,. To this end, let v = ( v l , v z ) be a
nontrivial vector o f T p ( M x H ) which is nonspacelike with respect to g. Then
since 3(v , v ) = g(v1, v l ) + f ( 4 2 1 ) ) . h(v2,v2) I 0 and f ( ~ ( v ) ) . h(v2, vz) > 0
with v2 # 0 , the nontriviality of v implies that vl # 0 and g(v l , v l ) 5 0. Thus
31(v, u) = gi(ui ,v i ) + f ( 4 ~ ) ) . h ( v 2 , ~ 2 ) < g(v1,vl) + f ( n ( v ) ) . h ( v 2 , ~ 2 ) I 0
which shows that 3 < 3, and establishes the proposition.
Geroch's Splitting Theorem (cf . Theorem 3.17) guarantees that any globally
hyperbolic space-time may be written as a topological product R x S where
S is a Cauchy hypersurface. Geroch's result suggests investigating conditions
on ( M , g ) and ( H , h ) which imply that the warped product ( M x f H , g ) is
globally hyperbolic. These conditions are given for dim M = 1 and dim M 2 2
in Theorems 3.66 and 3.68, respectively. In order to prove these results, it is
first necessary to show that a curve in a complete Riemannian manifold which
is inextendible in one direction must have infinite length.
Lemma 3.65. Let ( H , h ) be a complete Riemannian manifold, and let
y : [O, 1) -+ H be a curve of finite length in ( H , h). Then there exists a point
p E H such that y ( t ) -4 p as t -+ I - .
Proof. Let do denote the Riemannian distance function induced on H by
the Riemannian metric h. Let L = Lo(?) be the Riemannian arc length of y ,
and set K = { q E H : do(y(O),q) 5 L) . The Hopf-Rinow Theorem [cf. Hicks
(1965, pp. 163-164)] implies that K is compact. Fix a sequence (t,) in [O,1)
with t , -+ 1. Since d(y(O), y ( t ) ) 5 L ( y 1 [O,t]) 5 L for t E [0,1), we have
y[O, 1) C K. Thus by the compactness o f K , the sequence {y( t , ) ) has a limit
point p E K. I f lirnt,i- y ( t ) # p, there would then exist an E > 0 such that y
leaves the ball {m E M : d(p, m) 5 E ) infinitely often. But this would imply
that 7 has infinite length, in contradiction.
The following theorem may be obtained from the combination o f Corollary
3.56 and Lemma 3.65. The proof, which is similar to that o f Theorem 3.68,
will be omitted.
3.6 WARPED PRODUCTS 103
Theorem 3.66. Let ( H , h) be a Riemannian manifold, and let M = (a , b)
with -m 5 a < b I +co be given the negative definite metric -dt2. Then
the Lorentzian warped product ( M x f H , g ) is globally hyperbolic i f f ( H , h )
is complete.
Theorem 3.66 may be regarded as a "metric converse" to Geroch's splitting
theorem. I f f = 1 is assumed, so that the warped product ( M x f H , g ) is
simply a metric product ( M x H , g @ h) , Theorem 3.66 may be strengthened
to include geodesic completeness (c f . Definition 6.2 for the definition o f geodesic
completeness).
Theorem 3.67. Suppose that ( H , h ) is a Riemannian manifold and that
R x H is given the product Lorentzian metric -dt2 @ h. Then the following
are equivalent:
(1) ( H , h ) is geodesically complete.
(2 ) ( R x H , -dt2 @ h ) is geodesically complete
(3 ) ( R x H, -dt2 @ h ) is globally hyperbolic.
Proof. W e know that ( 1 ) i f f (3 ) from Theorem 3.66. Thus it remains to
show (1) i f f (2). But this is a consequence o f the fact that all geodesics o f
IR x H are either (up to parametrization) o f the form (At, c ( t ) ) , (Ao, c ( t ) ) , or
(At, ho), where A, A. E R are constants, ho E H , and c : J -+ H is a unit speed
geodesic in H . D
Suppose that a space-time ( M , g) o f dimension n 2 3 satisfies the timelike
convergence condition (i.e., has everywhere nonnegative nonspacelike Ricci
curvatures) and satisfies the generic condition (i.e., each inextendible non-
spacelike geodesic contains a point where the tangent vector W satisfies the
equation xF,d=l WCWdW[aRblcd[eWf l # 0 [cf. Chapter 21). Then i f ( M , g ) has
a compact Cauchy surface, the space-time ( M , g ) is geodesically incomplete.
Thus Theorem 3.66 may not be strengthened for arbitrary warped products
to include geodesic completeness. The "big bang" Robertson-Walker cosmo-
logical models (c f . Section 5.4) are examples o f globally hyperbolic warped
products which are not geodesically complete.
104 3 LORENTZIAN MANIFOLDS AND CAUSALITY
In contrast, let (W x H, -dt2 6+ h ) be a product space-time of the form
considered in Theorem 3.67. Fix any bo E H. Then y( t ) = ( t , bo) is a timelike
geodesic with R(y l ( t ) , v ) = 0 for all v E T-,(t)(W x H ) for each t E W. Thus
(R x H, -dt2 @ h ) fails to satisfy the generic condition.
If dim M = 1 and M is homeomorphic to W, we have just given necessary
and sufficient conditions for the warped product M x f H to be globally hyper-
bolic. If M = S 1 , we remarked above that (M x f H , g ) is nonchronological no
matter which Riemannian metric h is chosen for H. Thus no warped product
space-time ( S 1 x f H, 3 ) is globally hyperbolic.
We now consider the case dim M 2 2.
Theorem 3.68. Let ( M , g) be a space-time, and let (H, h) be a Riemann-
ian manifold. Then the Lorentzian warped product ( M x f H , g ) is globally
hyperbolic iff both of the following conditions are satisfied:
( 1 ) ( M , g) is globally hyperbolic.
(2) ( H , h ) is a complete Riemannian manifold.
Proof. (+) Suppose first that ( M x f H , g ) is globally hyperbolic. Fixing
b E H , we may identify ( M , g ) with the closed submanifold 77-'(b) = M x {b)
since the projection map r : 7/-1(b) -+ M is an isometry, Lemma 3.60 implies
that under this identification, the set J f (p l ) n 3-(91) in M corresponds to
77-l(b) n J+((pl, b)) n J - ( ( Q , b)) in ( M x f H ) for any pl and ql in M . Since
?-l(b) is closed and ( M x f H, 3 ) is globally hyperbolic, 77-'(b) n J f ( ( p l , b) ) n J-((91, b)) is compact in ( M x f H). Hence J+(pl) n J-(ql) is compact in
M . Because ( M x f H , 3 ) is globally hyperbolic, it is also strongly causal.
Thus ( M , g ) is strongly causal by Proposition 3.62. Hence ( M , g) is globally
hyperbolic as required.
Now we show that (M x f H,G) globally hyperbolic implies that ( H , h ) is
a complete Riemannian manifold. We will suppose that ( H , h ) is incomplete
and derive a contradiction to the global hyperbolicity of ( M x j H , g ) . For
this purpose, fix any pair of points pl and ql in M with pl << ql, and let
7 1 : [0, L] --t M be a unit speed future directed timelike curve in M from
pl to ql. Set a: = sup{f(y l ( t ) ) : t E [O, L ] ) where f : M -+ ( 0 , m ) is the
3.6 WARPED PRODUCTS
FIGURE 3.10. In the proof of Theorem 3.68, the curve c : [0, P ) -+
H is a geodesic which is not extendible to t = /3 < m. The curves
T ( t ) = ( y l ( t ) , c ( t ) ) and p(t) = ( y l ( L - t ) , c ( t ) ) are inextendible
nonspacelike curves in ( M x f H, 3 ) and hence do not have compact
closure.
given warping function. Since y1([0, L ] ) is a compact subset of M , we have
o < a < m .
Assuming that ( H , h ) is not complete, the Hopf-Rinow Theorem ensures the
existence of a geodesic c : [0,/3) -+ H with h(cl(t) , c l ( t ) ) = l / a which is not
extendible to t = /3 < m. By changing c(0) and reparametrizing c if necessary,
we may suppose that 0 < /3 < L/2. Define a future directed nonspacelike curve
7 : [0, /3) -+ M x H and a past directed nonspacelike curve 5 : [0, /3) -+ M x H
by ~ ( t ) = (y l ( t ) , c( t ) ) and ;I;(t) = (yl ( L - t ) , c( t ) ) , respectively. For each t with 0 5 t < /3, we have y l ( t ) << y l ( L - t ) in ( M , g ) since t < L - t . Hence
106 3 LORENTZIAN MANIFOLDS AND CAUSALITY
by Lemma 3.60, we have (y l ( t ) , c ( t ) ) << ( y l ( L - t ) , c ( t ) ) in M x f H . Thus
(p l , c(1)) ,< ~ ( t ) << T ( t ) < (91, c(0)) for all 0 5 t < P (c f . Figure 3.10). It
follows that ?([O, P ) ) is contained in J+ ( ( p l , c(0))) n J - ((ql , ~ ( 0 ) ) ) . Since
c = r] o 7 does not have compact closure in H , the curve 7 : [O, P ) -+ M x f H
does not have compact closure in J+ ((pl , c(0))) n J - ((ql , ~ ( 0 ) ) ) . But since
( M x f H , g ) is globally hyperbolic, the set J + ( ( ~ ~ , c ( o ) ) ) n J - ( ( q l , c ( ~ ) ) ) is
compact, in contradiction.
(e) Suppose now that ( M , g ) is globally hyperbolic. Assuming that the
warped product ( M x f H , i j ) is not globally hyperbolic, we must show that
( H , h ) is not complete. Since ( M , g ) is strongly causal, ( M x f H , i j ) is also
strongly causal by Proposition 3.62. Hence since ( M x f H , i j ) is not globally
hyperbolic, there exist distinct points (p l , b l ) and (p2, b2) in M x f H such
that J+((pl , b l ) ) n J-((pz, b2)) is noncompact. There is then a future directed
nonspacelike curve y : [O, 1) -+ J+((pl, b l ) ) n J-((p2, b2)) which is future inex-
tendible in ( M x f H . 3 ) . Let y ( t ) = (ul ( t ) ,u2( t ) ) , where u1 : [O, 1) --+ M
and u2 : [O, 1) -+ H . Then u1 : [O, 1 ) -+ M is a future directed non-
spacelike curve contained in J+(pl) n JW(p2). Since ( M , g ) is globally hy-
perbolic, J+(pl) n JW(p2) is compact. Hence i f we set a0 = i n f { f ( m ) : m E
J+(pl) n J-(pz)), then c r ~ > 0. Also since (M,g) is strongly causal, no future
directed, future inextendible, nonspacelike curve may be future imprisoned in
the compact set J+(pl) n J-(p2) (c f . Proposition 3.13). Hence there exists a
point r E J+(pl) n J-(pz) with lim,,l- u l ( t ) = r . W e may then extend u1
to a continuous curve u1 : [O, 11 -+ M by setting u l (1 ) = r. Since the curve
y = (u1,u2) was inextendible to t = 1, it follows that u z ( t ) cannot converge
to any point of H as t -+ I-. By Lemma 3.65, either ( H , h ) is incomplete or
u2 has infinite length. As u1 : [ O , l ] -+ M is a nonspacelike curve defined on a
compact interval, ul has finite length in (M,g) . Since f ( u l ( t ) ) > cro > 0 for
all t E [0, I ] and
it follows that u2 has finite length in ( H , h). Thus ( H , h ) is incomplete as
required. U
3.6 WARPED PRODUCTS 107
Cauchy surfaces may be constructed for globally hyperbolic Lorentzian
warped products as follows.
Theorem 3.69. Let ( H , h ) be a complete Riemannian manifold. Let
( M x f H , g ) be the Lorentzian warped product of ( M , g ) and ( H , h ) .
(1 ) I f M = (a , b) with -m < a < b 5 +co is given the metric -dt2, then
{ p l ) x H is a Cauchy surface o f ( M x f H , i j ) for each pl E M .
(2 ) I f ( M , g ) is globally hyperbolic with Cauchy surface S1 , then S1 x H
is a Cauchy surface of ( M xf H , g ) .
Proof. Since the proofs o f (1 ) and (2) are similar, we shall only give the proof
o f (2) . In this case, S1 x H is an achronal subset of ( M x j H , 3 ) . To show
S1 x H is a Cauchy surface, we must show that every inextendible nonspacelike
curve in M x f H meets S1 x H. Now given (pl,p2) E ( M x H ) - (S1 x H ) ,
either every future directed, future inextendible, nonspacelike curve in ( M , g)
beginning at pl meets S1 or every past directed, past inextendible, nonspacelike
curve starting at pl meets S1. Since the two cases are similar, we will suppose
the former holds and then show that every future directed, future inextendible,
nonspacelike curve y : [O, 1) --+ M x f H with y(0) = (pl ,pz) meets S1 x H.
Thus suppose that y : [O,l) -+ M x f H is a future directed, future inex-
tendible, nonspacelike curve with y(0) = (pl ,pz) which does not meet S1 x H .
Decompose y ( t ) = (u l ( t ) ,ua( t ) ) with ul : [O, 1 ) -+ M and u2 : [O, 1) + H .
Since S1 is a Cauchy surface for ( M , g) and ( M , g ) is globally hyperbolic,
the set J+(pl) O J-(S1) is compact [cf. Beem and Ehrlich (1979a, p. 163)).
As in the proof o f Theorem 3.68, the strong causality o f ( M , g) implies that
there exists a point r E J+(pl) n J-(S1) with limt,l- u l ( t ) = r. Since
J+(pl) n J - ( S l ) is compact, the warping function f : M -+ (0, co) achieves a
minimum cro > 0 on J+(pl) n J- (S1) . As in the proof o f Theorem 3.68, this
then implies that u2 : [O, 1) -+ H has finite length. Since ( H , h ) is complete,
by Lemma 3.65 there exists a point b f H with l imthl- u2(t) = b. Setting
y(1) = ( r , b), we have then extended y to a nonspacelike future directed curve
y : [O, 11 -+ M x f H , contradicting the inextendibility o f y. Hence y must
meet S1 x H as required. 0
108 3 LORENTZIAN MANIFOLDS AND CAUSALITY
We now consider the nonspacelike geodesic completeness of the class of
Lorentzian warped products of the form = (a, b) x f H with g = -dt2 @
f h. Here a space-time is said to be null (respectively, timelike) geodesically
incomplete if some future directed null (respectively, timelike) geodesic cannot
be extended to be defined for arbitrary negative and positive values of an affine
parameter (cf. Definitions 6.2 and 6.3). Since we are using the metric -dt2
on (a, b), the curve c(t) = (t, yo) with yo E H fixed is a unit speed timelike
geodesic in (z, 3 ) no matter which warping function is chosen. Consequently,
if a > -oo or b < +co, then ( M , g ) is timelike geodesically incomplete for
all possible warping functions f . Moreover, if a and b are both finite and if y
is any timelike geodesic in z = (a , b) x f H, then L(y) 5 b - a < oo. Thus
if a and b are finite, all timelike geodesics are past and future incomplete.
Nonetheless, if the warping function f is chosen suitably, (%,g) may be null
geodesically complete even if a and b are both finite. This will be clear from
the proof of Theorem 3.70 below.
If z = W x f H with 3 = -dt2 @ fh , then any timelike geodesic of the
form c(t) = ( t , yo) is past and future timelike complete. However, warped
product space-times = W x f H may be constructed for which all nonspace-
like geodesics except for those of the form t + (t, yo) are future incomplete.
One such example may be given as follows. Busemann and Beem (1966) stud-
ied the space-time = = {(x, y) € W2 : y > 0) with the Lorentzian metric
ds2 = y-2(dx2 - dy2). Busemann and Beem (1966, p. 245) noted that all
timelike geodesics except for those of the form t + (t, yo) are future incom-
plete. Setting t = In y, this space-time is transformed into the Lorentzian
warped product R x f R with 3 = -dt2 @ fdt2, where f (t) = e-2t. Since the
map F : ( S , ds2) -+ (W x f W,g) given by F(x, y) = (x, ln y) is a global isome-
try, all timelike geodesics of (W x f W, B) except for those of the form t -+ (t, yo)
are future incomplete. It will also follow from Theorem 3.70 below that all
null geodesics are future incomplete. Similarly, if (Wn, h) denotes Rn with the
usual Euclidean metric h = dxI2 + dxZ2 + . . - + dxn2, then (W x f Wn, ?j) with
g = -dt2 @ f h and f (t) = e-2t is a space-time with all nonspacelike geodesics
future incomplete except for those of the form t -+ (t, yo).
3.6 WARPED PRODUCTS 109
In order to study geodesic completeness, it is necessary to determine the
Levi-Civita connection for a Lorentzian warped product metric. For this pur-
pose, we will consider the general warped product ( M x f H , g e fh) where
f : M -+ (0, co), (H, h) is Riemannian, and (M, g) is equipped with a metric of
signature (-, +, . . . , +). Let V1 denote the Levi-Civita connection for (MI g)
and V2 denote the Levi-Civita connection for (H, h). Given vector fields
XI, Yl on M and X2, Y2 on H , we may lift them to M x H and obtain the vec-
tor fields X = (XI, 0) + (0, X2) = (XI, X2) and Y = (Yl, 0) + (0, Yz) = (Yl , Y2)
on M x H. Recall that the connection 7 for ( M x f H , g @ f h ) is related to
the metric 3 = g @ f h by the Koszul formula
[cf. Cheeger and Ebin (1975, p. 2)]. Using this formula and setting 4 = In f ,
we obtain the following formula for 7 for X and Y as above:
Here grad4 denotes the gradient of the function 4 on (M,g), and we
are identifying the vector VklYl 1, E T,M with the vector (V&,Yl, 0,) E
T(,,,)(M x HI, etc. We are now ready to obtain the following criterion for null geodesic incom-
pleteness of Lorentzian warped products z = (a, b) x f H [cf. Beem, Ehrlich,
and Powell (1982)l. Throughout the rest of this section, let wo denote an
interior point of (a, b).
Theorem 3.70. Let z = (a, b) x f H be a Lorentzian warped product
with Lorentzian metric = -dt2 @ f h where -co 5 a < b 5 +m, (H, h) is
an arbitrary Riemannian manifold, and f : (a, b) -+ (0, m). Set S(t) = m. Then if lim,-.+ JtwO S(s)ds respectively, limt+a- ~ ( s ) d s ] is finite, every [ future directed null geodesic in ( X , ? j ) is past [respectively, future] incomplete.
Proof. Let yo be an arbitrary future directed null geodesic in (?i?,?j). We
may reparametrize yo to be of the form y(t) = (t,c(t)), where y is a smooth
110 3 LORENTZIAN MANIFOLDS AND CAUSALITY
null pregeodesic. Accordingly, there exists a smooth function g(t) such that
[cf. Hawking and Ellis (1973, p. 33)]. However, since yl(t) = d l%[ , +cl(t) and
?j(yl, yl) = - 1 + g(cl, c') = 0, we obtain using formula (3.17) that
Equating terms with a d l & component, we obtain the formula
Thus ~ 7 t y 1 1 t = (1/2)[ln f (t)]' y'(t) = [lnS(t)ll yl(t). If we define p : (a, b) -+ R
by
~ ( t ) = Jr s ( ~ ) d ~ , wo
then pl(t) = S( t ) > 0 so that p-' exists. Moreover, from the classical theory
of projective transformations we know that the curve yl(t) = y o p-'(t) =
(p-'(t), c o p-l(t)) is a null geodesic [cf. Spivak (1970, pp. 6-35 ff.)]. Let
A = lim p(t) and B = lim p(t). t-a+ t-b-
Since p is monotone increasing, we have that p : (a, b) -+ (A, B ) is a bijection.
Hence p-' : (A, B) -+ (a, b) and thus yl = y o p-l : (A, B ) --+ ?@. Therefore if
A is finite, yl is past incomplete, and if B is finite, yl is future incomplete as
required.
It is immediate from Theorem 3.70 that if a and b are finite and the warp
ing function f : (a, b) -+ (0, m ) is bounded, then (?@,g) is past and future
null geodesically incomplete. Thus, assuming that a and b are finite, one-
parameter families ( a , g ( s ) ) = (?@, -dt2 @ f(s)h) of past and future null
geodesically incomplete space-times may easily be constructed. Choosing the
one-parameter family of functions f (s) : (a, b) 4 (0, m ) suitably, the curve
3.6 WARPED PRODUCTS 111
s + g(s) = -dt2 @ f (s)h of metrics will not be a continuous curve in ~ o r ( M )
in the fine C' topologies. Thus the space-times (2, g(0)) and ( M, g(s)) may
be far apart in ~ o r ( M ) for s # 0.
Notice that if the Riemannian manifold (H, h) is geodesically incomplete,
then = (a, b) x H may be null geodesically incomplete even if both integrals
in Theorem 3.70 diverge. On the other hand, if the completeness of (H, h) is
assumed, the following necessary and sufficient condition for the null geodesic
incompleteness of '7i? = (a, b) x H may be obtained from the proof of Theorem
3.70.
Remark 3.71. Let ?@ = (a, b) x f H be a Lorentzian warped product with
Lorentzian metric 3 = -dt2 @ fh , where (H, h) is a complete Riemannian
manifold and - m 5 a < b 5 +m. Let S(t) = as above. Then
( M , 3 ) is past (respectively, future) null geodesically incomplete if and only if
lim LwO S(S) ds is finite (respectively, t-a+
In Powell (1982), a more comprehensive study is made of nonspacelike ge-
odesic completeness of Lorentxian warped products, beginning with the ob-
servation that a geodesic in M x f H projects to a pregeodesic in (H, h) and - continuing with the observation that if 7 and ,B are unit speed geodesics in
(H, h) and (y, 5) is a pregeodesic of M x f H, then (y, g) is also a pregeodesic
of M x f H . Then Powell shows that if the Lorentzian warped product M x f H
is timelike (respectively, null or spacelike) geodesically complete, then (H, h)
is Riemannian complete and further (M,g) is timelike (respectively, null or
spacelike) complete.
A second aspect of Powell (1982) is the study of timelike geodesic com-
pleteness for warped products of the form 2 = (a, b) x f H with metrics
g = -dt2@ f h , corresponding to Theorem 3.70 and Remark 3.71 above for null
completeness. As remarked above, the completeness of the timelike geodesics
of the form T(t) = (t, yo) for some yo E H, which Powell terms "stationary,"
is entirely dependent on whether a = -co or a is finite, and/or b = +co or b
is finite. Also, if a and b are both finite, then all timelike geodesics are hoth
past and future incomplete independent of choice of warping function. How-
112 3 LORENTZIAN MANIFOLDS AND CAUSALITY
ever, for a suitable warping function, even if a = -oo and b = +m, it can
be arranged for all non-stationary timelike geodesics to be incomplete. More
precisely, Powell shows that if (H, h) is complete and wo E (a, b), then all fu-
ture directed non-stationary timelike geodesics are future (respectively, past)
complete if and only if
4 (t) dt = +m, respectively, J w O (a) dt = +m. L (mi)
From this result and our above results on null completeness, relationships may
be derived between null and timelike completeness for this class of warped
products. (In general, these types of geodesic completeness are logically inde-
pendent (cf. Theorem 6.4). In particular, null completeness of R x f H does
not imply timelike completeness.) However, if the warping function f for - M = R x ~ f H is bounded from above and (H, h) is Riemannian complete, then
the timelike and null completeness are equivalent.
In singularity theory in general relativity, conditions on the curvature ten-
sor of (%,-ji) which are discussed in Chapters 2 and 12, called the generic
condition and the timelike convergence condition, are considered. These two
conditions guarantee that if a nonspacelike geodesic y may be extended to
be defined for all positive and negative values of an affine parameter and
d i m z > 3, then y contains a pair of conjugate points. Hence these cur-
vature conditions may be combined with geometric or physical assumptions,
such as ( 2 , g ) is causally disconnected or ( M , g ) contains a closed trapped
set, to show that ( 2 , g ) is nonspacelike geodesically incomplete (cf. Section
12.4). Since ( M , g ) satisfies the generic condition and strong energy condi-
tion if all nonspacelike Ricci curvatures are positive, it is thus of interest to
consider conditions on the warping function f of a Lorentzian warped product
which guarantee that ( z , g ) has everywhere positive nonspacelike Ricci cur-
vatures. The assumption d i m m > 3 made in singularity theory is necessary
for null conjugate points to exist since no null geodesic in any two-dimensional
Lorentzian manifold contains a pair of conjugate points.
We now give the formulas for the curvature tensor R and Ricci curvature
tensor Ric for the warped product space-time ( M x f H , g ) where 3 = g @ fh.
3.6 WARPED PRODUCTS 113
As above, let V' [respectively, V2] denote the covariant derivative of (M,g)
[respectively, (H, h)]. Also let r$ = In f and recall that grad4 denotes the
gradient of 4 on (M,g). As before, we will decompose tangent vectors x in
TF(M x H) as x = (x1,x2). Let R' [respectively, R ~ ] denote the curvature
tensor of (M,g) [respectively, (H, h)]. Given tangent vectors X I , yl E TpM,
define the Hessian tensors H+ and h+ by
and
We will also write 1 1 grad + / I 2 = g(grad 4, grad 4). Using the sign convention
for the curvature tensor and substituting from formula (3.17), one obtains the
formula
where x, y, 2 E Tlp,,)(M x H). . Suppose now that dim M = m and dim H = n. To calculate the Ricci
curvature at jj = @,q) E M x H, let {el,e2,. . . , em) be a basis for TpM
with g(el,el) = -1, g(ej,ej) = 1 for 2 < j < m, and g(ei,ej) = 0 if i # j .
Also, let {em+l,. . . ,en+,) be a 7j-orthonormal basis for T,H. Then for any
x, y E TF(M x H), we have
114 3 LORENTZIAN MANIFOLDS AND CAUSALITY
The d'Alembertian 0 4 of 4 may also be calculated as
Using (3.21), it then follows that
dim H - g(x2, 1/21 [ io4(p ) + 7
dim H -- dim H 2 h + ( x 1 , ~ 1 ) - - - p 1 ( 4 ) ~ 1 ( 4 )
where x = ( x l , y l ) , y = ( y l , y2) E T(,,,) ( M x H ) , and ~ i c l and Ric2 denote
the Ricci curvature tensors of ( M , g ) and ( H , h ) , respectively.
W e now restrict to the case hl = (a , b) x f H with warped product metric
i j = -dt2 @ fh. In this case, O4(t ) = -4I1(t) and 11 gad$(t)l12 = -[4'(t)I2.
Thus we obtain from (3.22) for 3 = (0, v ) E T(t,,)((a, b) x H ) that
(3.23) Ric(i?, i?) = R ~ c ~ ( v , v ) + g(v, v)
I f x = bldtl, + v E T(,,,)((a, b) x H ) with v E T,H, we obtain
1 dim H (3.24) Ric(x, x ) = ~ i c ' ( v , v ) + g(v, v ) {5411(t) + - 4 14'(t)12)
dim H - --[4'(t)12) dim H . 4
Both bracketed terms in formulas (3.23) and (3.24) will be positive provided
that
(3.25) -[4'(t)I2 dim H < 2d1'(t) < -[4'(t)12
for all t E (a , b). Thus i f Ric2(v,v) 2 0 for all v E T H and condition (3.25)
holds, the space-time (v,ij) will have everywhere positive Ricci curvatures.
A globally hyperbolic family o f such space-times is provided by warped prod-
ucts = = (0, co) x f H , where ( H , h ) is a complete Riemannian manifold of
3.6 WARPED PRODUCTS 115
nonnegative Ricci curvature and 3 = -dt2 @ f h with f ( t ) = t' for a fixed
constant r 6 R satisfying (21 dim H ) < r < 2. I f ( H , h ) is taken to be W 3 with
the usual Euclidean metric and r = 413, we recover the Einstein-de Sitter
universe of cosniology theory [cf. Hawking and Ellis (1973, p. 138), Sachs and
W u (1977a, Proposition 6.2.7 ff.)].
W e may also obtain the following condition on 4 = In f for positive non-
spacelike Ricci curvature i f the Ricci tensor of ( H , h ) is bounded from below.
Proposition 3.72. Let 3 = (a , b) x f H with n = dim H 2 2, g = -dt2
fh, and 4 = In f . Suppose that there exists some constant X E R such that
Ric2(v, v ) > Xh(v,v) for all v E T H . Then i f
(3.26) 24I1(t) < min {- (q!~ ' ( t ) )~ , 4(n - l)-'Xe-@'('))
for all t E (a , b), the Lorentzian warped product (li?, 3 ) has everywhere posi-
tive nonspacelike Ricci curvature.
Proof. It suffices to show that Ric(x, x ) > 0 for all nonspacelike tangent
vectors x o f the form x = dldt + v E T ( M x H ) , v E T H . Since 3 (x , x) 5 0
and i j(a/dt , d l d t ) = - 1, we have p = g(v, v ) 5 1. Hence 0 5 ,B 5 1. Then
h(v, v ) = Be-$ and we obtain from (3.24) that
Thus Ric(x, x ) > 0 provided 4" < G(P) for all /3 E [O,l], where
Calculating G1'(P), one finds that G1(,B) does not change sign in [0, I]. Thus
G(,B) obtains its minimum on [ O , l ] for p = 0 or /3 = 1. Hence Ric(x, x) > 0
provided that 4" < min{G(O), G(P)) , which yields inequality (3.26).
W e now consider the scalar curvature o f warped product manifolds o f the
form = R x f H , 3 = -dt2 @ f h . W e will let n = dim H below. Given
( t ,p ) E M, choose el E T,H for 1 < j < n such that i f Zl = (0, e,) E T(t,p)%, - -
then { d l & = (d ld t , O,),El, . . . ,En) forms a 3-orthonormal basis for T( t ,p)M.
116 3 LORENTZIAN MANIFOLDS AND CAUSALITY
Hence {mel,. . . , m e , } forms an h-orthonormal basis for TpH. Thus
if r : 2 -+ W and TH : H -+ W denote the scalar curvature functions of ( 2 , 3 )
and (H, h) respectively, we have
and
Now formulas (3.23) and (3.24) above simplify to
and
for 1 5 j 5 n. Consequently, we obtain the formula
1 1 r( t , p) = - r ~ (p) + n+I1(t) + (n2 + n) [+'(t)12. f (4
Recalling that $(t) = In f (t), this may be rewritten as
where dim H = n as above. In particular, in the case that n = 3 as in general
relativity, we obtain the simpler formula
Example 3.73. With the formulas of this section in hand, we are now
ready to give an example of a 1-parameter family gA of nonisometric Einstein
metrics for Wn+' such that for X = 0, (Wn+l,go) is Minkowski space-time of
dimension n + 1. Let (Wn, h) be Euclidean n-space with the usual Euclidean
metric h = dx12 + dx22 + - . . +dzn2, and put MA = lipn+' = W x f Wn with the
3.7 SEMI-RIEMANNIAN LOCAL WARPED PRODUCT SPLITTINGS 117
Lorentzian metric ijx = -dt2 @ eXth, i.e., f (t) = ext. By Theorem 3.70, for all
X > 0 the space-time (Rn+',?jx) is future null geodesically complete but past
null geodesically incomplete, and for all X < 0, the space-time (Rn+1,3x) is
past null geodesically complete but future null geodesically incomplete. Using
formulas (3.28), (3.29), and (3.30), we obtain
and
Thus if X # 0, (zA,?jx) is an Einstein space-time with constant positive scalar
curvature.
Example 3.74. Let li?x = (0,co) x f x3, where gA = -dt2 @ f h with
f (t) = At, X > 0, and h the usual Euclidean metric on LR3. It is then immediate
from formula (3.31) that r(gX) = 0 for all X > 0. Since d( t ) = ln(Xt), it may
be checked using formulas (3.28) and (3.29) that (z~, 3,) is neither Ricci flat
nor Einstein for any X > 0. Also we have for any X > 0 that
for all t > 0. It follows that the space-times ( z A , ?jA) are "inextendible across"
(0) x R3 (cf. Section 6.5). Also, ( R x , g x ) is future null geodesically complete
by Theorem 3.70.
3.7 Semi-Riemannian Local Warped Produc t Spli t t ings
In each of the exact solutions to Einstein's equations which are presented
as warped product manifolds, the warped product decomposition emerges as a
natural mathematical expression of assumed physical symmetries. Moreover,
formulas for warped product curvatures (cf. Proposition 3.76) indicate that any
semi-Riemannian manifold (M, g) must possess certain measures of symmetry
and flatness in order to be (locally or globally) isometric to a warped product
118 3 LORENTZIAN MANIFOLDS A N D CAUSALITY
B x F. In this section we identify geometric conditions on a semi-Riemannian
manifold (M,g) which are necessary and sufficient to ensure that (M,g) is
locally isometric to a warped product B xf F. We shall call such a local
isometry a 'Llocal warped product splitting."
There are a number of different "splitting" or decomposition theorems
throughout differential geometry. For example, the splitting theorem of Geroch
(1970a), Theorem 3.17 in this chapter, demonstrates that a globally hyperbolic
space-time may be written as a particular type of topological product but not
necessarily as a metric product. By contrast, a well-known result of de Rham
[cf. Kobayashi-Nomizu (1963, p. 187)] asserts that a complete simply con-
nected Riemannian manifold which has a reducible holonomy representation
is isometric to a Riemannian product. Along very different lines, Chapter 14
provides the following Lorentzian analogue of the Cheeger-Gromoll Splitting
Theorem: if (M, g) is a space-time of dimension n 2 3 which (1) is globally
hyperbolic or timelike geodesically complete, (2) satisfies the timelike con-
vergence condition, and (3) contains a complete timelike line, then (M, g) is
isometric to a product (R x V, -dt2 $ h ) , where (V, h ) is a complete Riemann-
ian manifold. Since a product manifold is trivially a warped product, either
of these last two results clearly provides sufficient conditions to ensure that a
manifold is globally isometric to a warped product. However, the examples
(S1 x H, 3) of non-globally hyperbolic warped product space-times discussed
in Section 3.6 indicate that causal assumptions such as global hyperbolicity
are not necessary for the existence of a global warped product splitting.
The global splitting question typically involves rather delicate topological
considerations; our focus in this section will be on the simpler local warped
product splitting question: given a semi-Riemannian manifold (M,g) and a
point p E M, what conditions are necessary and suficient for the existence of
a n open neighborhood U of p such that the submanzfold (U,g lu) is isometric
to a warped product B x F ? The following assumption will be needed.
Convention 3.75. It will be assumed throughout the remainder of this
section that the neighborhood U mentioned above is a connected, simply con-
nected, open set.
3.7 SEMI-RIEMANNIAN LOCAL WARPED P R O D U C T SPLITTINGS 119
The geometry of a warped product B x f F is expressed through the ge-
ometries of the base (B,gB) and fiber ( F , g F ) and various derivatives and
integrals of the warping function f E 5(B). We will consider warped products
M = B xf F where both (B, gs) and (F, gF) may be semi-Riemannian man-
ifolds (thus generalizing the Lorentzzan warped products of Definition 3.51).
The symbols T : M -+ B and a : M -+ F denote the standard projections.
As in portions of Section 3.6, we will find it convenient to consider the square
root S of the warping function f . Throughout this sectzon, we wzll adhere to
the conventzon that S ( b ) = for b E B where M = B x f F denotes a
warped product wzth metric tensor
The function S will be called the root warpzng functzon.
As defined in Section 3.6, the lzft of f E 5(B) to a function j E S(M) is
defined by the formula 7 = f o T. The lifted function will simply be denoted
by f as well, when no ambiguity results. A vector field V E X(B) is lifted to
M by defining E X(M) in such a manner that at each (p, q) E M, T/(p, q)
is the unique vector in the tangent space T(,,,]M such that both d ~ ( ? ) = V
and do(?) = 0. Similar definitions apply for lifts from F to M. We shall
also denote lifted vector fields without the tildes, and we write V E C(F) to
denote a vector field on M lifted from F. More generally, vectors tangent to
leaves B x q are called honzontal while those tangent to fibers p x F are called
vertzcal. Lifts of covariant tensors on B and F are now defined in the obvious
way through the use of the pullbacks T* and a*.
Let D denote the Levi-Civita connection on M, and use V to denote the
Levi-Civita connections on both B and F. The curvature tensors on B x f F =
M may be characterized through their actions on lifted horizontal and vertical
vector fields. In the following proposition, the symbols BR and F~ denote the
lifts to M of the Riemannian curvature tensors on B and F , respectively. The
symbol H S will be used to denote 5, the lift to M of the Hessian of S. Note - that in general H S ( X , Y) = H'(x, Y) only for horizontal vector fields X , Y.
For notational simplicity, the bracket notation ( , ) will be occasionally used
120 3 LORENTZIAN MANIFOLDS AND CAUSALITY
to denote the metric g on M; the metrics on B and F will always be denoted
by g~ and g ~ .
Some basic curvature formulas for warped product manifolds are now given
in Proposition 3.76 for ease of reference. The standard reference for this ma-
terial is O'Neill (1983).
Proposition 3.76. Let the semi-Riemannian warped product M = B x f F
have Riemannian curvature tensor R, Ricci curvature Ric, and root warping
function S = a. Assume X, Y, Z E C ( B ) and U, V, W E C(F).
(1) If h f 5(B), then the gradient of the lift h o 7r of h to M = B x f F is - the lift to M of the gradient of h on B, i.e., grad (71) = grad h.
IV
(2) DxY f L(B) is the lift of VxY on B, i.e., D ~ F = (VxY).
(3) DxV = D v X = (9) V.
(4) Ric(X, Y) = l?.ic(X, Y) - ($) H'(x, Y) where d = dim F.
(5) Ric(V, X ) = 0.
(6) Ric(V, W) = Ric(V, W) - (V, W)Sd
where Sn = 9 + (d - 1)-, d = dimF, and AS is the
Laplacian of the root warping function S on B.
Consider first the local warped product splitting question in dimension two.
Assume a semi-Riemannian surface (M,g) is a warped product so that in the
appropriate local coordinates (yl, y2) adapted to B and F , the metric is given
by
(3.34) g = cldyl 8 dyl + c2[s(y1)I2dy2 8 dy2
where ci = r t l , i = 1,2. It is immediate that d/dy2 is a local Killing field (recall
the elementary fact that a coordinate vector field d/dxk is Killing if and only
if & = 0 for all i, j). Each (d/dyl)-integral curve is a geodesic of M (more
generally, each leaf B x q of a warped product is a totally geodesic submanifold).
Thus d/dy2 restricted to y is a Jacobi field for each such coordinate geodesic y,
and of course, {dl&', d/dy2) = 0 along y. Further, direct calculation shows
that d/dyl is ir-rotational-that is, curl(d/dyl) = 0, where the curl of a vector
field is the skew-symmetric (0,2) tensor defined through the formula
(3.35) [curlX](Y, 2 ) = (DUX, 2 ) - (Y, D z X )
3.7 SEMI-RIEMANNIAN LOCAL WARPED PRODUCT SPLITTINGS 121
for all vector fields X , Y, and 2. The unit vector field i (d/dy2) does not have
vanishing curl, in general. These observations lead to the following result.
Lemma 3.77. Given a two-dimensional semi-Riemannian manifold (M, g)
and a point p E M, p has a neighborhood U such that (U, g lu) is isometric to
a warped product if and only if there exists a non-vanishing Killing field on
an open neighborhood of p.
Proof. The local existence of a nonvanishing Killing field about any point
of a two-dimensional semi-Riemannian warped product was noted above.
Conversely, suppose there exists a neighborhood U about p E M having a
nonvanishing Killing field V. If the Killing field is null, then it is well known
that (U, g lu) is isometric to (a portion of) Minkowski two-space RT and hence
is (trivially) a warped product.
If the Killing field V is not null, then we may complete the classical con-
struction of local geodesic (or Fermi) coordinates (xl, x2) [cf. do Carmo (1976)]
such that V = d/dx2 on U and x1 measures g-arc length along the geodesics
orthogonal to the integral curves of V. In these coordinates the metric assumes
the form
ds2 = €1 dzl @ dxl + €2 F'(xl, x2)dx2 8 dx2
where ci = &I, i = 1,2. Since V = d/dx2 is Killing, we must have = 0,
yielding
ds2 = €1 dxl 8 dxl + €2 F'(x1)dx2 8 dx2
and leading to the desired local warped product representation. 0
In generalizing the preceding result to higher dimensions, the existence of a
Killing field V must be supplemented by an integrability condition which allows
the construction of leaves B x q orthogonal to V. This integrability condition
holds trivially in dimension two but need not hold in higher dimensions where,
in general, a Killing field need not be irrotational. It is also necessary to
include the additional assumption that the Killing field be nonnull.
Recall that a space-time (M, g) is called static if there exists on M a nowhere
zero timelike Killing field X such that the distribution of (n- 1)-planes orthog-
onal to X is integrable. The following result parallels the formal construction
124 3 LORENTZIAN MANIFOLDS AND CAUSALITY
and a : (x1,x2,. . . ,xn) --t (0,0,. . . ,O,xn). For arbitrary X,Y f X(U) with
coordinate basis expansions X = Xi(d/dxi), Y = Yi(d/dxi), we have
The "dimensional dual" of the preceding result asks for conditions necessary
and sufficient to ensure that (M, g) is locally isometric to a warped product
B x f F with dim B = 1. One answer to this question involves a construction
which bears strong similarities to the formal development of the Robertson-
Walker cosmological models [cf. O'Neill (1983)l. Recall that a signal feature
of Robertson-Walker space-time is the presence of a proper time synchroniz-
able geodesic observer field U . Since the observer field U is irrotational, the
infinitesimal rest spaces of U may be integrated to provide local rest spaces.
Through a construction quite similar to that of the preceding lemma, it is pos-
sible to verify the following [cf. Easley (1991)]: given an n-dimensional (n 2 3)
semi-Riemannian manifold and point p E M I the point p has a neighborhood
U M such that (U,glu) is isometric to a warped product B x j F with
dim B = 1 and dim F = (n - 1) if and only if (1) there exists an irrotational
unit vector field U (either spacelike or timelike) on a neighborhood of p such
that (2) the flow J[r induced by U acts as a positive homothety on the local
(n - 1)-dimensional submanifolds which are everywhere orthogonal to U near
p. Condition (2) is of course equivalent to a number of different geometric
conditions expressible in terms of curvature and the connection.
The local splitting problem in the context of Ricci flat (M,g) has an ex-
tremely restricted class of solutions. We consider first the only case in dimen-
sion four which will not be subsumed under more general results, namely, the
case when M is locally isometric to a warped product with base and fiber each
3.7 SEMI-RIEMANNIAN LOCAL WARPED PRODUCT SPLITTINGS 125
of dimension two. The following computational lemma is a necessary prelim-
inary step. In the following result, the symbols "K and B K will be used to
denote the sectional curvatures of F and B, respectively. Since these objects
are bnctions on the surfaces F and B, they may be unambiguously lifted to
M as well.
Proposition 3.79. Let M = B x f F be a 4-dimensional semi-Riemannian
warped product with dim B = dim F = 2 and root warping function S = fl. For M to be Ricci fiat, it is necessary and sufficient that
(1) F have constant sectional curvature F K ,
(2) = s A s + gB(grad S, grad S) = FK on B, where F~ is the
constant value from (1) and A denotes the Laplacian on B, and
(3) Dx (grad S ) = (p ) X for all X E L(B) .
Further, if M is Ricci flat, then
A s - B K o n B , (4) -3- -
and the sectional curvatures and root warping function are related as follows:
(5) K . S3 = 2 C,w on B, where CM is a const ant, and
(6) FK = B K . S2 + g~ (grad S, grad S) on B.
Proof. Using Proposition 3.76, we see that M is Ricci flat if and only if
(3.36) Ric(X, Y) - ~H'(X, Y) = 0 and
(3.37) Ric(V, W) - (V, W)S$ = 0
for all X , Y E C ( B ) and V, W E C(F), where SQ = * and A
denotes the Laplacian on B.
On the semi-Ftiemannian surface B we have
with the analogous formula holding on the surface F (here the symbol Ric
denotes the Ricci tensor on B and not its lift to M). Projecting equations
(3.36) and (3.37) onto B and F respectively, we see that M is Ricci flat if and
126 3 LORENTZIAN MANIFOLDS AND CAUSALITY
only if the following two conditions hold:
(3.38) B~ .gB(X,Y) = $H'(x,Y), and
(3.39) F K . g ~ ( v W) = s ~ ~ F ( v , W)S'
for all X , Y E X(B) and V, W E X(F). Equation (3.39) must hold as we
project from each fiber p x F into F. Since S is constant on fibers we see that
equation (3.39) will hold if and only if
F~ = constant, and
f 2 ~ g = S A S + g ~ ( g r a d S , g r a d S ) = F ~ OnB.
Rewriting the Hessian as HS(x , Y) = gB(Vx(grad S), Y), equation (3.38)
is equivalent to
which in turn holds if and only if
Vx(grad S ) = ( B ~ . $) X for all X E X(B).
That conditions (I), (2) and (3) are necessary and sufficient for M to be Ricci
flat now follows. Proposition 3.76-(2) is used to obtain the lifted form of
condition (3).
From condition (3) it follows that for a local orthonormal frame field Eo, El
on B with ~i = gp,(Ei, Ei),
Thus the Laplacian of S is given by
By combining (3.40) with condition (21, we see that
(3.41) F~ = K s 2 + g~ (grad S, grad S )
3.7 SEMI-RIEMANNIAN LOCAL WARPED PRODUCT SPLITTINGS 127
where F~ denotes the constant value of the sectional curvature of F. Differ-
entiating both sides of equation (3.41) produces
0 = s 2 ( x B K ) + B ~ 2 ~ ~ ~ + 2gB(vx(grad S) , grad S)
= S 2 ( x B K ) + B ~ 2 ~ ~ ~ + S . B ~ ( ~ ~ )
= s 2 ( x B ~ ) + 3 . B ~ ~ ( ~ ~ ) .
Thus, 3 . *K(xS)
X(BK) = - ( ) for all x t X(B)
yielding
and this holds for all lifts X E C(B). It follows that S3 . B K is constant on B
(and hence its lift is constant on M).
It is now possible to characterize all (2 x 2) Ricci flat warped products
M = B x j F with base B of constant curvature.
Corollary 3.80. If M = B x f F is a four-dimensional Ricci flat warped
product with base B a surface of constant curvature, then M is simply a
product manifold B x G where B and G are flat two-manifolds. Thus the
metric on M is semi-Euclidean.
Proof. Assume B has constant curvature. Proposition 3.79-(5) shows that
the root warping function S, and hence also the warping function f , must be
constant on B, and thus M may be v~ewed as a product manifold. Equations
(2) and (4) of Proposition 3.79 now imply that F and B are flat. O
The following result deals with (2 x 2) warped products B x f F where the
curvature B~ is not constant.
6 Proposition 3.81. Let M = B x j F be a Ricc~ flat semi-Riemannian 1 B warped product with dim B = dim F = 2, and assume that the sectional cur-
$ vatureBK of the base B isnot constant. IfgradS isnonnuil and (grad S)I, # 0 F
128 3 LORENTZIAN MANIFOLDS AND CAUSALITY
a t a given point p E B, then there exist local coordinates (t, r ) on a neighbor-
hood U C B of p such that the following conditions hold on U.
(1) The metric on B has coordinate expression ds2 = E(r)dt2 + G(r)dr2,
where G(r) = ( F K - %)-I , E(T) = iG- ' , CM is a constant, and
F K is the constant value of the sectional curvature on F.
(2) The root warping function has the form S(t, T) = T.
(3) The sectional curvature on B is given by B ~ ( ~ , t) = 9. The sign of E in condition (1) depends upon the signature of B.
Proof. Assume grad S is nonnull and (grad S)I, # 0. By continuity, we may
find a neighborhood U of p on which the gradient of S is non-vanishing. Let
co = S-'(k) n U, k E W+, denote a single level curve of S in U. Consider
geodesics intersecting co orthogonally; these geodesics are also integral curves
of grad S, and the orthogonal trajectories of these geodesics are level sets of
S. Applying the classical geodesic coordinate construction, we introduce local
coordinates (u, u) such that the geodesics are the u = constant curves and the
orthogonal trajectories are the u = constant curves. In these coordinates the
metric assumes the form
Rescale coordinates, introducing r = T(U) and t = t(v) such that
r(u) = S(u) and Da,at(a/at) = 0.
Thus, r merely traces the values of the warping function S, and t is an affine
coordinate along the geodesics. It should be noted that this will imply T > 0.
In (t, r ) coordinates we have metric tensor components
with the root warping function given simply by
3.7 SEMI-RIEMANNIAN LOCAL WARPED P R O D U C T SPLITTINGS 129
Furthermore, since
the gradient of S is given in (T, t ) coordinates by
Equation (5) of Proposition 3.79 now shows that the curvature on the surface
B is a function solely of r and is given by
Now gB(grad S, grad S ) = &, so formula (6) of Proposition 3.79 yields
We have determined the metric coefficient G:
It remains to determine the form of the metric coefficient E ( t , r ) . We first
show that E is a function only of the variable r and then derive the function
E(r).
Note that
so that in particular = 0. However, since (t, r ) are orthogonal coordinates,
we have 1
E . G . ~ ; , = l ~ t . ~ .
It follows that Et = 0 and E is a function solely of r .
130 3 LORENTZIAN MANIFOLDS AND CAUSALITY
Now using Proposition 3.79 with grad S = &&, we obtain
S d Dalat (& g) = ( B K . Z) z, or equivalently
1 CM a giving Dalat ( a l d r ) = - - 7-2 at'
G . C M d a t 1 = 7 z.
However, in the orthogonal ( t , T ) coordinate system, we have
It follows that
Er - ~ C M = G(T)- and thus
E T-2 '
With the substitution y = F~ - %, this becomes
In I El = I J 3, giving ?f
ln]El = f In lyl + c, or finally
.=fa. ( F K _ + )
where a = eC is positive. In orthogonal coordinates with e = and
g = m, the curvature is given by the classical formula
Since E and G are functions of r only, this reduces to the following simple
formula. We are ignoring the case-by-case analysis of the signs involving the
3.7 SEMI-RIEMANNIAN LOCAL WARPED PRODUCT SPLITTINGS 131
absolute values since the end result will show such considerations to be unnec-
essary.
Since B K = 9, we see that a = 1, and the sign in this final term is deter-
mined by the signature of B and the sign of G ( r ) . We have therefore shown
that
Observe that dt is a local nonvanishing Killing field on U in the above lemma,
which is valid for spaces of arbitrary signature. Let us now fix the signature of
all spaces under consideration to the Lorentzian signature (-, +, +, +). Corol-
lary 3.80 and Proposition 3.81 now have the following import: a Ricci flat
(2 x 2) warped product M is completely specified (locally) by the choice of
base ( B , g B ) and the constants F~ and CM in Proposition 3.79. Since we are
working locally, we may assume the fiber F of constant sectional curvature F K
is a subset of
(1) the sphere S 2 ( p ) if " K = $, (2) Euclidean space Ift2 if F~ = 0, or
( 3 ) hyperbolic space H 2 ( p ) if F K = - 7. Theorem 3.82. Assume ( M , g) is a Ricci Aat four-manifold of Lorentzian
signature (-, +, +, +), and let p E M be any point of M. The point p has a
neighborhoodU such that (U, g l u ) is isometric to a (2 x 2) warped product with
both parameters (CM, F ~ ) positive and nonconstant sectional curvature B K r k on B if and only if M is locally isometric to an open subset of Schwarzschild
CM "space-time with mass Mo = -- T ( F K ) ~ '
i Proof. Assume M admits such a local warped product structure. Since we
b a are working locally, we may assume the fiber F is the 2-sphere of radius 1 r JF7;;'
132 3 LORENTZIAN MANIFOLDS AND CAUSALITY
Let du2 denote the canonical metric on the unit 2-sphere. Proposition 3.81
implies the metric may be expressed on U in the form
with E(r) = ( F ~ - %). Upon making the change of variables u = --$-- and v = tm, the metric
F K
becomes
thus demonstrating the first claim. The converse is now clear. 13
This result should be contrasted with Birkhoff's Theorem [cf. Hawking and
Ellis (1973, p. 372)j on spherically symmetric solutions to Einstein's vacuum
field equations, which also offers an alternate proof of Theorem 3.82.
Theorem (Birkhoff). Any C2 solution of Einstein's empty space equa-
tions which is spherically symmetric in an open set V is locally equivalent to
part of the maximally extended Schwarzschild solution in V.
The typical construction of Schwarzschild space-time [cf. O'Neill (1983)l
follows the proof of Birkhoff's theorem. A number of strong physical assump
tions, not the least of which is spherical symmetry, lead one inevitably to
Schwarzschild space-time as the unique model. What we have shown is that
a Ricci flat space-time which possesses enough symmetry to be expressed lo-
cally as a (2 x 2) warped product must have spherical, planar, or hyperbolic
symmetry. In the first case we obtain the conclusion to Birkhoff's Theorem:
M is locally isometric to a portion of Schwarzschild space-time.
The following result may be established by an argument similar to the one
employed in the proof of Proposition 3.79 [cf. Easley (1991)].
Theorem 3.83. Let M = B x f F be a semi-Riemannian warped product
with dim B = 1 and dim F = n 2 2. For M to be Ricci flat, i t is necessary
and sufficient that the following two conditions hold.
(1) The root warping function S satisfies (grad S, grad S ) = Co, a constant.
3.7 SEMI-RIEMANNIAN LOCAL WARPED PRODUCT SPLITTINGS 133
(2) (F, g ~ ) is an Einstein manifold with Ric = 2(grad S, grad S)gF =
CO. gF.
This result indicates that a Ricci flat manifold ( M , g ) can admit only one
of a restricted class of local warped product splitting at p f M if we require
dim B = 1. If dim F = 2, for example, it follows that F is a surface of constant
curvature F K since FRic = F ~ g ~ in this case. If we also assume F K = 0,
we have S = Jf = constant, and the warped product is merely a standard
product manifold with a semi-Euclidean metric.
If dim F = 3, it follows that the Einstein manifold ( F , ~ F ) has constant
curvature [cf. Petrov (1969, p. 77) 1. If (F, g ~ ) is Riemannian, we may consider
F a subset of
(1) the sphere S3(r) if F~ = 5' (2) Euclidean space IR3 if FK = 0, or
(3) hyperbolic space ~ ~ ( r ) if F~ = - 7 1
with analogous restrictions holding if g~ is indefinite. An exact solution to
Einstein's equations which uses F = S3 is the spatially homogeneous Taub-
NUT model [cf. Hawking and Ellis (1973, p. 170)].
CHAPTER 4
LORENTZIAN DISTANCE
With the basic properties of Riemannian metrics in mind (cf. Chapter I) , it
is the aim of this chapter to study the corresponding properties of Lorentzian
distance and to show how the Lorentzian distance is related to the causal
structure of the given space-time. We also show that Lorentzian distance
preserving maps of a strongly causal space-time onto itself are diffeomorphisms
which preserve the metric tensor.
While there are many similarities between the Riemannian and Lorentzian
distance functions, many basic differences will also be apparent from this chap-
ter. Nonetheless, a duality between "minimal" for Riemannian manifolds and
"maximal" for Lorentzian manifolds will be noticed in this and subsequent
chapters.
4.1 Basic Concepts and Definitions
Let (M, g) be a Lorentzian manifold of dimension n > 2. Given p, q E M
with p < q, let fl,,, denote the path space of all future directed nonspacelike
curves y : [0, I] + M with $0) = p and $1) = q. The Lorentzian arc length
functional L = L, : Clp,q -' R is then defined as follows [cf. Hawking and Ellis
(1973, p. 105)]. Given a piecewise smooth curve y E Q p , , , choose a partition
0 = to < t l < .. . < t,-l < tn = 1 such that y 1 (t,.t,+l) is smooth for each
i = 0 , 1 , . . . , n - 1. Then define
It may be checked as in elementary differential geometry [cf. O'Neill (1966,
pp. 51-52)] that this definition of Lorentzian arc length is independent of
136 4 LORENTZIAN DISTANCE
FIGURE 4.1. The timelike curve y from p to q is approximated
by a sequence of curves {y,) with yn + y in the C0 topology but
L(yn) --t 0.
the parametrization of y. Since an arbitrary nonspacelike curve satisfies a
local Lipschitz condition, it is differentiable almost everywhere. Hence the
Lorentzian arc length L(7) of 7 may still be defined using (4.1). Alternative
but equivalent definitions of L(y) for arbitrary nonnull nonspacelike curves
may be given by approximating y by C' timelike curves [cf. Hawking and Ellis
(1973, p. 214)] or by approximating y by sequences of broken nonspacelike
geodesics {cf. Penrose (1972, p. 53)j. The Lorentzian arc length of an arbitrary
null curve may be set equal to zero.
Now fix p, q E M with p << q. If y is any timelike curve from p to q, then
L(y) > 0. On the other hand, y may be approximated by a sequence (7,)
of piecewise smooth "almost null" curves y, : [O ,1 ] + M with yn(0) = p and
yn(l) = q such that yn + y in the Co topology, but L(yn) -+ 0 (cf. Figure 4.1).
This construction shows, moreover, that given any p, q E M with p << q, there
4.1 BASIC CONCEPTS AND DEFINITIONS 137
are curves y E Q,,q with arbitrarily small Lorentzian arc length. Hence, the
infimum of Lorentzian arc lengths of all piecewise smooth curves joining any
two chronologically related points p << q is always zero. However, if p and q lie
in a geodesically convex neighborhood U, the future directed timelike geodesic
segment in U from p to q has the largest Lorentzian arc length among all
nonspacelike curves in U from p to q. Thus it is natural to make the following
definition of the Lorentzian distance function d = d(g) : M x M + W U {m)
of (M, g).
Definition 4.1. (Lorentzian Distance Function) Given a point p in M ,
if q 9 J+(p), set d(p, q ) = 0. If q E J+(P), set d(p, q) = sup{L,(y) : y E n,,,).
From the definition. it is immediate that
(4.2) d(p, q) > 0 if and only if q E I'(p).
Thus the Lorentzian distance function determines the chronological past and
future of any point. However, the Lorentzian distance function in general fails
to determine the causal past and future sets of p since d(p,q) = 0 does not
imply g E J+(p) - I+ (p). But at least if q E J+(p) - Ii(p), then d(p, q) = 0.
We emphasize that the Lorentzian distance d(p, q) need not be finite. One
way that the condition d(p, q) = m may occur is that tirnelike curves from p
to q may attain arbitrarily large arc lengths by approaching certain boundary
points of the space-time. In Figure 4.2, two points with d(p,q) = co are
shown in a Reissner-Nordstrijm space-time with e2 = m2 [cf. Hawking and
Ellis (1973, p. 160)].
A second way Lorentzian distance may become infinite is through causality
violations. Recall that a space-time is said to be vicious at the point p E M
if I+(p) n I-(p) = M and totally vicious if I+(p) n I V ( p ) = M for all p E M.
Lemma 4.2. Let (M, g) be an arbitrary space-time.
(1) I fp E If(p), thend(p,p) = co. Thusforeachp E M , eitherd(p,p) = 0
or d(p,p) = co.
(2) (M, g) is totally vicious iff d(p, q) = m for all p, q E M.
(3) If (M, g) is vicious a t p, then (M,g) is totally vicious.
138 4 LORENTZIAN DISTANCE
Proof. (1) Suppose p E I+(p). Then we may find a closed timelike curve
y : [0,1] -+ M with y(0) = y(1) = p. Since y is timelike, L(y) > 0. If
u, E R , , is the timelike curve obtained by traversing y exactly n times, then
L(u,) = nL(y) -+ ca as n --t ca. Thus d(p,p) = m .
(2) Suppose (M,g) is totally vicious. Fix p, q E M, and let n > 0 be any
positive integer. Since p E I+(p), we may find yl E R,,, with L(yl) 2 n by
part (1). Since q E I+(p), there is a timelike curve 7 2 from p to q. Then
y = yl * yz E R,,, is a timelike curve with length L(y) = L(y1) + L(yz) > n.
Hence d(p, q) = m.
Conversely, suppose d(p, q) = m for all p,q E M. Fixing r E M , we have
d(r,p) > 0 and d(p,r) > 0 for all p E M. Thus by (4.2), it follows that
I + ( r ) n I-(r) = M.
(3) Inspired by Lemma 4.2-(2), T. Ikawa and H. Nakagawa (1988) proved
(3) using the Lorentzian distance function. Subsequently, B. Wegner (1989)
noted that the following elementary argument yields the desired result. Let
q E M = I+(p)nI-(p). Then q E I+(p) so I-(p) C I-(q) by the transitivity of
<. Similarly, q E I-(p) implies I+(p) 5 I+(q). Hence, M = I+(p) n I-(p) C + - 0
By Definition 4.1, if I+(p) # M then there are points q E M with d(p, q) = 0
but p # q. Hence unlike the Riemannian distance function, the Lorentzian
distance function usually fails to be nondegenerate. Indeed, we have seen that
d(p, p) > 0 is possible. But if (M, g) is chronological, then d(p, p) = 0 for all
p E M. Also, the Lorentzian distance function tends to be nonsymmetric.
More precisely, the following may be shown for arbitrary space-times.
Remark 4.3. If p # q and d(p, q) and d(q,p) are both finite, then either
d(p,q) = 0 or d(q,p) = 0. Equivalently, if d(p,q) > 0 and d(q,p) > 0, then
d(p, 9) = 4 9 , P) = a.
Proof. If d(p, q) > 0 and d(q,p) > 0, we may find future directed timelike
curves yl from p to q and yz from q to p, respectively. Define a sequence (7,)
by yn = yl * (72 * y ~ ) ~ E Rp,q. AS n --+ m , L(yn) -+ m , whence d(p,q) is
infinite. Similarly, d(q, p) = m .
4.1 BASIC CONCEPTS AND DEFINITIONS 139
FIGURE 4.2. A Reissner-Nordstrom space-time with e2 = m2 is
shown. By taking timelike curves y from p to q close to Jf and
3-. we can make L(y) arbitrarily large. Thus d(p, q) = m , which
means an accelerated observer may take arbitrarily large amounts
of time in going from p to q.
A further consequence of Definition 4.1 is that if y : [O, ca) -+ (M, g) is any
future directed, future complete, timelike geodesic in an arbitrary space-time
(M, g ) , then limt,c, d(r(O), y(t)) 2 limt,m L(y I 10, t ] ) = m. By contrast,
complete Riemannian manifolds (N,go) may contain (nonclosed) geodesies
a : [0, m ) (NIgo) for which sup{do(u(0),u(t)) : t 2 0) is finite. Fur-
ther assumptions are needed for Riemannian manifolds to guarantee that
limt,, d(u(O), u(t)) = co for all geodesics a : [O, m) --+ (N, go) [cf. Cheeger
and Ebin (1975, pp. 53 and 151)].
140 4 LORENTZIAN DISTANCE
While the Lorentzian distance function fails to be symmetric and nondegen-
erate, at least a reverse triangle inequality holds (cf. Figure 1.3). Explicitly,
(4.3) If P < r < q, then d(p, q) 2 d(p, r ) + d(r, q).
We now discuss some properties of the Lorentzian distance that make it a
useful tool in general relativity and Lorentzian geometry. First, the Lorentzian
distance function is lower semicontinuous where it is finite [cf. Hawking and
Ellis (1973, p. 215)].
Lemma 4.4. For Lorentzian distance d, if d(p,q) < m, p, -+ p, and
qn -+ q, then d(p,q) 5 liminf d(p,,q,). Also, if d(p,q) = m, p, -+ p, and
q, -+ q, then limn,, d(p,, q,) = m.
Proof. First consider the case d(p, q) < m. If d(p, q) = 0, there is nothing
to prove. If d(p,q) > 0, then q E I+(p) and the lower semicontinuity follows
from the following fact. Given any 6 > 0, a timelike curve y of length L > d(p, q) - €12 from p to q and sufficiently small neighborhoods Ul of p and U2
of q may be found such that y may be deformed to give a timelike curve of
length L' 2 d(p, q) - c from any point r of Ul to any point s of U2.
Suppose now that d(p,q) = oo but liminfd(p,,q,) = R < m. Since
d(p, q) = m there exists a timelike curve y from p to q of length L(y) > R + 2. This implies that there exist neighborhoods Ul and Uz of p and q,
respectively, such that y can be deformed to give a timelike curve of length
L' > R + 1 from any point r of Ul to any point s of U2. This contradicts
lim inf d(p,, q,) = R. 0
In general, the Lorentzian distance function fails to be upper semicontinu-
ous. We give an example of a space-time (M, g) containing an infinite sequence
{p,) with p, -+ p and a point q E I+(p) such that d(p,,q) = m for all large
n but d(p, q) < ca (cf. Figure 4.3).
For globally hyperbolic space-times, on the other hand, the Lorentzian
distance function is finite and continuous just like the Riemannian distance
function.
Lemma 4.5. For a globally hyperbolic space-time (M,g), the Lorentzian
distance function d is finite and continuous on M x M.
4.1 BASIC CONCEPTS AND DEFINITIONS
FIGURE^.^. L e t M b e { ( x , y ) : O < y < 2 ) - { ( x , l ) : - 1 5 x 5 1 )
with the identification (x, 0) - (x, 2) and the flat Lorentzian metric
ds2 = dx2 - dy2. Let p = (O,O), q = (0,1/2), and p, -+ p as shown.
Then p, E I+(pn) and hence d(pn,pn) = m for all n. For large n
we have q E I+(p,) and thus d(p,,q) = m. On the other hand,
d(p, q) = 112 which yields d(p, q) < lim inf d(p,, q). This space-
time is not causal. However, the distance function may also fail to
be upper semicontinuous in causal space-times (cf. Figure 4.6).
Proof. To prove the finiteness of d, cover the compact set J+(p) n J-(q)
with a finite number of convex normal neighborhoods B1, B2, . . . , B, such that
no nonspacelike curve which leaves any Bi ever returns and such that every
nonspacelike curve in each Bi has length at most one. Since any nonspacelike
curve y from p to q can enter each Bi no more than once, L(y) 5 m. Hence
d(p1 q) 5 m.
If d failed to be upper semicontinuous at (p,q) E M x M, we could find
a 6 > 0 and sequences ipn) and {q,) converging to p and q respectively,
such that d(p,, q,) 2 d(p, q) + 26 for all n. By definition of d(p,, q,), we
may then find a future directed nonspacelike curve y, from p, to q, with
L(y,) 2 d(p, q) + 6 for each n. By Corollary 3.32, the sequence {y,) has a
142 4 LORENTZIAN DISTANCE
nonspacelike limit curve y from p to q. By Proposition 3.34, a subsequence
{y,) of {y,) converges to y in the C0 topology. Hence L(y) 2 d(p, q) + 6 by
Remark 3.35. But this contradicts the definition of Lorentzian distance. Thus
d is upper semicontinuous a t (p, q). U
We now define the following distance condition [cf. Beem and Ehrlich (1977,
Condition 4)].
Definition 4.6. (Finite Distance Condition) The space-time (M, g) sat-
isfies the finite distance condition if d(g)(p, q) < m for all p, q f M.
Lemma 4.5 then has the following corollary.
Corollary 4.7. If (M, g) is globally hyperbolic, then (M,g) satisfies the
finite distance condition and d(g) : M x M + R is continuous.
If (MI g) is globally hyperbolic, all metrics in the conformal class C(M, g)
are globally hyperbolic. Hence all metrics in C(M,g) satisfy the finite distance
condition. We will examine the converse of this statement in Section 4.3,
Theorem 4.30.
Since the given topology of a smooth manifold coincides with the metric
topology induced by any Riemannian metric, it is natural to consider the sets
{m f I+(p) : d(p, m) < E) for a Lorentzian manifold. However, as Minkowski
space shows, these sets do not form a basis for the given manifold topology
(cf. Figure 4.4). Indeed, this same example shows that no matter how small
E > 0 is chosen, the sets {m E J f (p) : d(p, m) 5 E) may fail to be compact
and fail to be geodesically convex as well as failing to be diffeomorphic to the
closed n-disk.
The sphere of radius E for the point p f M is given by K ( ~ , E ) = {q E M :
d(p,q) = E). This set need not be compact. However, the reverse triangle
inequality and (4.2) imply that K(p, E) is achronal for all finite E > 0 and all
p~ M .
In arbitrary space-times, neither the future inner ball
4.1 BASIC CONCEPTS AND DEFINITIONS
FIGURE 4.4. The set B + ( p , ~ ) = {q E I+(p) : d(p,q) < E)
in Minkowski space-time does not have compact closure, is not
geodesically convex, and does not contain p. Furthermore, sets
of the form B+(p, E) do not form a basis for the manifold topol-
ogy. But in general, if (M,g) is a distinguishing space-time with
a continuous Lorentzian distance function, then a subbasis for the
manifold topology is given by sets of the form B+(p, E) and B-(p, E)
[cf. Proposition 4.311. Hence these sets do form a subbasis for the
given topology of Minkowski space-time.
nor the past inner ball
B-(P, €1 = {q f I-(P) : d(q, P) < €1
need be open. On the other hand, when the distance function d : M x M -+
R U {m) is continuous, these inner balls must be open. In Section 4.3 we will
show that for distinguishing space-times with continuous distance functions,
the past and future inner balls form a subbasis for the manifold topology.
144 4 LORENTZIAN DISTANCE
A different subbasis for the topology of any strongly causal space-time
(M,g) with a possibly discontinuous distance function d = d(g) : M x M -+
WU{co) may be obtained by using the outer balls O+(p, E) and 0-(p, E) rather
than the inner balls.
Definition 4.8. (Outer Balls O+(p, E), 0-(p, e)) The outer ball O+fp, E)
[respectively, 0 - ( p , ~ ) ] of I+(p) [respectively, I-(p)] is given by
respectively,
O-(P,E) = {q E M : d(q,p) > E)
(cf. Figure 4.5)
Since the Lorentzian distance function is lower semicontinuous where it is
finite, the outer balls O+(p, E) and 0-(p, E) are open in arbitrary space-times.
The reverse triangle inequality implies that these sets also have the property
that if m, n f O+(p, E ) [respectively, m, n E 0-(p, E)] and m ,< n, then any
future directed nonspacelike curve from m to n lies in Ot(p, E ) [respectively,
0-(p, E)]. Moreover, we have
Theorem 4.9. Let (M, g) be strongly causal. Then the collection
forms a basis for the given manifold topology.
Proof. Let m f M be given, and let U be any open neighborhood containing
m. We may find a local causality neighborhood Ul with m E Ul C U, i.e.,
no nonspacelike curve which leaves Ul ever returns. Choose pl,p2 E Ul with
PI << m (< p2 such that I+(pl)fl I-(pz) C Ul. By the chronology assumptions
on PI and p2, we have d(p1, m) > 0 and d(m,pz) > 0. Choose constants el, t2
with 0 < €1 < d(p1,m) and 0 < €2 < d(m,pz). Then m E O + ( p l , ~ l ) n 0-(pz, €2). Since 0' (PI, €1) C I+(pl) and 0 - ( p z , ~ z ) G I-(pz), we also have
O+(PI, €1) n 0-(pz, €2) E I+(pl) n I-fpz) c UI c u as required.
4.1 BASIC CONCEPTS AND DEFINITIONS 145
FIGURE 4.5. The outer balls Of (p, E) = {q E M : d(p, q) > E)
and 0-(p, E) = {q f M : d(q,p) > E) are open in arbitrary space-
times. Furthermore, O+(p, <) and 0-(p, <) are always subsets of
I+(p) and I-(p), respectively. If (M,g) is strongly causal, the
outer balls O+(p, E ) and 0-(p, E) with p E M and E > 0 arbitrary
form a subbasis for the manifold topology.
For complete Riemannian manifolds, any two points may be joined by a
minimal (distance-realizing) geodesic segment. We now examine the dual of
this property for space-times.
In Hawking and Ellis (1973, p. 110), a timelike geodesic y from p to q is said
to be maximal if the index form of y is negative semidefinite. This definition
implies that if the geodesic y is not maximal, there exist variations of y which
yield curves from p to q "close" to y having longer Lorentzian arc length than
y. If y is maximal in this sense, however, no small variation of y keeping p
and q fixed will produce timelike curves u from p to q with L(u) > L(y).
146 4 LORENTZIAN DISTANCE
Nonetheless, there may still exist a timelike geodesic a1 in M from p to q
("far" from y) with d(p,q) = L(o1) > L(y). Thus maximality as defined
by Hawking and Ellis does not imply "maximality in the large." To study
"maximality in the large," we adopt, in analogy to the concept of minimality
in Riemannian geometry, a definition of maximality valid for all curves in the
path space O,,, [cf. Beem and Ehrlich (1977, Definition I)]. The motivation
for our definition is Theorem 4.13 below [cf. Beem and Ehrlich (1979a, p. 166)]
and its applications, particularly the construction of geodesics as limit curves
of sequences of "almost maximal" curves in Chapter 8 and the definition of
the Lorentzian cut locus in Chapter 9.
Definition 4.10. (Maximal Curve) Let p, q E M with p < q, p # q. The
curve y E Op,, is said to be maximal if L(y) = d(p,q).
An immediate consequence of the reverse triangle inequality (4.3) is
Remark 4.11. If y : [O, 11 -+ M in R,,, is maximal, then for all s , t with
0 5 s < t 5 1, we have d(y(s), y(t)) = L(y j [s, t]) .
The following result, stated somewhat differently in Penrose (1972, Propo-
sition 7.2), is the analogue of the principle in Riemannian geometry that "lo-
cally" geodesics minimize arc length [cf. Bishop and Crittenden (1964, p. 149,
Theorem 2)].
Proposition 4.12. Let U be a convex normal neighborhood centered at
point p E M. For q E J f ( p ) , Jet jjij denote the unique nonspacelike geodesic
c : [O, 11 -+ U in U with c(0) = p and c(l) = q. If y is any future directed
nonspacelike curve in U from p to q with L(y) = d(p, q), then y coincides with
jjij up to parametrization.
Proof. For q E I+(p) and d(p,q) > 0, Penrose (1972, p. 53) shows, using a
synchronous coordinate system, that if y is any causal trip in U from p to q
other than ?5-Q, then L(y) < L(jjij) = d(p, q). This may be obtained equivalently
using the Gauss Lemma (cf. Corollary 10.19 of Section 10.1). Hence the result
is established if d(p, q) > 0.
Suppose now that d(p, q) = 0, and let y be any nonspacelike curve in U from
p to q. Then L(y) 5 d(p, q) = 0. Thus y : [O,1] -+ M is a null curve. Suppose
4.1 BASIC CONCEPTS AND DEFINITIONS 147
that y(t) $ Int(@). Let yl be the unique null geodesic in U from p to y(t),
and let yz be the unique null geodesic in U from y(t) to q. By Proposition
2.19 of Penrose (1972, p. 15), y l * ~ ~ is either a smooth null geodesic or p << q.
Since d(p, q) = 0, p << q is impossible. Hence yl * 7 2 is a smooth null geodesic
which, by convexity of U , must coincide with ?5-Q up to parametrization. 17
Proposition 4.12 has the following important consequence.
Theorem 4.13. If y E Op,, satisfies L(y) = d(p,q), then y may be
reparametrized to be a smooth geodesic.
Proof. Fix any point y(t) on y. We may find 6 > 0 such that a con-
vex neighborhood centered at y(t + 6) contains y([t - 6, t + 61). By Remark
4.11 the curve y I [t - 6, t + 61 is maximal. Thus Proposition 4.12 implies that
y I [t - 6, t + 61 may be reparametrized to be a smooth geodesic. As t was
arbitrary, the theorem now follows.
As an illustration of the use of Definition 4.10 and Theorem 4.13, we give a
simple proof of a basic result in elementary causality theory [cf. Penrose (1972,
Proposition 2.20)] that is usually obtained by different methods.
Corollary 4.14. If p < q but i t is not the case that p << q, then there is a
maximal null geodesic from p to q.
Proof. The causality assumptions on p and q imply that d(p, q) = 0. Now
let y be a future directed nonspacelike curve from p to q. By definition of
Lorentzian distance, d(p,q) 2 L(y) 2 0. Thus L(y) = d(p,q) = 0 and y is
maximal. By Theorem 4.13, y may be reparametrized to a smooth geodesic
c : \O, 11 --t M from p to q. Since L(c) 5 d(p, q ) = 0, the geodesic c must be a
null geodesic.
Note in Corollary 4.14 that since the null geodesic is maximal, it cannot t contain any null conjugate points to p prior to q [cf. O'Neill (1983, p. 404)j. A I sometime useful special case of Corollary 4.14 occurs when p = q is assumed.
k- i; In this case, one may deduce that if the space-time (M,g) is chronological I t but not causal, then there exists a smooth null geodesic 0 : [O, 11 -i (M, g)
/ with P(0) = @(I) and p ( 1 ) = Ap(0) for some A > 0. (If p ( 1 ) and p ( 0 )
148 4 LORENTZIAN DISTANCE
were not proportional, then ,O could be deformed near P(0) to be future time-
like, contradicting the condition d(m, m) = 0 for all m in M, since (M, g) is
chronological [cf. Proposition 2.19 of Penrose (1972, p. 15)]. Even more dra-
matically, a closed timelike curve could be produced to violate chronology [cf.
Proposition 10.46 of O'Neill (1983, pp. 294-295)].)
As a second application of the elementary properties of the distance func-
tion, we give a proof of the existence of a smooth closed timelike geodesic
on any compact space-time having a regular cover with a compact Cauchy
surface. Using infinite-dimensional Morse theory, it may be shown [cf. Klin-
genberg (1978)] that any compact Riemannian manifold admits a t least one
smooth closed geodesic. However, the method of proof relies crucially on the
positive definiteness of the metric and thus is not applicable to Lorentzian
manifolds. Nonetheless, one may obtain the following theorem of Tipler by
direct methods [cf. Tipler (1979) for a stronger result].
Theorem 4.15. Let (M,g) be a compact space-time with a regular cov-
ering space which is globally hyperbolic and has a compact Cauchy surface.
Then (M, g) contains a closed timelike geodesic.
Proof. Since M is compact, there exists a closed, future directed, timelike
curve y : [0,1] + M. Set p = y(0) = y(1). Let T : G -+ M denote the given
covering manifold, and let 7 : [0, I] + M be a lift of y, i.e., ~ o y ( t ) = y(t) for all
t E [0, 11. Then 7 is a future directed timelike curve in z. Put pl = y(0) and
p2 = r(1). Then the global hyperbolicity of implies pl and p2 are distinct
points which cannot lie on any common Cauchy surface. Since T : G -+ M
is regular, there must be a deck transformation 11, : G -t % taking pl to pz
[cf. Wolf (1974, pp. 35-38, 60)j. Choose a compact Cauchy surface S1 of G containing pl, and define S 2 = $I(&). Since (%,7j) is globally hyperbolic, the - - distance function d = d(Z) : M x M + WU (m) is finite-valued and continuous.
Thus we may define a continuous function f : S1 -t W by f (s) = d(s, $(s)).
Since f 071) > 0, we have A = sup{d(s, +(s)) : s E S1) > 0. Moreover, since
S1 is compact, A < w and there exists an rl E S1 with d(rl111,(rl)) = A. Let
E : [0, I] -+ M be a timelike geodesic segment with Z(0) = TI, Z(1) = $(rl),
and L(E) = d(rl,$(rl)) = A. This geodesic exists since ( z , ~ ) is globally
4.1 BASIC CONCEPTS AND DEFINITIONS 149
hyperbolic. Because 5 = ~ ' g , it follows that c = ?r oE : [0, I] -+ M is a timelike
geodesic. Since Z(0) = r l and E(1) = +(rl) , we also have c(0) = c(1). If c
were not smooth at c(O), we could deform c to a timelike curve u : [O, 11 + M - with L,(o) > L(c), a(0) = u(1) E ~ ( S I ) , which lifts to a curve 3 : [O, 11 -+ M
with a(0) E S1 and Z(1) = q(Z(0)) E S2 . But then L z (a) = L,(u) > L,(c) =
L a (Z) = A, in contradiction. 0
More recently, G. Galloway (1984b) has obtained, by elementary geometric
arguments, the existence of a closed timelike geodesic from any stable nontrivial
free homotopy class of closed timelike curves on an arbitrary compact space-
time. Galloway's approach is based on geodesic convexity methods which were
earlier used to obtain a result, basic to the development of global Riemannian
geometry during the early part of this century, given first by J. Hadamard for
surfaces and then by E. Cartan for general Riemannian manifolds of higher
dimension. Their result concerns the existence, within any nontrivial free ho-
motopy class of curves on a compact Riemannian manifold, of a shortest curve
in the given homotopy class, which must then be a nontrivial smooth closed
geodesic. From a directly geometric viewpoint, certain basic ideas involved in
the existence of this geodesic are the following.
Let Lo > 0 denote the infimum of all lengths of curves in this homotopy
class. Choose a minimizing sequence {ck) with lim L(ck) = LO in the given free
homotopy class. By compactness and convexity radius arguments, one covers
the given manifold by a finite number of geodesically convex neighborhoods
(each having compact closure in a larger geodesically convex neighborhood).
Using these neighborhoods in succession, each ck may be approximated by
a piecewise smooth geodesic, or equivalently, may be viewed as an N-tuple
of successive points pl(k),p2(k), . . . ,pN(k), where a uniform bound needs to
be given for the number N of points required. Since a geodesic segment in
a convex neighborhood is the shortest curve between any two of its points,
this approximation procedure produces a shorter curve than ck, but one which
is also still homotopic to ck, and hence is in the given free homotopy class.
By compactness, a diagonalizing argument gives points pi, pz, . . . , p ~ which
are limits of a subsequence of all of the above sequences. Joining p, to p,+l
150 4 LORENTZIAN DISTANCE
successively produces a piecewise smooth geodesic c in the given free homotopy
class which realizes the minimal length Lo. Now the geodesic c must in fact be
smooth a t the pi's, or an even shorter closed curve in the free homotopy class
could be produced by the usual "rounding off the corners" procedure. Since
Lo > 0, this closed geodesic c must be nontrivial, i.e., not a "point curve." A
detailed discussion of the steps involved in the above argument may be found
in Spivak (1979, p. 358).
Now in carrying these ideas over to the space-time setting, it is clear that
"minimal geodesic segment" should be replaced by "maximal timelike geodesic
segment." Technical difficulties arise, however, since the set of unit timelike
tangent vectors based at a given point is not a compact set. Hence, problems
can arise with timelike tangent vectors tending toward a null direction when
trying to do subsequence arguments. Equivalently, in the above context, it is
necessary to prevent timelike geodesic segments joining pi(k) to pi+l(k) from
converging to a null geodesic segment from pi to pi+l when the diagonalization
procedure is carried out. Indeed, Galloway (198613) gives an example of a
compact space-time which contains no closed timelike geodesics but which
contains a closed null geodesic.
In view of these difficulties, Galloway (1984b) considers timelike free ho-
motopy classes which are "stable" for a given compact space-time (M,go).
Here a given free timelike homotopy class C for (M, go) is said to be stable if
there exists a "wider" Lorentzian metric g for M , i.e., go < g in the sense of
the discussion following Remark 3.14, such that if L, denotes the Lorentzian
arc length for (M,g), then the given timelike homotopy class C satisfies the
condition
sup L,(c) < +m. C€E
Galloway (1984b) shows that this concept of stability gives the control needed
to force convergence arguments to be successful and hence obtains the theorem
that for any compact Lorentzian manifold, each stable free timelike homotopy
class contains a longest closed timelike curve which is of necessity a closed
timelike geodesic. Galloway also shows how his result may be used to recover
Tipler's Theorem 4.15 given above and gives a criterion for a free homotopy
4.2 DISTANCE PRESERVING AND HOMOTHETIC MAPS 151
class of closed timelike curves to be stable. Finally, we note that in Galloway
(198613) it is shown by covering space arguments that every compact two-
dimensional Lorentzian manifold contains a closed timelike or null geodesic.
Here one uses dimension two in the essential way that a closed timelike curve for
(M, g) corresponds to a closed spacelike curve (hence a spacelike hypersurface)
for (M, -g), which is also a Lorentzian manifold since dim(M) = 2. Thus,
certain techniques in general relativity involving spacelike hypersurfaces may
be applied to the existence problem.
4.2 Distance Preserving a n d Homothet ic M a p s
Myers and Steenrod (1939) and Palais (1957) have shown that i f f is a dis-
tance preserving map of a Riemannian manifold (Nl, gl) onto a Riemannian
manifold (N2, g2), then f is a diffeomorphism which preserves the metric ten-
sors, i.e., f *gz = 91. In particular, every distance preserving map of (Nl,gl)
onto itself is a smooth isometry. In this section we give similar results for
Lorentzian manifolds following Beem (1978a).
Recall that a diffeomorphism f : (M1,gl) + (M2,gz) of the Lorentzian
manifold (M1,gl) onto the Lorentzian manifold (M2,gz) is said to be homo-
thetic if there exists a constant c > 0 such that gz(f,v, f,w) = cgl(v, w) for
all v, w E TpMl and all p E MI. In particular, if c = 1, then f is a (smooth)
isometry. The group of homothetic transformations is important in general
relativity since it has been shown to be the group of transformations which
preserves the causal structure for a large class of space-times [cf. Zeeman
(1964, 1967), Gobel (1976)j.
We will let dl denote the Lorentzian distance function of (Ml, gl) and d2
denote the Lorentzian distance function of (M2, 92) below. The distance ana-
logue of a smooth homothetic map is defined as follows.
Definition 4.16. (Distance Homothety) A map f : ( M I , gl) --+ (M2,gz) is said to be distance homothetic if there exists a constant c > 0 such that
d2(f(p), f(q)) = cdl(p,q) for all p,q E 1M. If c = 1, then f is said to be
distance preserving.
152 4 LORENTZIAN DISTANCE
It is important to note that for arbitrary Lorentzian manifolds, distance
preserving maps are not necessarily continuous. For if (M,g) is a totally
vicious space-time, we have seen that d@, q) = 03 for all p, q E M [cf. Lemma
4.2-(2)]. Hence any set theoretic bijection f : M + M is distance preserving
but need not be continuous.
T h e o r e m 4.17. Let ( M I , g l ) be a strongly causal space-time, and let
(M2,g2) be an arbitrary space-time. If f : (Ml ,g l ) -+ (M2, g 2 ) is a dis-
tance homothetic map (not assumed to be continuous) of Ml onto M2, then f
is a smooth homothetic map. That is, f is a diffeomorphism, and there exists
a constant c > 0 such that f *g2 = cg1. In particular, every map of a strongly
causal space-time ( M , g) onto itself which preserves Lorentzian distance is an
isometry.
Corollary 4.18. If ( M I g) is a strongly causal space-time, then the space of
distance homothetic maps of ( M , g) equipped with the compact-open topology
is a Lie group.
Proof of Corollary 4.18. Since ( M , g ) is strongly causal, this group coincides
by Theorem 4.17 with the space of smooth homothetic maps of M onto itself
which preserve the time orientation. But this second group is a Lie group. CI
The proof of Theorem 4.17 will be broken up into a series of lemmas.
L e m m a 4.19. Let ( M l , g l ) and (Mz,g2) be space-times, and consider a
map f : ( M l , g l ) --+ (M2,g2) which is onto but not necessarily continuous. If
f is distance homothetic, then
Proof. First (1) holds since d2(f (p), f (9)) = cdl(p, q), and P << q [respec-
tively, f (p) << f (q)] iff dl(p, q) > 0 [respectively, d2(f ( P I , f ( 9 ) ) > 01. Since (1)
implies p << r << q iff f (p) << f ( r ) << f (q), statement (2) foll~ws. 0
The importance of (2) stems from the fact that if ( M , g) is strongly causal,
the sets {I+(p) n I - (q) : p, q E M ) form a basis for the topology of M . Recall
4.2 DISTANCE PRESERVING AND HOMOTHETIC MAPS 153
that a map f : Ml + M2 is said to be open if f maps each open set in MI t o
an open set in M2.
Lemma 4.20. Let ( M I , g l ) be strongly causal and let (M2,g2) be an ar-
bitrary space-time. If f is a (not necessarily continuous) distance homothetic
map of ( M I , g l ) onto (M2,g2), then f is open and one-to-one.
Proof. The openness of f is immediate from part (2) of Lemma 4.19. It
remains to show that f is one-to-one. Assume there are distinct points p and
q of MI with f ( p ) = f(q). Let U(p) be an open neighborhood of p with
q $! U(p) and such that no nonspacelike curve intersects U(p) more than once.
Choose r l , 7-2 E U(p) with rl << p << 7-2. Clearly, q 6 I f ( r l ) n I - ( r 2 ) . It follows
from Lemma 4.19 that f (r1) << f (p) = f ( q ) << f (r2) implies rl << q << 7-2.
This yields q f I+(r l ) n I-(r2) which is a contradiction. O
Applying Lemma 4.20 to f and f-' we obtain
Proposi t ion 4.21. Let ( M I , g l ) be strongly causal, and let ( M 2 , g2) be an
arbitrary space-time. Let f be a (not necessarily continuous) map of MI onto
M2. I f f is distance homothetic, then f is a homeomorphism and (M2,g2) is
strongly causal.
Proof. The relation f -' is a function since f is one-to-one by Lemma 4.20.
Furthermore, f -' is continuous since Lemma 4.20 shows f is an open map.
In order to complete the proof it is sufficient to show M2 is strongly causal
since Lemma 4.20 will then imply f -' is an open map, whence f is continuous.
Given pi E M2, let p = f-'(p'). If r' << p' << q', then Lemma 4.19 applied to
the distance homothetic map f -' yields f -'(rl) << p << f - l (q l ) . Let U1(p') be
an open neighborhood of p'. Choose V1(p') 2 U'(pl) with the closure of V'(pl)
a compact set contained in an open convex normal neighborhood W'(p l ) of p'.
We may assume that (W' (p f ) ,g2 I,,(,,)) is globally hyperbolic. Let {r;) and
(9;) be sequences in V1(p') such that r; -+ P I , q; --, p', and r; << << q;
for all n. Assume the strong causality of M2 fails at pl. This means that for
each n, the set I+(r;) n I-(9;) cannot be contained in the convex normal
neighborhood W1(p') because otherwise the sets I f (r;) n I-(9;) would give
arbitrarily small neighborhoods of p' which each nonspacelike curve intersects
154 4 LORENTZIAN DISTANCE
at most once. Choose a sequence o f points (2;) contained in the boundary
o f V t (p ' ) with zk E I+(&) n I-(qh) for each n. The sequence (26) has an
accumulation point z because the closure o f V t (p t ) is compact. Furthermore,
f E f - l ( I+(rh)nI-(qh)) = I+(f - ' ( rL))nI- ( f The continuity
o f f -' implies that f -l(r;) -+ p and f -+ p. The strong causality o f
MI yields that the sets I+( f - l ( rk ) ) n I - ( f - l (q&)) are approaching the point
p. Thus, f - I ( & ) -+ p which means f - l ( z ) = p = f -'(pt). This contradicts
the one-to-one property of f - l . Consequently, M2 must be strongly causal,
and the proposition is established. 13
Consider the strongly causal space-time M . Given p E M , let U(p) be
a convex normal neighborhood of p. The set U(p) may be chosen so small
that whenever q, r E U(p) with q < r , the distance d(q, r ) is the length o f the
unique geodesic segment a(q , r ) from q to r which lies in U(p). Furthermore,
U ( p ) may be chosen such that i f q, z,.r E U(p) with q << z << r , then the
reverse triangle inequality d(q, r ) 2 d(q, z ) +d(z , r ) is valid with strict cquality
i f and only i f z is on the geodesic segment from q to r in U(p). Thus, timelike
geodesics in a strongly causal space-time are characterized by the space-time
distance function, and it follows that distance homothetic maps take timelike
geodesics to timelike geodesics.
Lemma 4.22. I f f is a distance homothetic map defined on a strongly
causal space-time, then f maps null geodesics to null geodesics.
Proof. Let U(p) be a convex normal neighborhood o f p as in the above
paragraph, chosen sufficiently small such that f (U(p)) lies in a convex nor-
mal neighborhood o f f ( p ) . Let a ( q , r ) be a null geodesic in U(p) . Choose
q, - q and r , -- r with q, << T, for all n. Proposition 4.21 then im-
plies that f (q,) -+ f (q) and f (r,) -+ f ( r ) . The map f takes the time-
like geodesic a(q,,r,) with endpoints q, and r, to the timelike geodesic
a( f (q,), f (r,)). Since the geodesics a(q,, r,) converge to a (q , r ) and the
geodesics a( f (q,), f (r,)) converge to a( f (qf , f ( r ) ) , it follows that f maps
r, a ( f (q)i f ( r ) ) .
4.2 DISTANCE PRESERVING AND HOMOTHETIC MAPS 155
Proof of Theorem 4.17. The fact that f is a diffeomorphism follows from
a result proved by Hawking, King, and McCarthy (1976) which states that a
homeomorphism which maps null geodesics to null geodesics must be a diffeo-
morphism. Since Ml and M2 are strongly causal, for each p E Ml there exists
a convex normal neighborhood Ul (p) such that for q E U1 (p) with p << q, the
lengths o f the timelike geodesics a(p , q) joining p to q and a( f (p ) , f (q ) ) join-
ing f (p ) to f (q) are given by dl(p, q) and d2( f ( p ) , f ( q ) ) , respectively. Using
d2(f ( P ) , f (q ) ) = cdl(p ,q) , it follows that f maps gl onto the tensor c-2gz.
It is well known that i f a complete Riemannian manifold is not locally flat,
then it admits no homothetic maps that are not isometries [cf. Kobayashi and
Nomizu (1963, p. 242, Lemma 2)] . An essential step in the proof consists o f
showing for arbitrary complete Riemannian manifolds that any homothetic
map which is not an isometry has a unique fixed point. This may be done
by using the triangle inequality for the Riemannian distance function and the
metric completeness o f any geodesically complete Riemannian manifold.
In view o f Theorem 4.17 above, it is then o f interest to consider the analo-
gous question of the existence o f nonisometric homothetic maps o f a Lorentzian
manifold [cf. Beem (1978b)j. In what follows, we will use the standard ter-
minology o f proper homothetic map for a homothetic map which is not an
isometry.
W e first note that R2 with the Lorentzian metric ds2 = dxdy provides an
example of a globally hyperbolic geodesically complete space-time that admits
a fixed-point free, proper homothetic map. For fixing any ,b' # 0 and choosing
any c > 0, the map f (x: y) = ( x -k p, cy) is a fixed-point free homothetic
map with homothetic constant c. Thus the existence o f a fixed point for a
proper homothetic map must be assumed for geodesically complete Lorentzian
manifolds unlike the Riemannian case.
Now suppose f is a proper homothetic map o f a space-time ( M , g ) such
that f (p ) = p for some p E M . Then f,p : T,M -+ T,M has at least one
nonspacelike eigenvector [cf. Beem (1978b, p. 319, Lemma 3)]. This eigenvec-
tor may be null, however. For example, composing the Lorentzian "boost"
156 4 LORENTZIAN DISTANCE
isometry F of (W2, ds2 = dx2 - dy2),
F(x, y) = (x cosh t + y sinh t, x sinh t + y cosh t)
with t > 0 fixed, and a dilation T(x, y) = (cx, y) with c > 0 and c # 1 yields a
proper homothetic map f of Minkowski space-time fixing the origin such that
f*(,,,, has null vectors for eigenvectors.
But if f,p : T p M -+ T p M is a proper homothetic map which has a timelike
eigenvector with eigenvalue X < 1, it may be shown that (M, g) is Minkowski
space-time [cf. Beem (197813, p. 319, Proposition 4)]. Also, iff is a homothetic
map with a fixed point p such that all eigenvalues of f,p are real and all have
absolute value less than one, then (M,g) is Minkowski space-time [cf. Beem
(197813, p. 316, Theorem I)].
We now give an example of a n o d a t space-time admitting a global homo-
thetic flow. Let M = W3 with the metric g = ds2 = exzdxdy + dz2. Thus
are tangent vectors a t (x, y, z), we have
It may then be checked that while (M, g) is not flat, the map 4, : (W3, ds2) -+
(W3, ds2) given by
&(x, y, z) = (etx, e-3ty, e?z)
is a proper homothety with g(&v, q5t.w) = e-2tg(v, w) for each fixed nonzero
t .
We now show, however, that this space-time is null geodesically incomplete.
Let X = d/dx, Y = d/by, and Z = d/az. Then all inner products vanish
except for g(X, Y) = ezz/2 and g(Z, Z) = 1; furthermore, [X, Y] = [X, Z] = 0.
Hence using the Koszul formula
4.3 LORENTZIAN DISTANCE FUNCTION AND CAUSALITY 157
we obtain the following formulas for the Levi-Civita connection of ( R ~ , ds2):
x x x VxY = --ex"Z, V x Z = -X, and V y Z = -Y.
4 2 2
Thus the only nonzero Christoffel symbols are I'il = z, r:2 = I':, = -$ex",
and I'i3 = I'i, = = I?j2 = $. Hence if y(t) = (x(t), y(t), ~ ( t ) ) is a geodesic,
the usual system of second order differential equations
for y reduces to the following system:
It may finally be checked that the curve y : (-1, w) + (LR3,ds2) defined by
y(t) = (ln(1 + t), 0 , l ) satisfies this system of differential equations and hence
is the unique null geodesic in (W3,ds2) with yl(0) = d / d ~ l ~ ~ , ~ , , ) . Thus this
space-time is null geodesically incomplete.
4.3 The Lorentzian Distance Function and Causality
In this section we study the relationship between the continuity and finite-
ness of the Lorentzian distance function d = d(g) : M x M -+ W U {a) for
(M, g) and the causal structure of (M, g). The most elementary properties,
extending Lemma 4.2 above, are summarized in the following lemma. Re-
call that Lor(M) denotes the space of all Lorentzian metrics for M. The C0
topology on Lor(M) was defined in Section 3.2.
Lemma 4.23.
11) d(p,q) > 0 iff 9 f I+(P). (2) The space-time (M, g) is t o t d y vicious iff d(p, q) = co for all p, q E M .
158 4 LORENTZIAN DISTANCE
(3) The space-time (M, g) is chronological iff d is identically zero on the
diagonal A(M) = {(p,p) : p E M ) of M x M.
(4) The space-time (M,g) is future [respectively, past] distinguishing iff
for each pair of distinct p, q E M , thcrc is some x E M such that
exactly one of d(p, x) and d(q, x) [respectively, d(x, p) and d(x, q)] is
zero.
( 5 ) The space-time (M,g) is stably causal iff there exists a neighborhood
U of g in the fine C0 topology on Lor(M) such that d(gl)(p,p) = 0 for
all g' E U and p E M.
Proof. Similar to Lemma 4.2 and Remark 4.3.
Recall that the Lorentzian distance function in general fails to be upper
semicontinuous. Thus the continuity of d(g) should have implications for the
causal structure of (M,g). An example is the following result first stated in
Beem and Ehrlich (1977, p. 1130). Here d is regarded as being continuous at
(p, q) E M x M with d(p,q) = co because d(p,, q,) -+ w for all sequences
p, -+ p and q, -+ q (cf. Lemma 4.4).
Theorem 4.24. Let (M,g) be a distinguishing space-time. If d = d(g) :
M x M + R U {w) is continuous, then (M, g) is causally continuous.
Proof. We need only show that I+ and I- are outer continuous. Assume - I+ is not outer continuous. There is then some compact set K C M - I+(p)
and some sequence p, -+ p such that K n I+(p,) # 0 for all n. Let q, E
K n I+(pn) and let {qm) be a subsequence of {q,) such that {q,) converges
to some point q of the compact set K. Then q, -+ q and q, E I+(p,)
imply there must be a sequence {q;) converging to q such that qk E I+(p,) - for each m. Since M - I+(p) is an open neighborhood of q, there is some
r E M - I+(p) with q << r . For sufficiently large m we then have q i << r
and hence p, << qk << r . Thus d(p,, r ) 2 d(p,, qk) + d(qk, r). Using the
lower semicontinuity of distance and the causality relation q << r, we obtain
0 < d(q,r) 5 liminf d(qk,r) . Consequently, d(p,,r) 2 d(q,r)/2 > 0 for
all sufficiently large m. However, since r 4 I+(p), we have d(p,r) = 0, and
hence d(p,r) # lirnd(p,,r). Thus if d is continuous, I+ is outer continuous.
4.3 LORENTZIAN DISTANCE FUNCTION AND CAUSALITY 159
FIGURE 4.6. Let (M,g) denote Minkowski space-time with the
point r deleted. Choose p, q E M such that in Minkowski space-
time the point r is on the boundary of I+(p) n I-(q) as shown.
Let {q,) be a sequence of points approaching q with q << q, for
each n. There is a smooth conformal factor 0 : M -+ (0, m) such
that R = 1 on I+(p) n I-(q) and yet d(Qg)(p, q,) 2 2d(g)(p, q) for
each n. The function R will be unbounded near the deleted point
T. Since d(g)(p, q) = d(Rg)(p,q) < liminf d(Rg)(p,q,), the ca~lsally
continuous space-time (M, Rg) has a Lorentzian distance function
which is discontinuous at (p, q) E M x M .
A similar argument shows that I- is outer continuous. Thus continuity of d
implies that (M, g) is causally continuous. fl
160 4 LORENTZIAN DISTANCE
The following example shows that the converse of Theorem 4.24 is false. Let
(M, g) denote Minkowski space-time with a single point removed. The space-
time (M,Rg) will be causally continuous for any smooth conformal factor
R : M -+ (0,m). However, R may be chosen such that d = d(Rg) is not
continuous (cf. Figure 4.6).
We now turn to a characterization of strongly causal space-times in terms
of the Lorentzian distance function. The definition of convex normal neigh-
borhood was given in Section 3.1. Given any spacetime (M, g), we have
Definition 4.25. (Local Distance Function) A local distance function
(D, U) on (M, g) is a convex normal neighborhood U together with the distance
function D : U x U -+ R induced on U by the space-time (U, gIu).
More explicitly, given p, q E U, then D(p, q) = 0 if there is no future di-
rected timelike geodesic segment in U from p to g. Otherwise, D(p, q) is the
Lorentzian arc length of the unique future directed timelike geodesic segment
in U from p to q.
We will let I+(p, U) (respectively, J+(p, U)) denote the chronological (re-
spectively, causal) future of p with respect to the space-time (U, glU).
Lemma 4.26. Let (M, g) be a space-time, and let U be a convex normal
neighborhood of (M, g). Assume that D : U x U -t IF% is the distance function
for (U, glu). Then D is a continuous function on U x U and D is differentiable
on U+ = {(p, q) E U x U : q E I+(p, U)).
Proof. Given p,q E U with q E J+(p,U), let cpq : [0, I] -+ U denote the
unique nonspacelike geodesic segment with cpq(0) = p and %,(I) = q. We
then have D(p, q) = [-g(ci,(~), C ~ , ( O ) ) ] ~ / ~ and D(P, 9)' = [-g(ciq(0), c;,(O))].
Rom the differentiable dependence of geodesics on endpoints in convex neigh-
borhoods, it is immediate that D is continuous on U x U and that D is differ-
entiable on U+. U
Minkowski space-time shows that D fails to be differentiable across the null
cones and thus fails to be smooth on all of U x U. It is not hard to see that the
local distance function (D, U) uniquely determines the Lorentzian metric g on
U. Consequently if {U,) is a covering of M by convex normal neighborhoods
4.3 LORENTZIAN DISTANCE FUNCTION AND CAUSALITY 161
with associated local distance functions {(D,, U,)), then {(D,, U,)) uniquely
determines g on M.
We now characterize strongly causal space-times in terms of local distance
functions [cf. Beem and Ehrlich (1979c, Theorem 3.4)].
Theorem 4.27. A space-time (M, g) is strongly causal if and only if each
point r E M has a convex normal neighborhood U such that the local distance
function (D, U) agrees on UX U with the distance function d = d(g) : M x M -+
R u {m).
Proof. If (M, g) is strongly causal and r E M, then there is some convex nor-
mal neighborhood U of r such that no nonspacelike curve which leaves U ever
returns. The local distance function for U then agrees with d = d(g) 1 (U x U).
Conversely, assume that strong causality breaks down at some point r E M.
Let U be a convex normal neighborhood of r such that D(p, q) = d(p, q)
for all p, q f U. There exists a neighborhood W C U of r such that any
future directed nonspacelike curve y : (0,1] -+ U with y(1) E W and y past
inextendible in U contains some point not in J f (W, U). Since strong causality
fails to hold at r, there is a future directed timelike curve y, : [O, 11 --+ M with
T' = yl(0) E W, y1(1/2) 4 U, and yl( l ) E W. By construction of W, there
is some point p E 71 n U with p $2 J+(r1, U). Hence D(rl ,p) = 0. However,
d(rr,p) > 0 since d(rr,p) is at least as large as the length of yl from r' to
p. Thus D(rl,p) # d(rr,p). Taking the contrapositive then establishes the
theorem. (7
Corollary 4.28. If (M, g) is strongly causal, then d is continuous on some
neighborhood of A(M) = {(p,p) : p E M) in M x M. Also, given any
point m E M, there exists a convex normal neighborhood U of m such that
d 1 (U x U) is finite-valued.
We now give a characterization of globally hyperbolic space-times among
all strongly causal space-times using the Lorentzian distance function. For
this purpose, it is first necessary to show that the usual definition of globally
hyperbolic may be weakened.
162 4 LORENTZIAN DISTANCE
Lemma 4.29. Let ( M , g) be a strongly causal space-time. If J+(p)n J-(q)
has compact closure for all p, q E M , then ( M , g) is globally hyperbolic.
Proof. I t is only necessary to show J+(p) n J-(q) is always closed. Assume
T E (J+(p) n J- (q ) ) - (J+(p) n J-(q)) . Choose a sequence {T,) o f points
in J+(p) n J-(q) with r , -+ r . For each n let yn : [O, 1) --, M be a future
directed, future inextendible, nonspacelike curve with p = y,(O) and r, E y,,
q E y,. By Proposition 3.31, there is some future directed, future inextendible,
nonspacelike limit curve y : [O, 1) -+ M of the sequence (7,). Furthermore,
p = y(0). The limit curve y cannot be future imprisoned in any compact subset
o f M because ( M , g ) is strongly causal (c f . Proposition 3.13). Consequently,
there is some point x on y with x @ J+(p) fl J-(q). The definition o f limit
curve yields a subsequence (7,) o f {y,) and points x, E y, with x , -t x.
Since x @ J+ (p) n J - (q) , we have x , 4 J+(p) n J-(q) for all large m. Using
y, C J+(p), it follows that x , @ J-(q) for large m. Hence q lies between p
and x, on y, for large m. Let y [ p , x ] (respectively, y,[p, x,]) denote the
portion o f y (respectively, 7,) from p to x (respectively, x,). By Proposition
3.34 we may assume, by taking a subsequence o f {y,[p, x,]) i f necessary,
that {y,[p,xm]) converges to y [ p , x ] in the C0 topology on curves. Hence
q E y,[p, x,] for large m implies q E y[p, x ] . Also r , -+ r and T, 6 q which
yield r E y [ p , q ] . Thus r E J+(p) n J-(q), in contradiction. O
Recall from Definition 4.6 above that a space-time ( M , g ) is said to satisfy
the finite distance condition i f and only i f d(g)(p, q) < m for all p, q E M . This
condition may be used to characterize globally hyperbolic space-times among
strongly causal space-times [cf. Beem and Ehrlich (197913, Theorem 3.5))
Theorem 4.30. The strongly causal space-time ( M , g) is globally hyper-
bolic i f f (M,gr ) satisfies the finite distance condition for dl g' € C ( M , g).
Proof. I t has already been remarked that i f ( M , g ) is globally hyperbolic,
then all metrics in C ( M , g) satisfy the finite distance condition (c f . Corollary
4.7).
Conversely, assume that ( M , g) is not globally hyperbolic. Lemma 4.29
implies that there exist p,q E M such that J+(p) n J- (q ) does not have
4.3 LORENTZIAN DISTANCE FUNCTION AND CAUSALITY 163
compact closure. Let h be an auxiliary geodesically complete positive definite
metric on M , and let do : M x M -+ B be the Riemannian distance function
induced on M by h. The Hopf-Rinow Theorem implies that all subsets o f M
which are bounded with respect to & have compact closure. Thus J+(p) n J- (q ) is not bounded. Hence, for each n we may choose p, E J+(p) n J-(q)
such that do(p,p,) > n. Choose p' and q' with p' << p << q << q'. W e wish to
show there exists a conformal factor 0 such that d(Rg)(pl, q') = w. For each
n > 1, choose y, to be a future directed timelike curve from p' to pn such that
yn[1/2, 3/41 5 { r E M : n-1 < da(p, r ) < n ) . For each n > 1, let R, : M -+ IW be a smooth function such that O,(x) = 1 i f x 4 { r : n - 1 < &(p, r ) < n ) and
such that the length of yn[1/2, 3/41 is greater than n for the metric R,g. Let
R = n, R,. This infinite product is well defined on M since for each x E M
at most one o f the factors R, is not unity. Then d(Rg) (p', p,) > n for each
n > 1. Hence d(flg)(pl, 9') = as d(Rg)(p1, q') L d(Rg)(pl, pn) + d ( R g ) ( ~ n , q') for each n. 0
W e now turn to the proof that for distinguishing space-times with continu-
ous distance functions, the future and past inner balls form a subbasis for the
given manifold topology. Recall that
and
Thus defining f i : M -+ R for i = 1,2 by f i(q) = d(p,q) and f2(q) = d(q,p),
we have B+(P, E ) = f ~ ' ( ( 0 , E ) ) and B-(p, E ) = f ~ ' ( ( 0 , E ) ) . Hence i f ( M , g) has
a continuous distance function, the inner balls B+ (p, E) and B- (p , E ) of M are
open in the manifold topology.
Proposition 4.31. Let ( M , g) be a distinguishing space-time with a con-
tinuous distance function. Then the collection
forms a basis for the given manifold topology o f M
164 4 LORENTZIAN DISTANCE
Proof. The above arguments show that sets of the form B+(p, c l ) n B V ( q , c2)
are open in the manifold topology. Thus given an arbitrary point r E M
and an arbitrary open neighborhood U ( r ) o f r in the manifold topology, it is
sufficient to show that there exist p, q E M and €1, €2 > 0 with T E B+(p, el) n B-(q, € 2 ) U(r ) .
Theorem 4.24 yields that ( M , g ) is causally continuous and hence also
strongly causal. Thus we may choose a convex normal neighborhood V of
r with V U ( r ) such that no nonspacelike curve which leaves V ever returns
and such that d : V x V + W U {oo) is finite-valued (cf . Corollary 4.28). Fix
p, q E V with p << r << q. Then r E I+(p) n I - (q) C V since no nonspace-
like curve from p to q can leave V and return. Letting €1 = d(p,r) + 1 and
€2 = d(r, q) + 1, we obtain
which establishes the proposition. 0
W e conclude this section with a characterization o f totally geodesic timelike
submanifolds in terms o f the Lorentzian distance function. An analogous result
holds for submanifolds o f (not necessarily complete) Riemannian manifolds
[cf. Gromoll, Klingenberg, and Meyer (1975, p. 159)].
Let ( M , g) denote an arbitrary strongly causal space-time. Suppose that
i : N + M is a smooth submanifold and set i j = i*g. Recall that ( N , i j ) is said
t o be a timelike submanifold of ( M , g) i f 3 l p : TpN x TpN + W is a Lorentzian
metric for each p f N . As usual, we will identify N and i ( N ) . Let Z , L
and 2, d denote the arc length functionals and Lorentzian distance functions
of ( N , i j ) and ( M , g ) , respectively. Then i f y is a smooth curve in ( N , i j ) , we
have Z(r) = L(y) . Note also that i f q E I+(p, N ) , then p << q in ( M , g), and
i f q E J+(p, N ) , then p < q in ( M , g ) . Thus it follows immediately from the
definitions o f d and d that
- (4.4) d (m, n) 5 d(m, n) for all m, n E N .
Wi th this remark in hand, we are ready to prove the following result.
4.3 LORENTZIAN DISTANCE FUNCTION AND CAUSALITY 165
Proposition 4.32. Let ( N , 3 ) be a totally geodesic timelike submanifold
of the strongly causal space-time ( M , g ) . Then given any p E N , there exists
a neighborhood V of p in N such that d 1 ( V x V ) = 2 I (V x V ) .
Proof. First, let W be a convex neighborhood of p in ( M , g ) such that every
pair o f points m , n E W are joined by a unique geodesic of ( M , g) lying in W
and i f m 6 n, then this geodesic is maximal in ( M , g ) . We may then choose
a smaller neighborhood Vo of p in M with Vo C W such that i f V = Vo n N ,
then V is contained in a convex normal neighborhood U o f p in N with U
contained in W .
Suppose first that m , n E V and n E J+(m, N ) . Since V C U , there exists
a nonspacelike geodesic y o f (N,ij) in U from m to n . Also, as N is totally
geodesic, y is a nonspacelike geodesic in ( M , g). Because y is contained in U
and U E W , y is maximal in (M,g) . W e thus have d ( m , n ) > Z ( y ) = L(T) =
d(m, n ) . In view o f (4.4), we obtain d(m, n ) = d(m, n ) as required.
It remains to consider the case that m , n E V and n # J+(m, N ) . Thus - d(m, n ) = 0 by definition. Suppose that d (m, n ) > 0. Then there exists a
timelike geodesic yl o f ( M , g ) in W from m to n. On the other hand, since
m , n E U , there exists a geodesic 7 2 o f ( N , i j ) from m to n lying in U which
must be spacelike since n # J+(m, N ) . Since (N,?j) is totally geodesic, 7 2 is
also a spacelike geodesic o f ( M I g) from m to n which lies in U C W . Thus we
have distinct geodesics yl and 72 in W from m to n , in contradiction. Hence
d ( m , n ) = 0 =Z(m,n) as required. Cl
W e now prove the converse of Proposition 4.32.
Proposition 4.33. Let ( N , i j ) be a timelike submanifold of the strongly
causal space-time ( M , g ) . Suppose that for all p E N there exists a neighbor-
hood V o f p in N such that d / ( V x V ) = 2 I (V x V ) . Then ( N , g ) is totally
geodesic in ( M , g).
Proof. It suffices to fix any p E N and show that the second fundamental
form S, vanishes at p (cf . Definition 3.48). Since any tangent vector in TpN
may be written as a sum o f nonspacelike tangent vectors, it is enough to
show that S,(v, w) = 0 for all nonspacelike tangent vectors in T p N . Also, as
166 4 LORENTZIAN DISTANCE
Sn(- v, w) = -Sn(v, w), it suffices to show that Sn(v, w) = 0 for all future
directed nonspacelike tangent vectors in T,N.
Thus let v E T,N be a future directed nonspacelike tangent vector. Let
y denote the unique geodesic in (N,?j) with y'(0) = v. Also let V be a
neighborhood of p in N on which the distance functions a and d coincide.
Choose t > 0 such that if m = y(t), then m E V and z(p, m) = Z(y 1 [0, t ] ) is
finite. We then obtain
d ( ~ , m ) L L(T / 10, t ] ) = Z(y 1 10, t ] ) = a(p, m).
But since m E V we have d(p,m) = z(p,m), whence L(y I [O,t]) = d(p,m).
Hence y I [O, t ] is a geodesic in (M,g) by Theorem 4.13. Thus we have shown
that if u E TpN is any future directed tangent vector, the geodesic in (M,g)
with initial direction u is also a geodesic in (N,?j) near p. Therefore Sn(u, u) =
0 for all future directed nonspacelike tangent vectors. Since the sum of two
nonparallel future directed nonspacelike tangent vectors is future timelike, it
follows by polarization that S,(u, w) = 0 for all future directed nonspacelike
tangent vectors u, w E TpN, as required. 0
Combining Propositions 4.32 and 4.33 yields the following characterization
of totally geodesic timelike submanifolds of strongly causal space-times in
terms of the Lorentzian distance function.
Theorem 4.34. Let (M, g) be a strongly causal space-time of dimension
n 2 2, and suppose that (N, i'g) is a smooth timelike submanifold of (M,g),
i.e., ?j = i*g is a Lorentzian metric for N . Then (N,?j) is totally geodesic
iff given any p E N, there exists a neighborhood V of p in N such that the
Lorentzian distance functions 2 of ( N , 3 ) and d of ( M , g ) agree on V x V.
4.4 Maximal Geodesic Segments and Local Causality
We have seen in this chapter that certain causality conditions placed on a
space-time are related to pleasant local or global behavior of the Lorentzian
distance function. For instance, we saw that if (M,g) is globally hyperbolic,
then all metrics conformal to g for M are not only continuous but also satisfy
the finite distance condition as well. Further, the finite distance condition for
4.4 MAXIMAL GEODESIC SEGMENTS, LOCAL CAUSALITY 167
all metrics conformal to a given strongly causal metric for a space-time forces
the conformal class of space-times to be globally hyperbolic. If a space-time
is strongly causal, then we noted that for any given p in A4 there exists a
neighborhood U of p in which the local distance function of (U, glu) coincides
with the global Lorentzian distance function on U x U. Hence the Lorentzian
distance function is forced to be finite-valued on U x U.
For the purposes of this section, it will be convenient to reformulate Defi-
nition 4.10 slightly as follows.
Definition 4.35. (Mmimal Segment) A future directed nonspacelike
curve c : la, b] -, (M, g) is said to be a maximal segment provided that
L(c) = d(c(a), c(b)) and hence L(c 1 [ s , t ] ) = d(c(s), c(t)) for all s , t with
a s s l t l b .
Evidently, the Lorentzian distance function restricted to the image of a
maximal segment must be finite-valued and continuous. In the terminology
of Definition 4.35, the previous Proposition 4.12 may be rephrased as follows.
Suppose that (M, g) is strongly causal. Given p in M , let U be a local causality
neighborhood of p, i.e., a causally convex neighborhood about p which is also
a geodesically convex normal neighborhood. Then any nonspacelike geodesic
segment lying in U is a maximal segment in the space-time (M, g).
In Chapter 14 the Lorentzian Splitting Theorem for timelike geodesically
complete (but not necessarily globally hyperbolic) space-times is studied.
Since global hyperbolicity is not assumed and hence the global continuity
and finite-valuedness of the space-time distance function may not be taken
for granted, it is necessary to make a careful study of the space-time distance
function and asymptotic geodesics in a neighborhood of a given timelike line.
What emerged in a series of papers, as summarized in the introduction to
Chapter 14 which will not be repeated here, is that the existence of a global
maximal timelike geodesic has important implications for the distance function
and the Busemann function of the timelike line in a neighborhood of the given
line [cf. Eschenburg (1988), Galloway (1989a), Newman (1990), Galloway and
Horta (1995)l. Especially, Newman (1990) made a thorough study of the geo-
desic geometry in the case that timelike geodesic completeness, but not global
168 4 LORENTZIAN DISTANCE
hyperbolicity, is assumed. In this section we give certain elementary prelim-
inaries which will be germane to Chapter 14 but which fit into the spirit of
this chapter. It is interesting that the reverse triangle inequality plays an im-
portant role, hence this material is decidedly non-Riemannian. Also the lower
semicontinuity of the Lorentzian distance function for an arbitrary space-time
(cf. Lemma 4.4) is useful here.
Note as an immediate first example that if a space-time (M,g) contains
a single maximal segment, then (M,g) cannot be totally vicious, since the
Lorentzian distance function is finite-valued on the particular segment while
totally vicious space-times satisfy d(p, q) = +m for all p, q in M. Newman
(1990) noted the more interesting consequence that the existence of a maximal
timelike segment c : [a, b] -4 (M,g) implies that strong causality holds a t all
points of c((a, b)). Since strong causality is an open condition [cf. Penrose
(1972, p. 30)], this thus yields an open neighborhood U of c((a, b)) for which
strong causality at q is valid for all q in U . Hence, not only does the existence
of a local causality neighborhood in a strongly causal space-time guarantee
the local existence of maximal segments, but a kind of converse holds: the
existence of a maximal timelike segment implies that some local region of
(M,g) containing all interior points of the given maximal timelike segment
must be strongly causal.
For our use in Chapter 14, these basic consequences of the existence of
maximal segments will be treated in the present section. We begin with a
maximal null segment.
Lemma 4.36. Let c : [O, 11 -4 (M,g) be a maximal null geodesic segment,
and let I(c) denote the domain of c extended to be a future inextendible null
geodesic emanating from c(0). Then
(1) For any s, t with 0 5 s < t 5 1 and r E J+(c(s)) fl J-(c(t)), we have
hence T lies on c(I(c));
4.4 MAXIMAL GEODESIC SEGMENTS, LOCAL CAUSALITY 169
(2) For any p, q in J+(c(s)) n J-(c(t)) with p ,< q, we have d(p, q) = 0;
and
(3) Chronology holds at all points of J+(c(O)) n J- ( ~ ( 1 ) ) .
Proof. (1) By assumption, c(s) < r < c(t) and d(c(s), c(t)) = 0, so that the
reverse triangle inequality d(c(s), c(t)) 2 d(c(s), r ) +d(r, c(t)) implies equation
(4.5). Further, since c(s) < r < c(t) is assumed, there exist future causal curves
cl from c(s) to r and cz from r to c(t) which by (4.5) must both be maximal
null geodesic segments. Since c(s) and c(t) are not chronologically related,
the concatenation of cl and cz must constitute a single null geodesic by basic
causality theory [cf. Penrose (1972, Proposition 2.19)], whence r E c(I(c)).
(2) This is immediate from the reverse triangle inequality applied to
(3) Condition (3) now follows since if chronology fails to hold, then d(p,p) is
infinite which contradicts (2) applied with q = p.
We should caution that I(c) is not necessarily equal to [O, +m) (cf. Lemma
7.4). In the case of a maximal timelike segment, the reverse triangle inequality
yields the finiteness of Lorentzian distance from points in some neighborhood
of the segment despite the possible general lack of finiteness of distance for
chronologically related points (cf. Figure 4.2).
Similar arguments to those used in Lemma 4.36 yield the following finiteness
of the distance function in the causal hull of any causal maximal segment.
Lemma 4.37. Let c : [O,1] -, (M,g) be a causal maximal segment. Then
(1) Given any p, q in J+(c(O)) n J - ( ~ ( 1 ) ) with p 6 q, the distance d(p, q)
is finite.
(2) Chronology holds at all points of J+(c(O)) n J - ( ~ ( 1 ) ) .
Having dealt with chronology, let us now consider the stronger requirement
of causality. The cylinder M = S' x W with metric ds2 = dBdt contains closed
null geodesics c : [0, +m) -t M which satisfy d(c(O), c(t)) = 0 for all t 2 0.
Hence, the existence of a maximal null geodesic does not imply more than
local chronology.
170 4 LORENTZIAN DISTANCE
Lemma 4.38. Let c : [0, I] + (M, g) be a maximal timelike segment. Then
causality holds a t all points of c([O, 11).
Proof. Suppose that cl is a closed nonspacelike curve beginning and ending
at c(t1) with tl > 0. Since chronology holds at c(t1) by Lemma 4.37, cl must
be a maximal null segment (cf. Corollary 4.14). Consider the composite causal
curve y = (c I [0, tl]) * cl from c(O) to c(tl). Since 7 is a causal curve from
c(0) to c(tl) which is a timelike followed by a null geodesic, first variation
"rounding the corner" arguments produce a causal curve yl from c(0) to c(tl)
which is longer than y, hence longer than ~110, t l] [cf. O'Neill (1983, p. 294)).
But this contradicts the maximality of c 1 [O, tl].
If t l = 0, apply the samc type of argument to the composition of the closed
null geodesic cl followed by c 1 [O,l].
Now we turn to a result with a somewhat more difficult proof first given
in Newman (1990, p. 166) and an alternate proof suggested in Galloway and
Horta (1995). In the course of the proof, it will be helpful to employ a charac-
terization of the failure of strong causality given by Kronheimer and Penrose
[cf. Penrose (1972, p. 31)j.
Lemma 4.39. Let p E (M, g). Then strong causality fails a t p if and only
if there exists a point q E J - ( p ) with q # p (which may be chosen arbitrarily
closely to p) such that x <( p and q << y together imply x << y for all x, y in
(M,g).
Proposition 4.40 [Newman (1990)j. Let c : [0, a] + (M, g) be a rnax-
imal timelike geodesic segment. Then for any to with 0 < to < a, strong
causality holds a t p = c(t0).
Proof. For convenience, we will assume that c is parametrized by unit speed
and also, given p = c(to), take q E J-(p) in Lemma 4.39 sufficiently close to p
that q E I+(c(O)). Now assume that strong causality fails to hold a t p.
We first need to establish that d(q,p) = 0, whence q is the initial point
of a maximal null segment from q to p. Suppose that d(q,p) > 0. Choose a
sequence {y,) C I+(q) with y, --, q and a sequence {t,) with 0 < t, < to
and t, -4 to. Put x, = c(t,), whence {x,) C I-(p) and x, -+ p. By the
4.4 MAXIMAL GEODESIC SEGMENTS, LOCAL CAUSALITY 171
lower semicontinuity of distance at (q,p), we have d(y,, x,) > 0 for some n
sufficiently large. Hence y, << x,. On the other hand, by Lemma 4.39 we
also have x, << y,. Thus x, << x,, which contradicts Lemma 4.37-(2). Thus
d(q,p) = 0 and the geodesic X from q to p is a maximal null segment.
We now show how a timelike curve from c(O) to c(a) of length greater than
a may be constructed, contradicting the maximality of the timelike segment c.
Consider first the concatenation ,O = X * c I [to, a] which is a future causal curve
from q to c(a). Since /3 consists of a null followed by a timelike geodesic and
L(0) = L(c I [to, a]) = a - to, rounding the corner at p produces a causal curve
from q to c(a) of length greater than L(0) = a - to. (Take a convex normal
neighborhood V centered at p, and join a point close to p on the null geodesic
X to c(to + 6), for 6 sufficiently small, by a timelike geodesic segment lying in
V.) Hence, there exists a constant E > 0 such that d(q, c(a)) > (a - to) + 46.
By Lemma 4.4, we may find a neighborhood U 5 I'(c(0)) of q such that
for every q' in U. Choose n sufficiently large that y, E U and also that
x, = c(t,) satisfies L(c 1 [0, t,]) = d(c(O), c(t,)) = t, 2 to - E. Fixing this n,
let cl be a future timelike curve from x, to y, guaranteed by Lemma 4.39.
Also given y,, in view of inequality (4.6) we may find a causal curve c2 from
y, to c(a) with L(cz) 2 a - to + 26. Now let y = c 1 jO, t,] * cl * c2, which is a
future causal curve from c(0) to c(a). On the other hand, we have the length
estimate
L(r) 1 L(c I [O, t,l) + L(cz)
Z (to - E) + (a - to + 26)
But this inequality contradicts the maximality of the timelike segment c.
Hence, strong causality must hold at c(to). 0
CHAPTER 5
EXAMPLES OF SPACE-TIMES
In this chapter we present a variety of examples of space-times. Some
of these space-times are important for physical as well as mathematical rea-
sons. In particular, Minkowski space-time, Schwarzschild space-times, Kerr
space-times, and Robertson-Walker space-times all have significant physical
interpretations.
Minkowski space-time is simultaneously the geometry of special relativ-
ity and the geometry induced on each fixed tangent space of an arbitrary
Lorentzian manifold. Thus Minkowskian geometry plays the same role for
Lorentzian manifolds that Euclidean geometry plays for Riemannian mani-
folds. Minkowski space-time is sometimes called fE~t space-time. But more
generally, any Lorentzian manifold on which the curvature tensor is identically
zero is flat.
The Schwarzschild space-times represent the spherically symmetric, empty
space-times outside nonrotating, spherically symmetric bodies. Since suns
and planets are assumed to be slowly rotating and approximately spherically
symmetric, the Schwarzschild space-times may be used to model the gravita-
tional fields outside of these bodies. These space-times may also be used to
model the gravitational fields outside of dead (i.e., nonrotating) black holes.
The usual coordinates for the Schwarzschild solution outside a massive body
are ( t , r, 0, &), where t represents a kind of time and r represents a kind of
radius [cf. Sachs and Wu (1977a, Chapter 7)]. This metric has a special radius
r = 2m associated with it. Points with r = 2m correspond to the surface of a
black hole. I t was once thought that the metric was singular at r = 2m, but
it is now known that the usual form of the Schwarzschild metric with r > 2m
may be analytically extended to points with 0 < r < 2m. In fact, there is a
174 5 EXAMPLES OF SPACE-TIMES
maximal analytic extension of Schwarzschild space-time [cf. Kruskal (1960)l
which contains an alternative universe lying on the "other side" of the black
hole.
The gravitational fields outside of rotating black holes apparently corre-
spond to the Kerr spacetimes [cf. Hawking and Ellis (1973, pp. 161, 331),
Carter (1971b), O'Neill (1995)j. These spacetimes represent stationary, ax-
isymmetric metrics outside of rotating objects. The Kerr and Schwarzschild
space-times are asymptotically flat and correspond to universes which are
empty, apart from one massive body. Thus while these metrics may be rea-
sonable models near a given single massive body, they cannot be used as large
scale models for a universe with many massive bodies.
The usual "big bang" cosmological models are based on the Robertson-
Walker space-times. These space-times are foliated by a special set of space-
like hypersurfaces such that each hypersurface corresponds to an instant of
time. The isometry group I ( M ) of a Robertson-Walker space-time (M,g)
acts transitively on these hypersurfaces of constant time. Thus Robertson-
Walker universes are spatially homogeneous. Furthermore, they are spatially
isotropic in the sense that for each p E M, the subgroup of I ( M ) fixing p is
transitive on the directions at p which are tangential to the hypersurface of
constant time through p. In our discussion of Robertson-Walker space-times,
we will use Lorentzian warped products Mo x f H , as described in Section
3.6. The cosmological assumptions made about Robertson-Walker universes
imply that (H, h) is an isotropic Riemannian manifold. Hence the classifica-
tion of two-point homogeneous Riemannian manifolds yields a classification
of all Robertson-Walker space-times. We also show how the results of Sec-
tion 3.6 may be specialized to construct Lie groups with bi-invariant globally
hyperbolic Lorentzian metrics.
5.1 Minkowski Space-time
Minkowski space-time is the manifold M = Rn together with the metric
5.1 MINKOWSKI SPACE-TIME
FIGURE 5.1. Let (M, g) be Minkowski space-time. The null cone
at p has a future nappe and a past nappe. The future [respectively,
past] nappe is also the horismos E+(p) [respectively, E-(p)] of p.
The chronological future I f (p) is an open convex set bounded by
E+(p). In more general space-times, I+(p) may fail to be convex
but is always open.
This space-time is time oriented by the vector field d/dxl. It is also glob-
ally hyperbolic and hence satisfies all of the causality conditions discussed in
Section 3.2.
The geodesics of Minkowski space-time are just the straight lines of the
underlying Euclidean space Rn. The affine parametrizations of these geodesics
in Minkowski space are even proportional to the usual Euclidean arc length
parametrizations in Rn. The null geodesics through a given point p in Minkow-
ski space form an elliptic cone with vertex p. The future directed null geodesics
starting at p thus form one nappe of the null cone of p. This nappe forms the
boundary in Rn of an open convex set which is exactly the chronological future
I+(p) of p. In Minkowski space, the causal future J+(p) of p is the closure of
5 EXAMPLES OF SPACGTIMES
FIGURE 5.2. The unit sphere K ( p , 1 ) corresponding to p is half of
a hyperboloid of two sheets. It is not compact, and p does not lie
in the convex open set bounded by K ( p , 1) .
I + ( p ) . The future horismos E f ( p ) = J + ( p ) - I + ( p ) is the nappe of the null
cone of p corresponding to the future (cf. Figure 5.1).
Minkowski space-time is a Lorentzian product (i.e., a warped product in
the sense of Definition 3.51 with f = 1). If W is given the negative definite
5.1 MINKOWSKI SPACE-TIME 177
FIGURE 5.3. Two-dimensional Minkowski space-time with one
point q removed is shown. The future horismos E + ( p ) of p is an
"L" shaped figure consisting of a half-closed line and a half-open
line segment. The causal future J + ( p ) is the union of I t ( p ) and
E+(p) . Notice that J f (p) is not a closed set nor is J+(p) equal to
the closure of If ( p ) .
metric -dt2 and Wn-' is given the usual Euclidean metric go, then ( R n =
R x Rn-l, -dt2 @go) is the n-dimensional Minkowski space-time.
Consider two points p = (pl ,p2, . .. ,pn) and q = ( q l , 92,. ... qn) in Minkow-
ski space-time. The chronological relation p << q holds whenever pl < ql and
( P I - q ~ ) ~ > (p2 - q2)2 i- . . - + (p , - qn)2 in R. If p << q, then the distance
from p to q is given by
n
( P I - q d 2 - ) i = 2 (pi - qi)2
The "unit sphere" in Minkowski space-time centered a t p is then K ( p , 1) =
! { q E A4 : d(p , g ) = 1) . However, this set is actually one sheet of a hyperboloid
1 of two sheets (ct Figure 5.2). t
. .
5 EXAMPLES OF SPACE-TIMES 5.2 SCHWARZSCHILD .4ND KERR SPACE-TIMES
i +
/--------
i- FIGURE 5.5. The Penrose diagram for Minkowski space-time is
i- shown.
FIGURE 5.4. Minkowski space-time is conformal to the open set Minkowski space-time and many other important space-times may be r e p
enclosed by the two null cones indicated. The vertices i+ and i - cor- resented by Penrose diagrams. A Penrose diagram is a two-dimensional r e p
respond to timelike infinity. All future directed timelike geodesics resentation of a spherically symmetric space-time. The radial null geodesics
go from i- to i+. The sets 3+ and 3- represent future and past are represented by null geodesics a t 145O. Dotted lines represent the origin
null infinity. Topologically, 3+ and 3- are each W x Sn-2. The (T = 0) of polar coordinates. Points corresponding to smooth boundary points
intersection of the two null cones is a set which is identified to a (cf. Section 12.5) which are not singularities are represented by single lines.
single point iO. The point i0 is called spacelike infinity. Double lines represent irremovable singularities (Figure 5.5; cf. Figure 4.2 of a Reissner-Nordstrom space-time with e2 = m2 for an example).
If we remove a point from Minkowski space-time, then it is no longer 5.2 Schwarzschild and Kerr Space-times
causally simple and hence no longer globally hyperbolic (cf. Figure 5.3).
I t is possible to conformally map all of Minkowski space-time onto a small In this section we describe the four-dimensional Schwarzschild and Kerr
open set about the origin. This is illustrated in Figure 5.4 [cf. Penrose (1968, solutions to the Einstein equations. Let R4 be given coordinates (t, r, 8, +),
p. 178), Hawking and Ellis (1973, p. 123)j. where (r, B,4) are the usual spherical coordinates on R3. Given a positive
180 5 EXAMPLES O F SPACE-TIMES
constant m, the exterior Schwarzschild space-time is defined on the subset
r > 2m of W4, a subset which is topologically W2 x S2. The Schwarzschild
metric for the region r > 2m is given in (t, r, 8 ,4) coordinates by the formula
Each element of the rotation group SO(3) for W3 induces a motion of the
Schwarzschild solution. Given qb E S0(3) , a motion ?C, of Schwarzschild space-
time may be defined by setting q ( t , r,$, 4) = (t, $(T, @, $)). Thus a t a fixed
instant t in time, the exterior Schwarzschild space-time is spherically symmet-
ric. The metric for this space-time is also invariant under the time translation
t -+ t + a. The coordinate vector field a/at is a timelike Killing vector field
which is a gradient, and the metric is said to be static. This space-time is
also Ricci flat (i.e., Ric = 0). Using the Einstein equations (cf. Chapter 2),
it follows that the energy-momentum tensor for the exterior Schwarzschild
space-time vanishes. Thus this space-time is empty.
The exterior Schwarzschild space-time may be regarded as a Lorentzian
warped product (cf. Section 3.6). For let M = {(t, r ) E W2 : r > 2m) be given
the Lorentzian metric
and let H = S2 be given the usual Riemannian metric h of constant sectional
curvature one induced by the inclusion S2 -+ W3. Define f : M --+ W by
f ( t , r ) = r2. Then (M x f H,ij) is the exterior Schwarzschild space-time,
where i j = g $ f h.
Physically, the exterior Schwarzschild solution represents the gravitational
field outside of a nonrotating spherically symmetric massive object. Compar-
ison with the Newtonian theory [cf. Einstein (1916, p. 819), Pathria (1974,
p. 217)] shows that m can be identified with the gravitational mass of the
massive body. The solution is not valid in the interior of the body.
The above form of the exterior Schwarzschild metric appears to have a
singularity at r = 2m. However, this is not a true singularity; the exterior
5.3 SPACES O F CONSTANT CURVATURE 181
Schwarzschild solution may be analytically continued across the surface given
by the equation r = 2m.
Kruskal(1960) investigated the maximal analytic extension of Schwarzschild
space-time. Suppressing 8 and 4, the following two-dimensional representation
of this maximal extension may be given (cf. Figure 5.6).
The gravitational field outside of a rotating black hole will not correspond
to the Schwarzschild solution. The generally accepted solutions of the Einstein
equations for rotating black holes are Kerr solutions. In Boyer and Lindquist
coordinates (t, r, 8 ,4 ) the Kerr metrics are given by [cf. Hawking and Ellis
(1973, p. 161), O'Neill (1995)l
where p2 = r2 + a2 cos2 0 and A = r2 - 2mr + a2. The constant m represents
the mass, and the constant m a represents the angular momentum of the black
hole [cf. Boyer and Price (1965), Boyer and Lindquist (1967)l. Tomimatsu
and Sato (1973) have given a series of exact solutions which include the Kerr
solutions as special cases.
5.3 Spaces of Constant Curva tu re
It is known that two Lorentzian manifolds of the same dimension which have
constant sectional curvature K are locally isometric [cf. Wolf (1974, p. 69)].
Thus any Lorentzian manifold of constant sectional curvature zero is locally
isometric to Minkowski space-time. In this section we will consider Lorentzian
model spaces which have constant nonzero sectional curvature.
We first define W: to be the standard semi-Euclidean space of signature
(-, . . , , -, +, . . . , +), where there are s negative eigenvalues and n - s positive
eigenvalues. Hence the semi-Euclidean metric on W: is given by
5 EXAMPLES OF SPACCTIMES
FIGURE 5.6. The Kruskal diagram for the maximal analytic ex-
tension of the exterior Schwarzschild space-time is shown. The ex-
tended space-time is the connected nonconvex region I U I1 U I' U 11'
bounded by the hyperbola corresponding to r = 0. The points of
this hyperbola are the true singularities of this space-time. The
lines a t f 45' separate the space-time into four regions. Region
I corresponds to the exterior Schwarzschild solution. Region I1 is
the "interior" of a nonrotating black hole. Region I' is isometric to
region I and corresponds to an alternative universe on the "other
side" of the black hole. There is no nonspacelike curve from region
I to region 1'.
5.3 SPACES OF CONSTANT CURVATURE 183
In particular, IR; is n-dimensional Minkowski space-time. We also define for
r > 0 [cf. Wolf (1974, Section 5.2)]
and
Topologically, S;" is R' x Sn-I and Hi' is S' x Rn-' [cf. Wolf (1974, p. 68)).
The semi-Euclidean metric on w;" (respectively, R;+') induces a Lorentzian
metric of constant sectional curvature K = r-' (respectively, K = -r-2) on
Sy (respectively, HT). The space-time Sf. is a Lorentzian analogue of the
usual Riemannian spherical space of radius r and has positive curvature T - ~ .
The universal covering manifold H;" of H;" is topologically Rn and is thus
a Lorentzian analogue of the usual Riemannian hyperbolic space of negative
curvature -rW2.
Definition 5.1. (de Sitter and anti-de Sitter Space-times) Let S;" and
Hf. be defined as above. Then SF- is called de Sitter space-time, and the
universal covering H r of H r is called (universal) anti-de Sitter space-time.
Remark 5.2.
(1) Sf . is simply connected for n > 2 and .rrl(S:) = Z.
(2) Sf. is globally hyperbolic and geodesically complete.
(3) H f . is nonchronological since y (t) = (T cos t , r sin t , 0 , . . . ,0) is a closed
timelike curve. Also @, while strongly causal, is not globally hyper-
bolic.
The de Sitter space-time represented in Figure 5.7 may be covered by global
coordinates (t,x,B,$) with -m < t < cm, 0 < x < T , 0 < B 5 T , and
0 < $ 5 27r. Here t is the coordinate on IR and (x, B,$) represent coordinates
on S 3 Icf. Hawking and Ellis (1973, pp. 125, 136)]. In these coordinates, the
metric for de Sitter space-time of constant positive sectional curvature l / r 2 is
given by
184 5 EXAMPLES OF SPACGTIMES
FIGURE 5.7. The n-dimensional de Sitter space-time with positive
constant sectional curvature r-2 is the set -x12 +x22 +- . . +x:+~ =
r2 in Minkowski space-time R;'~. The geodesics of S," lie on the
intersection of S," with the planes through the origin of R;".
This may be reinterpreted as a Lorentzian warped product metric (cf. Section
3.6) as follows. Let f : W + (0, m) be given by f (t) = r2 cosh2(t/r), and
let S3 be given the usual complete Riemannian metric of constant sectional
curvature one. Then the de Sitter space-time described in local coordinates
as above is the warped product (W x S3, -dt2 @ fh).
Universal anti-de Sitter space-time of curvature K = -1 may be given
coordinates (t', r , 8,4) for which the metric has the form
[cf. Hawking and Ellis (1973, pp. 131, 136)). Regarding -(dt')2 as a nega-
tive definite metric on W and dr2 + sinh2(r)(a2 + sin2 B as the complete
5.4 ROBERTSON-WALKER SPACGTIMES 185
Riemannian metric h of constant negative sectional curvature -1 on the hy-
perbolic three-space H = W3, this space-time may be represented as a warped
product of the form (R x f H, - f dt2@h), where the warping function is defined
on the Riemannian factor H (cf. Remark 3.53).
5.4 Robertson-Walker Space-times
In this section we discuss Robertson-Walker space-times in the framework
of Lorentzian warped products. These space-times include the Einstein static
universe and the big bang cosmological models of general relativity. In order
to give a precise definition of a Robertson-Walker space-time, it is necessary
to first recall some concepts from the theory of two-point homogeneous Rie-
mannian manifolds and isotropic Riemannian manifolds.
Let (H, h) be a Riemannian manifold. Denote by I ( H ) the isometry group
of (H, h) and by & : H x H -+ R the Riemannian distance function of (H, h).
Definition 5.3. (Homogeneous and Two-Point Homogeneous Manifolds)
The Riemannian manifold (H, h) is said to be homogeneous if I ( H ) acts tran-
sitively on H , i.e., given any p,q f H , there is an isometry 4 E I ( H ) with
4(p) = q. Further, (H, h) is said to be two-point homogeneous if given any
PI, ql, p2,92 E H with do(~1,ql) = do(p2, 4, there is a n isometry 4 E I ( H )
with +(PI) = p2 and 4(ql) = 92.
Since it is possible to choose p, = q, for z = 1 ,2 , a two-point homogeneous
Riemannian manifold is also homogeneous. Two-point homogeneous spaces
were first studied by Busemann (1942) in the more general setting of locally
compact metric spaces. Wang (1951, 1952) and Tits (1955) classified two-point
homogeneous Riemannian manifolds.
Notice that in Definition 5.3 it is not required that (H, h) be a complete
Riemannian manifold. Nonetheless, homogeneous Riemannian manifolds have
the important basic property of always being complete.
Lemma 5.4. If (H, h) is a homogeneous Riemannian manifold, then (H, hf
is complete.
186 5 EXAMPLES OF SPACE-TIMES
Proof. By the Hopf-Rinow Theorem, it suffices to show that ( H , h ) is
geodesically complete. Thus suppose that c : [a, 1) -+ H is a unit speed
geodesic which is not extendible to t = 1. Choosing any p E H , we may find
a constant a > 0 such that any unit speed geodesic starting at p has length
1 2 a. Set 6 = min{a/2, ( 1 - a ) / 2 ) > 0. Since isometries preserve geodesics, it
follows from the homogeneity o f ( H , h ) that any unit speed geodesic starting
at c(1 - 6) may be extended to a geodesic o f length 1 2 26. In particular,
c may be extended to a geodesic c : [a, 1 + 6 ) -4 H , in contradiction to the
inextendibility o f c to t = 1. 0
Remark 5.5. It is important to note that the conclusion o f Lemma 5.4 is
false in general for homogeneous Lorentzian manifolds [cf. Wolf (1974, p. 95),
Marsden (1973)l.
W e now recall the concept o f an isotropic Riemannian manifold. Given
p 6 ( H , h ) , the isotropy group I p ( H ) o f ( H , h ) at p is the closed subgroup
I p ( H ) = {4 E I ( H ) : $(p) = p ) o f I ( H ) consisting of all isometries of ( H , h )
which fix p. Given any 4 E I p ( H ) , the differential #,, maps TpH onto TpH
since 4 (p ) = p. As h(4,v, &v) = h(v , v ) for any v E T p H , the differential q5,,
also maps the unit sphere SpH = {v E T,M : h(v , v ) = 1) in T,H onto itself.
Definit ion 5.6. (Isotropic Riemannian Manifold) A Riemannian man-
ifold ( H , h) is said to be isotropic at p i f I p ( H ) acts transitively on the unit
sphere SpH o f T p H , i.e., given any v , w E S,H, there is an isometry 4 E I,(H)
with 4,v = w. The Riemannian manifold ( H , h ) is said to be isotropic i f it is
isotropic at every point.
W e now show that the class o f isotropic Riemannian manifolds coincides
with the class of two-point homogeneous Riemannian manifolds [cf. Wolf (1974,
p 289)].
Proposit ion 5.7. A Riemannian manifold ( H , h ) is isotropic iff it is two-
point homogeneous.
Proof. Recall that & denotes the Riemannian distance function of ( H , h).
First suppose that ( H , h ) is isotropic. Then for each p E H and each inex-
tendible geodesic c : (a , b) -+ H with c(0) = p, there is an isometry 4 E I,(H)
5.4 ROBERTSON-WALKER SPACE-TIMES 187
with $,cl(0) = -cJ(0). Hence by geodesic uniqueness, g!~(c(t)) = c(- t ) for
all t E (a , 6). This implies that the length of c 1 (a , 0] equals the length o f
c 1 [0,6). Since p may be taken to be any point o f the geodesic c, it follows
that a = -m and 6 = f m . Thus ( H , h ) is geodesically complete. Hence by
the Hopf-Rinow Theorem, given any two points pl, p2 E H , there is a geodesic
segment Q o f minimal length &(pl,pz) from pl to p2. Let p be the midpoint
o f Q. As ( H , h ) is isotropic, there is an isometry # E I,(H) which reverses Q.
I t follows that # ( P I ) = p2. Hence ( H , h ) is homogeneous. It remains to show
that i f P I , ql, p2, q2 E H with & ( P I , ql) = do(p2, q2) > 0 are given, we may
find an isometry 4 E I ( H ) with # ( P I ) = p2 and #(ql) = q2. Choose minimal
unit speed geodesics cl from pl to ql and c2 from p2 to 92. Since ( H , h ) is
homogeneous, we may first find an isometry 1I, E I ( H ) with lI,(pl) = p2. Then
as ( H , h ) is isotropic, we may find 77 E I,,(H) with v*((+ o c1)'(0)) = ~ ' ( 0 ) .
I t follows that 4 = 71 0 $ is the required isometry.
Now suppose that ( H , h ) is two-point homogeneous. Fix any p E M , and
let U be a convex normal neighborhood based at p. Choose a > 0 such that
expp(v) E U for all v E T,H with h(v ,v) 5 a . Now let v? w E TpH be any pair
o f nonzero tangent vectors with h(v , v ) = h(w, w) < a/2. Set ql = expp v and
q2 = exp, w. Then ql, q2 E U and d(p, q l ) = = = d(p, q2).
Since ( H , h ) is two-point homogeneous, there is thus an isometry $ E I ( H )
with 4 (p ) = p and $(ql) = q2. I t follows that $,v = w. The linearity o f
qap : TpH --t TpH for any 11 E I p ( H ) then implies that I,(H) acts transitively
on SpH. Thus ( H , h ) is isotropic at p. As the same argument clearly holds for
all p E H , it follows that ( H , h ) is isotropic as required.
Corollary 5.8. Any isotropic Riemannian manifold is homogeneous and
complete.
R e m a r k 5.9. ( 1 ) The two-point homogeneous Riemannian manifolds are
well known [cf. Wolf (1974, pp. 290-296)]. In particular, the odd-dimensional
two-point homogeneous (hence isotropic) Riemannian manifolds are just the
odd-dimensional Euclidean, hyperbolic, spherical, and elliptic spaces [cf. Wang
(1951, p. 47311.
(2) Astronomical observations indicate that the spatial universe is approxi-
188 5 EXAMPLES OF SPACE-TIMES
mately spherically symmetric about the earth. This suggests that the spatial
universe should be modeled as a three-dimensional isotropic Riemannian mani-
fold. Hence the possibilities are limited to the Euclidean, hyperbolic, spherical,
and elliptic spaces. However, if one only assumes local isotropy, there are more
possibilities [cf. Misner, Thorne, and Wheeler (1973, pp. 713-725)].
(3) Any three-dimensional isotropic Riemannian manifold (H, h) has constant
sectional curvature, and also dim I ( H ) = 6 [cf. Walker (1944)l.
We are now ready to define Robertson-Walker spacetimes using Lorentzian
warped products and isotropic Riemannian manifolds.
Definition 5.10. (Robertson-Walker Space-time) A Robertson-Walker
space-time (M, g) is any Lorentzian manifold which can be written in the form
of a Lorentzian warped product (Mo x f H,g) with Mo = (a ,b) , -co 5 a < b 5 -too, given the negative definite metric -dt2, with (HI h) an isotropic
Riemannian manifold, and with warping function f : Mo -+ (0, m).
In the notation of Section 3.6, we thus have g = -dt2 @ f h and Mo x f H
is also topologically the product Mo x H. Letting du2 denote the Riemannian
metric h for H and defining S(t) = m, the Lorentzian metric g for Mo x f H
may be rewritten in the more familiar form
The map n. : Mo x f H --+ R given by x(t ,z) = t is a smooth time function
on Mo x f H so that the Lorentzian manifold Mo x f H , of Definition 5.10
actually is a (stably causal) spacetime. Also each level surface n.-l(c) of the
map T : Mo x f H --+ Mo c R is an isotropic Riemannian manifold which is
homothetic to (HI h). Furthermore, the isometry group I ( H ) of (HI h) may be
identified with a subgroup T(H) of I(Mo x f H ) as follows. Given 4 E I(H),
defbe E F(H) by $(T, h) = (r,$(h)) for all (r, h) E Mo x H. With this
definition, F(H) restricted to the level surfaces s-'(c) of s acts transitively on
each level surface.
Since all isotropic Riemannian manifolds are complete, Theorem 3.66 im-
plies that all Robertson-Walker spacetimes are globally hyperbolic. From
5.4 ROBERTSON-WALKER SPACE-TIMES 189
Theorem 3.69 we also know that every level surface n.-'(c) = {c) x H is a
Cauchy surface for Mo x f H.
Next to Minkowski space Rn = R x Rn-' itself, the Einstein static universe
is the simplest example of a Robertson-Walker space-time.
Example 5.11. (Einstein Static Universe) Let Mo = R with the neg-
ative definite metric -dt2, and let H = Sn-' with the standard spherical
Riemannian metric. I f f : R -+ (0, m ) is the trivial warping function f = 1,
then the product Lorentzian manifold M = Mo x H = Mo x f H is the n-
dimensional Einstein static universe. If n = 2, then M is the cylinder R x S'
with Aat metric -dt2 + do2. If n 2 3, then this metric for M = R x Sn-' is
not flat since Sn-' has constant positive sectional curvature K = 1.
For the rest of this section we restrict our attention to four-dimensional
Robertson-Walker spacetimes. By Remark 5.9, these are warped products
Mo x f H , where (H, h) is Euclidean, hyperbolic, spherical, or elliptic of di-
mension three. In the first two cases, H is topologically R3. In the third case
H = S3, and in the last case H is the real projective three-space RP3. We
thus have the following
Corollary 5.12. All four-dimensional Robertson-Walker spacetimes are
topologically either R4, R x S3, or R x RP3.
Also by Remark 5.9, the sectional curvature K of (H, h) is constant. If K is
nonzero, the metric may be rescaled to be of the form ds2 = -dt2 + S2(t)du2
on M so that K is either identically $1 or -1. This is the form of the metric
usually studied in general relativity.
In physics, cosmological models are built from four-dimensional Robertson-
Walker space-times assumed to be filled with a perfect fluid. The Einstein
equations (cf. Chapter 2) are then used to find the form of the above warping
function S2(t). Among the models this technique yields are the big bang
cosmological models [cf. Hawking and Ellis (1973, pp. 134-138)]. These models
depend on the energy density p and pressure p of the perfect fluid a s well as
the value of the cosmological constant A in the Einstein equations. In the big
bang cosmological models, the inextendible nonspacelike geodesics are all past
190 5 EXAMPLES O F SPACE-TIMES
incomplete. The stability of this incompleteness under metric perturbations
will be considered in Section 7.3. Astronomical observations of clusters of
galaxies indicate that distant clusters of galaxies are receding from us. This
expansion of the universe suggests the existence of a "big bang" in the past and
also suggests that the universe is a warped product with a nontrivial warping
function rather than simply a Lorentzian product. Observations of blackbody
radiation support these ideas [cf. Hawking and Ellis (1973, Chapter lo)].
5.5 Bi-Invariant Lorentzian Metrics o n Lie Groups
The purpose of this section is to show how Theorems 3.67 and 3.68 of
Section 3.6 may be used to construct a large class of Lie groups admitting
globally hyperbolic, bi-invariant Lorentzian metrics.
We first summarize some basic facts from the elementary theory of Lie
groups. Details may be found in a lucid exposition by Milnor (1963, Part IV)
or at a more advanced level in Helgason (1978, Chapter 2). A Lie group is a
group G which is also an analytic manifold such that the mapping (g, h) -+
gh-I from G x G -+ G is analytic. This multiplication induces left and right
translation maps L,, R, for each g E GI given respectively by L,(h) = gh
and Rg(h) = hg. A Riemannian or Lorentzian metric ( , ) for G is then said
to be left invariant (respectively, right invariant) if (L,*v, L,,w) = (v, w)
(respectively, (Rgtv, R,,w) = (v, w)) for all g E G and v, w E TG. A metric
which is both left and right invariant is said to be bi-invariant. By an averaging
procedure involving the Haar integral, any compact Lie group may be given
a bi-invariant metric [cf. Milnor (1963, p. 112)]. In fact, the Haar integral
may be used to produce a bi-invariant Riemannian metric for G from any left
invariant Riemannian metric for G. Any Lie group may be equipped with a
left invariant Riemannian (or Lorentzian) metric by starting with a positive
definite inner product (respectively, inner product of signature n - 2) ( , )I, on the tangent space T,G to G at the identity element e E G, then defining
( , )I,:T,GxT,G+ W by
5.5 LIE GROUPS: BI-INVARIANT LORENTZIAN METRICS 191
Thus any compact Lie group is furnished with a large supply of bi-invariant
Riemannian metrics.
On the other hand, while (5.1) equips any Lie group with left-invariant
Lorentzian metrics, the standard Haar integral averaging procedure used for
Riemannian metrics fails to preserve signature (-, +, . - . , +), so it cannot be
used to convert left-invariant Lorentzian metrics into bi-invariant Lorentzian
metrics.
But we will see shortly that a large class of bi-invariant Lorentzian metrics
may be constructed for noncompact Lie groups of the form R x G, where G is
any Lie group admitting a bi-invariant Riemannian metric.
Before giving the construction, we need to discuss product Lie groups briefly.
Let G and H be two Lie groups. The product manifold G x H is then turned
into a Lie group by defining the multiplication by
It is immediate from (5.2) that if u = (g, h) E G X H , then the translation maps
L,, R, : G x H -+ G x H are given by L, = (L,, Lh), and R, = (R,, Rh),
i.e., L,(gl, hl) = (L,gl, Lhhl)> etc. Recall that T,(G x H) 2 T,G x ThH.
It is straighforward to check that for any u E G x H and any tangent vector
t = (v, w) f T,(G x H) 2 TgG x ThH, one has
and
Now if ( , )l is a Lorentzian metric for G and ( , j2 is a Riemannian
metric for H , the product metric (( , )) = ( , )1 @ ( , )2 is a Lorentzian
metric for G x H . Explicitly, recalling Definition 3.51, we have for tangent
vectors El = (vl, wl) and t2 = (vz, w2) in T,(G x H ) the formula
X ((Ei,E2)) = (v1,~2)1 + (~1,w2)2. i
It is then immediate from (5.3) and (5.4) that if ( , is a bi-invariant I
Lorentzian metric for G and ( , )2 is a bi-invariant Riemannian metric for i
H , then (( , )) is a bi-invariant Lorentzian metric for G x H. To summarize, P
192 5 EXAMPLES OF SPACE-TIMES
Proposition 5.13. Let (G, ( , be a Lie group equipped with a bi-
invariant Lorentzian metric, and let (H, ( , )2) be a Lie group equipped with
a bi-invariant Riemannian metric. Then the product metric (( , )) = ( , ) I @
( , )z is a bi-invariant Lorentzian metric for the product Lie group G x H.
Hence (G x H, (( , ))) is a Lorentzian symmetric space and, in particular, is
geodesically complete.
Proof. I t is only necessary to prove the last statement which is a standard
fact in Lie group theory. Recall that we must show that for each a E G x H, there exists an isometry I,, : G x H -+ G x H which fixes a and reverses the
geodesics through a. That is, if y is a geodesic in G x H with y(0) = u, we
must show that I,(y(t)) = y(-t) for all t. This is equivalent to showing that
I,. : T,(G x H ) -+ T,,(G x H ) is the map I,,(E) = -[ and also implies that
Is2 = Id.
We will follow the proof given in Milnor (1963, pp. 109, 112). First, if we
denote the identity element of G x H by e and define a map I, : G x H -t G x H
by Ie(a) = u-' , then I,, : T,(G x H ) -+ Te(G x H ) is given by I,-(v) = -v.
Thus I,. : Te(G x H ) -+ T,(G x H ) is an isometry of T,(G x H). To see that
I,. is an isometry of any other tangent space T,(G x H ) + To-I(G x H ) and
hence that I, : G x H 4 G x H is an isometry, we simply note that
Since (( , )) is bi-invariant, all left and right translation maps are isometries.
Then as
Ie. I,, = R,;l 1,Ie. IeLu;l 1, and I,, 1, is an isometry of T,(G x H), it follows that I,, : T,(G x H ) --+
To-I(G x H ) is also an isometry. The map I, : G x H -+ G x H is thus the
required geodesic symmetry at e.
We define the geodesic symmetry I, for any a E G by setting I,, =
R,I,R,-1. Since R, and R,-1 are isometries by the bi-invariance of (( , ))
and we have just shown that I, is an isometry, it follows that I, : G x H -,
G x H is an isometry, and obviously I,(a) = a since I,(h) = ah-'a. Finally,
5.5 LIE GROUPS: BI-INVARIANT LORENTZIAN METRICS 193
for any [ E T,,(G x H) we have
~ u . < = ~ u . (1,. ( ~ ~ ; l [ ) )
= R,,. (- R,:.<) since C E T, (G x H )
= -R,.R,;IJ
= - (R,R,,-I), ( = -[.
Thus I, reverses geodesics at a as required. We have therefore shown that
G x H is a symmetric space.
It may be shown that any symmetric space is geodesically complete as
follows. Let y be a geodesic in M, and set p = y(0). Supposing that q = y(A)
is defined, one may derive the formula [cf. Milnor (1963, p. log)]
provided that y(t) and y(t + 2A) are defined.
Thus if y is defined originally on an interval y : [O, X] -+ G x H , y may be
extended to a geodesic ;j: : [O,2X] -+ G x H by choosing q = $4'2) and putting
y(t) = I,Ip(y(t - A)) for t E [A, 24 . It is then clear that y may be defined on
(-co, co). Thus (G x H, (( , ))) is geodesically complete. 0
Now Proposition 5.13 has the apparent defect that the existence of Lie
groups (G, ( , ) I ) equipped with bi-invariant Lorentzian metrics is assumed.
It will now be shown how such Lie groups may be constructed by taking
products of the form (R x G, -dt2 @ ( , )) where (G, ( , )) is a Lie group
equipped with a Riemannian hi-invariant metric.
The Lie group structure on (R, -dt2) we will use is that induced by the
usual addition of real numbers. Accordingly, we will write (a, b) I-+ a + b for
the Lie group "multiplication" despite our use of the product notation above
for the group operation. Here R is the analytic manifold determined by the
chart t : IW -+ W, t(r) = r. Let a/at denote the corresponding coordinate vector
field on R. The left and right translation maps La, R, : R -+ lR are given by
L,(r) = a + r and R,(r) = r+a . It is easy to check that if v = X dl&/, E T,W, then L,*v and R,-v in T,+,(R) are given by L,*v = Ra-v = X d/&l,+,. Hence
194 5 EXAMPLES OF S P A C S T I M E S
-dt2(L,*v, La.v) = -dt2(~,-v , R,.v) = -A2 = -dt2(v,v) SO that -dt2 is left
and right invariant.
Let (G, ( , )) be a Lie group with a bi-invariant Riemannian metric. By
the proof given in Proposition 5.13, G is a complete symmetric space. Also
using (5.3) and (5.4), it is easily seen that the metric (( , )) = -dt2 @ ( , )
is a bi-invariant Lorentzian metric for W x G. (Here, if El = (A1 a/&],, vl)
and E2 = (Az dldtl, ,v2) with vl, v2 E TSG, the inner product ((El, &)) =
-A1A2 + (vl,v2).) Since (G, ( , )) is a complete Riemannian manifold, the
product (R x G, (( , ))) is globally hyperbolic by Theorem 3.67. We have
obtained
Theorem 5.14. Let (R, -dt2) be given the usual additive group structure
and let (G, ( , )) be any Lie group equipped with a bi-invariant Riemannian
metric. Then the product metric (( , )) = -dt2 @ ( , ) is a bi-invariant
Lorentzian metric for the product Lie group W x G. Thus (R x G, (( , ))) is
a geodesically complete, globally hyperbolic space-time.
Much research inspired by E. Cartan's work was done on Lie groups, ho-
mogeneous spaces, and symmetric spaces equipped with indefinite metrics be-
fore causality theory had assumed such a prominent role in general relativ-
ity. Thus most of this work was carried out not for Lorentzian metrics in
particular but rather for general semi-Riemannian metrics of arbitrary sig-
nature. Rather than attempting to give an exhaustive list of references, we
refer the reader to the bibliography in Wolf's (1974) text. Much of this re-
search has been concerned with the problem of classifying all geodesically
complete semi-Riemannian manifolds of constant curvature (the "space-form
problem"). Two papers dealing with semi-Ftiemannian Lie theory have been
written by Kulkarni (1978) and Nomizu (1979). Nomizu's paper deals specifi-
cally with Lorentzian metrics, considering the existence of constant curvature
left-invariant Lorentzian metrics on a certain class of noncommutative Lie
groups.
A more general treatment of the class of Lie groups which admit left invari-
ant Lorentzian metrics of constant sectional curvature was then given in Barnet
(1989). Also more recently, research in the Lie theory of semigroups, as re-
5.5 LIE GROUPS: BI-INVARIANT LORENTZIAN METRICS 195
ported in Hilgert, Hofmann, and Lawson (1989) or Lawson (1989), has sparked
renewed interest in the causality and differential geometry of Lorentzian Lie
groups on the part of the semigroup community. A representative research
paper where causality and Lie semigroups are considered is Levichev and
Levicheva (1992). Additional results for left invariant Lorentzian metrics on
Lie groups of dimension three have recently been obtained by Cordero and
Parker (1995b).
CHAPTER 6
COMPLETENESS AND EXTENDIBILITY
We mentioned in Chapter 1 that the Hopf-Rinow Theorem guarantees the
equivalence of geodesic and metric completeness for arbitrary Riemannian
manifolds. Further, either of these conditions implies the existence of min-
imal geodesics. That is, given any two points p,q E M, there is a geodesic
from p to q whose arc length realizes the metric distance from p to q. If M
is compact, it also follows from the Hopf-Rinow Theorem that all Riemann-
ian metrics for M are complete. In the noncompact case, Nomizu and Ozeki
(1961) established that every noncompact smooth manifold admits a complete
Riemannian metric. Extending their proof, Morrow (1970) showed that the
complete Riemannian metrics for M are dense in the compact-open topology
in the space of all Riemannian metrics for M [cf. Fegan and Millman (1978)l.
In the first three sections of this chapter we compare and contrast these
results with the theory of geodesic and metric completeness for arbitrary
Lorentzian manifolds. In Section 6.1 a standard example is given to show
that geodesic completeness does not imply the existence of maximal geodesic
segments joining causally related points. Then we recall that the class of
globally hyperbolic space-times possesses this useful property. In Section 6.2
we consider forms of completeness such as nonspacelike geodesic completeness,
bounded acceleration completeness (b. a. completeness), and bundle complete-
ness (b-completeness) that have been studied in singularity theory in general
relativity [cf. Clarke and Schmidt (1977), Ellis and Schmidt (1977)l. We also
state a corollary to Theorem 8 of Beem (1976a, p. 184) establishing the exis-
tence of nonspacelike complete metrics for all distinguishing space-times. In
Section 6.3 we discuss Lorentzian metric completeness and the finite compact-
ness condition.
198 6 COMPLETENESS AND EXTENDIBILITY
In the last three sections of this chapter we discuss extensions and local ex-
tensions of space-times. Since extendibility is related to geodesic completeness,
extendibility plays an important role in singularity theory in general relativity
[cf. Clarke (1973, 1975, 1976), Hawking and Ellis (1973), Ellis and Schmidt
(1977)l. In particular, one usually wants to avoid investigating space-times
which are proper subsets of larger space-times since such proper subsets are
always geodesically incomplete.
A space-time (M1,g') is said to be an extension of a given space-time
(M,g) if (M,g) may be isometrically embedded as a proper open subset of
(MI, g'). A space-time which has no extension is either said to be inextendible
[cf. Hawking and Ellis (1973)l or maximal [cf. Sachs and Wu (1977a, p. 29)).
A local extension is an extension of a certain type of subset of a given
space-time. In general, local inextendibility (i.e., the nonexistence of local
extensions) implies global inextendibility. Since questions of extendibility nat-
urally relate to the boundary of space-time, in Section 6.4 we briefly describe
the Schmidt b-boundary and the Geroch-Kronheimer-Penrose causal bound-
ary. In Section 6.5 two types of local extensions are defined and studied. If
a Lorentzian manifold has no local extensions of either of these two types,
it is shown to be inextendible. We also give a local extension of Minkowski
space-time which shows that while b-completeness forces a space-time to be
(globally) inextendible, b-completeness does not prevent a space-time from
having local extensions.
In Section 6.6 local extensions are related to curvature singularities. For
example, if (M,g) is an analytic space-time such that each timelike geodesic
y : [O, a) -+ M which is inextendible to t = a is either complete (in the sense
that a = m) or else corresponds to a curvature singularity, then (M,g) has
no analytic local b-boundary extensions. Also, the a-boundary of Scott and
Szekeres (1994) and its role in classifying singularities is discussed.
6.1 Existence of Maximal Geodesic Segments
The purpose of this section is twofold. First, we recall that for arbitrary
Lorentzian manifolds, geodesic completeness does not imply the existence of
6.1 EXISTENCE OF MAXIMAL GEODESIC SEGMENTS 199
FIGURE 6.1. The universal cover M = {(x,t) : -x /2 < r < 7r/2) of two-dimensional anti-de Sitter space is shown. The metric
is given by ds2 = sec2x (-dt2 + dx2). The points p and q are
chronologically related in M , yet no maximal timelike geodesic in
M joins p to q since all future directed timelike geodesics emanating
from p are focused at r.
maximal geodesic segments joining causally related pairs of points. Second,
we discuss the important and useful fact that distance realizing geodesics do
exist for the class of globally hyperbolic space-times.
The universal covering manifold (M, g) of two-dimensional anti-de Sitter
space provides an example that geodesic completeness does not imply that
every pair p, q E M with p << q may be joined by a timelike geodesic y with
L(y ) = d ( p , q) (cf. Figure 6.1). %call that if y is any future directed timelike
200 6 COMPLETENESS AND EXTENDIBILITY
curve from p to q with L(y) = d(p, q), then y may be reparametrized to a
timelike geodesic (cf. Theorem 4.13). Thus this same example shows that
geodesically complete space-times exist which contain points p << q such that
L(y) < d(p, 4) for all y f O,,,.
The space-time (M, g) may be represented by the strip M = {(x, t ) E R2 :
--7r/2 < x < r /2 ) in B2 with the Lorentzian metric ds2 = sec2 x (-dt2 + dx2)
[cf. Penrose (1972, p. 7)j. The points p and q in Figure 6.1 satisfy p (< q, yet
all future timelike geodesics emanating from p are focused again a t the future
timelike conjugate point T. Thus there is no timelike geodesic in M from p to
q. Hence there is no maximal timelike geodesic or maximal timelike curve from
p to q. Even more strikingly, it should be noted that an open set U C I+(p)
of points of the type of q as in Figure 6.1 may be found with the property
that none of the points of U are connected to p by any geodesic whatsoever,
despite the geodesic completeness of this space-time.
We now consider which space-times do have the property that every pair
of points p,q E M with q E J+(p) may be joined by a distance realizing
geodesic. If M = R2-((0, 0)) with the Lorentzian metric ds2 = dx2-dy2, then
p = (0, -1) and q = (0,l) are points in M with d(p, q) = 2 > 0 which cannot
be joined by a maximal timelike geodesic. [The desired geodesic would have
to be the curve y(t) = (O,t), -1 5 t 5 1, which passes through the deleted
point (0, O).] On the other hand, this space-time is chronological, strongly
causal, and stably causal. Thus it is reasonable to restrict our attention to the
class of globally hyperbolic space-times. For these space-times, Avez (1963)
and Seifert (1967) have shown that given any p, q E M with p < q, there is a
geodesic from p to q which maximizes arc length among all nonspacelike future
directed curves from p to q (cf. Theorem 3.18). In the language of Definition
4.10, this may be stated as follows.
Theorem 6.1. Let (M, g) be globally hyperbolic. Then given any p, q E M
with q E J+(p), there is a maximd geodesic segment y E i2,,,, i.e., a future
directed nonspacelike geodesic y from p to q with L(y) = d(p,q).
We sketch Seifert's (1967, Theorem 1) proof of this result [cf. Penrose (1972,
Chapter 6)]. Since (M,g) is globally hyperbolic, it may be shown that if
6.1 EXISTENCE O F MAXIMAL GEODESIC SEGMENTS 201
p < q, the nonspacelike path space a,,, is compact. On the other hand, since
(M,g) is strongly causal, the arc-length functional L : 52,,, --t R is upper
semicontinuous in the C0 topology (cf. Section 3.3). Thus there exists a curve
yo E a,,, with L(yo) = sup{L(y) : y E Q,,,). It follows from the variational
theory of arc length that if yo is not a reparametrization of a smooth geodesic,
a curve a E R,,, with L(a) > L(y) may be constructed, in contradiction.
Alternatively, if L(y0) = sup{L(y) : y E a,,,), then L(y0) = d(p,q) by the
definition of Lorentzian distance. Hence Theorem 4.13 implies that yo is, up
to reparametrization, a smooth geodesic.
In the case that p << q, the maximal curve yo may also be constructed using
the results of Section 3.3. Let h : M --t B be a globally hyperbolic time function
for (M, g). Choose to with h(p) < to < h(q). Then K = J+(p)fl J-(q)flh-'(to)
is compact, and any nonspacelike curve from p to q intersects K. By definition
of Lorentzian distance, we may find a curve y, E O,,, with
for each positive integer n. Let rn E x n K . Since K is compact, a subsequence
{T,(~)) converges to T E K. By Corollary 3.32, there is a nonspacelike limit
curve yo of the sequence {yn(j)) passing through T and joining p to q. Since
(M, g) is strongly causal, a subsequence of {y,(,)) converges to yo in the C0
topology by Proposition 3.34. Using Remark 3.35 and condition (6.1), we
obtain L(yo) > d(p, q). Hence by the definition of distance, L(yo) = d(p, q),
and yo may be reparametrized to a smooth geodesic by Theorem 4.13. If p ,< q
and d(p, q) = 0, we already know that there is a maximal null geodesic segment
from p to q by Corollary 4.14.
In connection with Theorem 6.1, it should be noted that global hyperbolicity
is not a necessary condition for the existence of maximal geodesic segments
joining all pairs of causally related points. For let M = {(x, y) E R2 : 0 < x < 10, 0 < y < 10) be equipped with the Lorentzian rr~etric it inherits as an
open subset of Minkowski space. Since the geodesics in M are just Euclidean
straight line segments, it is readily seen that maximal geodesics exist joining
any pair of causally related points. However, if p = (1 , l ) and q = (1,9), then
202 6 COMPLETENESS AND EXTENDIBILITY
J+(p) n 5-(q) is noncompact. Hence this space-time, while strongly causal,
fails to be globally hyperbolic.
6.2 Geodesic Completeness
We showed in Theorem 4.9 that for space-times which are strongly causal,
the Lorentzian distance function may be used to construct a subbasis for the
given manifold topology. Nonetheless, the sets {q E M : d(p, q) < R) fail
t o form a basis for the given manifold topology. Thus geodesic completeness
rather than metric completeness of spacetimes has usually been considered
in general relativity.
Let (M, g) be an arbitrary Lorentzian manifold.
Definition 6.2. (Complete Geodesic) A geodesic c in (M, g) with affine
parameter t is said to be complete if the geodesic can be extcnded to be
defined for -m < t < moo. A past and future inextendible geodesic is said
to be incomplete if it cannot be extended to arbitrarily large positive and
negative values of an affine parameter. Future or past incomplete geodesics
may be defined similarly.
An affine parameter t for the curve c is a parametrization such that c(t) sat-
isfies the geodesic differential equation V,.c1(t) = 0 for all t [cf. Kobayashi and
Nomizu (1963, p. 138)]. It is necessary to use the concept of an affine param-
eter since null geodesics, which have zero arc length, cannot be parametrized
by arc length. If s and t are two affine parameters for c, it follows from the
geodesic differential equations that there exist constants a , /3 E IR such that
s(t) = a t + P for all t in the domain of c. Hence completeness or incompleteness
as defined in Definition 6.2 is independent of the choice of affine parameter. In
particular, if c is an inextendible timelike geodesic parametrized by arc length
(i.e., g(cl(t), cl(t)) = -1 for all t in the domain of c), then c is incomplete if
L(c) < m. Even if L(c) = m, it may happen that c is incomplete. This occurs,
for example, when the domain of c is of the form (a, co), where a > c o .
Certain exact solutions to the Einstein equations in general relativity, like
the extended Schwarzschild solution, contain nonspacelike geodesics which be-
come incomplete upon running into black holes. Even though the existence of
6.2 GEODESIC COMPLETENESS 203
incomplete, inextendible, nonspacelike geodesics does not force a space-time
to contain a black hole, these examples suggest that nonspacelike geodesic
incompleteness might be used as a first order test for "singular space-times"
[cf. Hawking and Ellis (1973, Chapter 8), Clarke and Schmidt (1977), Ellis
and Schmidt (1977)j. Thus it is standard to make the following definitions in
general relativity. Recall that a geodesic is said to be inextendible if it is both
past and future inextendible.
Definition 6.3. (Geodesically Complete) A space-time (M,g) is said
to be timelike (respectively, null, nonspacelike, spacelike) geodesically wm-
plete if all timelike (respectively, null, nonspacelike, spacelike) inextendible
geodesics are complete. The space-time (M, g) is said to be geodesically com-
plete if all inextendible geodesics are complete. Also, (M, g) is said to be time-
like (respectively, null, nonspacelike, spacelike) geodesically incomplete if some
timelike (respectively, null, nonspacelike, spacelike) geodesic is incomplete. A
nonspacelike incomplete space-time is said to be a geodesically singular space-
time.
It was once hoped that timelike geodesic completeness might imply null ge-
odesic completeness, etc. However, Kundt (1963) gave an example of a space-
time that is timelike and null geodesically complete but not spacelike complete.
Then Geroch (1968b, p. 531) gave an example of a space-time conformal to
Minkowski two-space and thus globally hyperbolic which is timelike incomplete
but null and spacelike complete. Also Geroch remarked that modifications of
Kundt's and his examples gave space-times that were (1) incomplete in any
two ways but complete in the third way, (2) spacelike incomplete but null and
timelike complete, and (3) timelike incomplete but spacelike and null complete.
Then Beem (1976~) gave an example of a globally hyperbolic space-time that
was null incomplete but spacelike and timelike complete. These results may
be summarized as follows.
1 T h e o r e m 6.4. Timelike geodesic completeness, nu11 geodesic complete-
ness, and spacelike geodesic completeness are all logically inequivalent.
6 COMPLETENESS AND EXTENDIBILITY
FIGURE 6.2. Shown is Geroch's example of a space-time glob-
ally conformal to Minkowski two-space, which is null and spacelike
geodesically complete but timelike geodesically incomplete. Here
the positive t axis may be parametrized to be an incomplete t i m e
like geodesic since b(O, t) + 0 like tT4 as t -+ a.
In order to illustrate the constructions used in the proof of Theorem 6.4,
we now describe Geroch's example of a space-time which is null and spacelike
complete but timelike incomplete. Let (IR2, gl) be Minkowski two-space with
global coordinates (x, t) and the usual Lorentzian metric gl = ds2 = dx2 - dt2.
6.2 GEODESIC COMPLETENESS 205
Conformally change the metric gl to a new metric g = $gl for W2, where
4 : R2 --+ (0 , co) is a smooth function with the following properties (cf. Figure
6.2):
(1) $ (x , t )= 1 i f x s -1 o r x > 1;
(2) $(x, t) = $(-x, t) for all (x, t ) f W2; and
(3) On the t axis, d(0, t ) goes to zero like t-4 as t -4 a.
Since g is conformal to gl, the space-time (W2, g) is globally hyperbolic, and
null geodesics still have as images straight lines making angles of 45" with the
positive or negative x axis. By property (2), the reflection F ( x , t) = (-x, t)
is an isometry of Since the fixed point set of a n isometry is totally
geodesic, the t axis may be parametrized as a timelike geodesic. By condition
(3), this geodesic is incomplete as t + m. Thus (W2, g) is timelike incomplete.
But every null or spacelike geodesic which enters the region -1 5 x 5 1
eventually leaves and then remains outside this region. Thus condition (1)
implies that (EX2, g) is null and spacelike complete.
We now consider the converse problem of constructing geodesically complete
Lorentzian metrics for paracompact smooth manifolds. In order to preserve
the causal structure of the given space-time, we restrict our attention to global
conformal changes rather than arbitrary metric deformations.
For Riemannian metrics, Nomizu and Ozeki (1961) showed that an arbitrary
metric can be made complete by a global conformal change. On the other hand,
space-times exist with the property that no global conformal factor will make
these space-times nonspacelike geodesically complete. A two-dimensional ex-
ample with this property has been given by Misner (1967). In this example
there are inextendible null geodesics which are future incomplete and future
trapped in a compact set [cf. Hawking and Ellis (1973, pp. 171-172)j. Any
conformal change of this example will leave these null geodesics pointwise fixed
and future incomplete. Thus one may not establish an analogue of the Nomizu
and Ozeki result for arbitrary space-times.
However, the existence of nonspacelike complete Lorentzian metrics has
been shown for space-times satisfying certain causality conditions. Seifert
(1971, p. 258) has shown that if (M, g) is stably causal, then M is conformal
206 6 COMPLETENESS AND EXTENDIBILITY
to a space-time with all future directed (or all past directed) nonspacelike
geodesics complete. Also Clarke (1971) has shown that a strongly causal space-
time may be made null geodesically complete by a conformal factor. Beem
(1976a) studied space-times with the property that for each compact subset
K of M, no future inextendible nonspacelike curve is future imprisoned in
K. (Recall that the nonspacelike curve y is said to be future imprisoned in
K if there exists to E R such that y(t) E K for all t 2 to.) If (M,g) is
a causal space-time satisfying this condition, then there exists a conformal
factor Sl : M -t (0, GO) such that (M, Slg) is null and timelike geodesically
complete [Beem (1976a, p. 184, Theorem 8)]. This imprisonment condition is
satisfied if ( M , g ) is stably causal, strongly causal, or distinguishing. Hence
we may state the following result.
T h e o r e m 6.5. If ( M , g ) is distinguishing, strongly causal, stably causal,
or globally hyperbolic, then there exists a smooth conformal factor R : M -,
(0, m) such that the space-time (M, Rg) is timelike and null geodesically com-
plete.
I t is an open question as to whether Theorem 6.5 can be strengthened to
include spacelike geodesic completeness as well (cf. Corollary 3.46 for space-
times homeomorphic to R2).
Suppose that a space-time is defined to be nonsingular if it is geodesically
complete. Then "no regions have been deleted from the space-time manifold"
of a nonsingular space-time [Geroch (1968b, Property I)]. But Geroch (196813,
Property 2) suggested a second condition that nonsingular space-times should
satisfy, namely, "observers who follow 'reasonable' (in some sense) world lines
should have an infinite total proper time." Here a "world line" is a timelike
curve in (M,g). Then Geroch (1968b, pp. 534-540) constructed a geodesi-
cally complete space-time which contains a smooth timelike curve of bounded
acceleration but having finite length. Thus this example fails t o satisfy Ge-
roch's Property 2 even though all timelike geodesics have infinite length by the
geodesic completeness.
Accordingly, in addition to geodesically incomplete space-times, further
kinds of singular space-times have been studied in general relativity. In the rest
6.2 GEODESIC COMPLETENESS 207
of this section, we will discuss two of these additional types of completeness,
b.a. completeness (bounded acceleration completeness) and b-completeness
(bundle completeness).
The concept of b.a. completeness stems from the preceding example of Ge-
roch. For the purpose of stating Definition 6.6, we recall than any C2 time-
like curve may be reparametrized to a C2 timelike curve y : J --+ M with
g(yl(t), yl(t)) = -1 for all t f J.
Definition 6.6. (Bounded Acceleration) A C2 timelike curve y : J -+ M
with g(yl(t), yl(t)) = -1 for all t E J is said to have bounded acceleration if
there exists a constant B > 0 such that Ig(V,iyl(t), V , J ~ ' ( ~ ) ) I 5 B for all
t E J.
Here V is the unique torsion free connection for M defined by the metric g
[cf. Section 2.2, equations (2.16) and (2.17)). In particular, if y is a geodesic,
then y has zero and hence bounded acceleration. The requirement that y be
C2 makes it possible to calculate V,jyl.
Definition 6.7. (b.a. complete space-time) A space-time (M, g) is said
to be b.a. complete if all future (respectively, past) directed, future (respec-
tively, past) inextendible, unit speed, C2 timelike curves with bounded acceler-
ation have infinite length. If there exists a future (or past) directed, future (or
past) inextendible, unit speed, C2 timelike curve with bounded acceleration
but finite length, then (M,g) is said to be b.a. incomplete.
Geroch's example (1968b, pp. 534-540) shows that geodesic completeness
does not imply b.a. completeness. Furthermore, Beem (1976c, p. 509) has given
an example to show that even for globally hyperbolic space-times, geodesic
completeness does not imply b.a. completeness. Trivially, b.a. completeness
implies timelike geodesic completeness. On the other hand, the example of
Geroch given in Figure 6.2 may be modified by changing the sign of the met-
ric tensor to show that b.a. completeness does not imply spacelike geodesic
completeness.
A stronger form of completeness, b-completeness, does imply geodesic com-
pleteness and hence overcomes this last objection to b.a. completeness. The
208 6 COMPLETENESS AND EXTENDIBILITY
concept of b-completeness, which was first studied for Lorentzian manifolds by
Schmidt (1971), is intuitively defined as follows [cf. Hawking and Ellis (1973,
p. 259)). First, the concept of an affine parameter is extended from geodesics
to all Ci curves. Then a space-time is said to be b-complete if every C1 curve
of finite length in such a parameter has an endpoint.
We now give a brief discussion of b-completeness. First, it is necessary to
discuss the concept of a generalized afine parameter for any C1 curve y : J --+
M. Recall that a smooth vector field V along y is a smooth map V : J --t TM
such that V(t) E T7(tlM for all t E J. Such a smooth vector field V along
y is said to be a parallel field along y if V satisfies the differential equation
Vy,V(t) = 0 for all t E J (cf. Chapter 2).
A generalized affine parameter p = p(y, El, E2 , . . . , En) may be constructed
for y : J -+ M as follows. Choosing any to E J, let {el, e2, . . . ,en} be any
basis for Ty(t,lM. Let Ei be the unique parallel field along y with E,(to) = ei
for 1 5 i 5 n. Then {El(t), E2(t), . . . , En(t)) forms a basis for T7(tlM for
each t E J. We may thus write yl(t) = Cy=l Vi(t)E,(t) with Vi : J -+ R
for 1 5 i 5 n. Then the generalized affine parameter p = p(y, El,. . . , E n ) is
given by
The assumption that y is C1 is necessary in order to obtain the vector fields
{El, E2 , . . . , E n ) by parallel translation. It may be checked that y has finite
arc length in the generalized affine parameter p = p(y, El,. . . , En) if and
only if y has finite arc length in any other generalized affine parameter p = -
p ( y , E l , . . . , E n ) calculated from any other basis {E,):=, for TMI, obtained
by parallel translation along y [cf. Hawking and Ellis (1973, p. 259)j. Hence
the concept of finite arc length with respect to a generalized affine parameter
is independent of the particular choice of generalized affine parameter. I t thus
makes sense to make the following definition.
Definition 6.8. (b-complete space-time) The space-time (M, g) is said
to be b-complete if every C1 curve of finite arc length as measured by a gen-
eralized affine parameter has an endpoint in M.
6.3 METRIC COMPLETENESS 209
Suppose y : J + M is any smooth geodesic. Taking El(t) = yl(t) in the
above construction, for any choice of E2,. . . , E n we have generalized affine
parameter p(y, E l , E2 , . . . , En)(t) = t. Hence b-completeness implies geodesic
completeness. It is also known that b-completeness implies b.a. completeness.
Geroch's example (cf. Figure 6.2) with the sign of the metric tensor changed
shows that there are globally hyperbolic space-times which are b.a. complete
but not b-complete. Thus b.a. completeness does not imply b-completeness.
6.3 Met r i c Completeness
The Hopf-Rinow Theorem for Riemannian manifolds (N, go) implies that
the following are equivalent:
(1) N with the Riemannian distance function do : N x N + [O, a) is a
complete metric space, i.e., all Cauchy sequences converge.
(2) (N, do) is finitely compact, i.e., all do-bounded sets have compact clo-
sure.
(3) (N, go) is geodesically complete.
Here a set K in a Riemannian manifold (N,go) is said to be bounded if
sup{do(p, q) : p, q E K } < co. By the triangle inequality, this is equiva-
lent to the condition that K be contained inside a closed metric ball of finite
radius.
In Section 6.2 we considered the geodesic completeness of Lorentzian man-
ifolds. In this section we shall consider Lorentzian analogues of conditions (1)
and (2) above. From the very definition of Lorentzian distance [i.e., d(p, q) = 0
if q $! J+(p)], it is clear that attention should be restricted to timelike Cauchy
sequences.
Busemann (1967) studied general Hausdorff spaces having a partial order-
ing with properties similar to those of the chronological partial ordering p << q
of a space-time. Also, Busemann supposed that these spaces, which he called
timelike spaces, were equipped with a function which behaves just like the
Lorentzian distance function of a chronological space-time restricted to the set
{(p, q) E M x M : p < q}. For this class of nondifferentiable spaces, Busemann
observed that the length of continuous curves could be defined and, moreover,
6 COMPLETENESS AND EXTENDIBILITY
FIGURE 6.3. Shown is a sequence {x,) in Minkowski two-space
(IR2, ds2 = dx2 -dy2) with 2, >> p for all n, d(p, z,) + 0 as n -+ co,
but such that {x,) has no point of accumulation.
the length functional is upper semicontinuous in a topology of uniform con-
vergence [cf. Busemann (1967, p. lo)]. Busemann's aim in studying timelike
spaces was to develop a geometric theory for indefinite metrics analogous to
the theory of metric G-spaces [cf. Busemann (1955)j. In particular, Busemann
studied finite compactness and metric completeness for timelike spaces in the
spirit of (1) and (2) of the Hopf-Rinow Theorem.
Beem (1976b) observed that Busemann's definitions of finite compactness
and metric completeness for timelike G-spaces may be adapted to causal space-
times. First, timelike Cauchy completeness may be defined for causal space-
times as follows.
6.3 METRIC COMPLETENESS 211
Definition 6.9. (Timelike Cauchy Complete) The causal space-time
(M,g) is said to be timelike Cauchy complete if any sequence {x,) of points
with x, << x,+, for n, m = 1,2 ,3 , . . . and d(z,, x,+,) 5 B, [or else x,+, << x, and d(x,+,, x,) 5 B,] for all m 1 0, where B, -+ 0 as n -+ oa, is a con-
vergent sequence.
For Riemannian manifolds, finite compactness may be defined by requiring
that all closed metric balls be compact. On the other hand, we have noted
above (cf. Figure 4.4) that thesubsets {q E J+(p) : d(p, q ) 5 e ) of a space-time
are generally noncompact. Thus the Riemannian definition must be modified.
One possibility is the following [cf. Busemann (1967, p. 22)].
Definition 6.10. (Finitely Compact) The causal space-time (M,g) is
said to be finitely compact if for each fixed constant B > 0 and each sequence
of points {x,) with either p << q < x, and d(p, x,) 5 B for all n, or x, < q << p
and d(z,,p) < B for all n, there is a point of accumulation of {x,) in M.
It may be seen that without requiring p << q < x, (or x, < q << p) for some
q E M in Definition 6.10, Minkowski space-time fails to be finitely compact
(cf. Figure 6.3).
For globally hyperbolic space-times, a characterization of finite compactness
more reminiscent of condition (2) above for Riemannian manifolds may be
given.
Lemma 6.11. Let (M,g) be globally hyperbolic. Then (M, g) is finitely
compact iff for each real constant B > 0, the set {x E M : p << q 6 x, d(p, x) 5 B) is compact for any p,q E M with q E I+(p) and the set {x E M : x < q << p, d(x,p) 5 B) is compact for any p,q E M with p E I+(q).
Proof. This now follows easily because the sets J+(q) are closed and the Lor-
entzian distance function is continuous since (M, g) is globally hyperbolic.
Minkowski space-time is both timelike Cauchy complete and finitely com-
pact. More generally, it may be shown that these concepts are equivalent for
all globally hyperbolic space-times [cf. Beem (1976b, pp. 343-344)l.
212 6 COMPLETENESS AND EXTENDIBILITY
Theorem 6.12. If (M, g) is globally hyperbolic, then (M, g) is finitely com-
pact iff (M, g) is timelike Cauchy complete. Also, if (M,g) is globally hyper-
bolic and nonspacelike geodesically complete, then (M, g) is finitely compact
and timelike Cauchy complete.
Remark 6.13. Even for the class of globally hyperbolic space-times, finite
compactness, or equivalently, timelike Cauchy completeness, does not imply
timelike geodesic completeness. Indeed, Geroch's example given in Figure 6.2
is a timelike geodesically incomplete, globally hyperbolic space-time which is
finitely compact.
At times, it is important to consider the completeness of a submanifold as
well as that of the given manifold. Conditions which are sufficient to guarantee
the completeness of submanifolds of Lorentzian manifolds are, in general, more
complicated than corresponding conditions for submanifolds of Riemannian
manifolds [cf. Beem and Ehrlich (1985a,b), Harris (1987, 1988a,b, 1994)j. If
(H, h) is a complete Riemannian manifold and F : M + H is an embedding
of M with F(M) a closed subset of H, then M is a complete Riemannian
manifold using the induced metric. The converse of this result is false. For
example, let the curve c : (0, +m) -+ W2 be given by c(t) = (t,sin(l/t)). This
curve has an image which is a complete submanifold of the usual Euclidean
plane, but this image clearly fails to be a closed subset of R2.
Unlike the Riemannian case, closed embedded submanifolds of Lorentzian
manifolds need not be complete. To see this, it suffices to take a curve in
the Minkowski plane which is asymptotic to a null line quickly enough in at
least one direction along the curve. For example, the following map [cf. Harris
is an embedding of W1 as an incomplete closed spacelike submanifold of the
(x, y) plane with the usual Minkowski metric 77 = dx2-dy2. Harris (1988a) has
studied the completeness of embedded and immersed spacelike hypersurfaces of
Minkowski space. Among other things, he has shown that if such an immersed
hypersurface in (n + 1)-dimensional Minkowski space (i.e., Ln+') is complete,
6.3 METRIC COMPLETENESS 213
then its image must be a closed achronal subset which is diffeomorphic to
Rn and it must be embedded as a graph of a function defined on a spacelike
hyperplane. A more detailed resume of Harris's work on spacelike completeness
together with additional references is given in Harris (1993, 1994).
Separate investigations had earlier been conducted in the case of constant
mean curvature spacelike hypersurfaces S of Ln+' which are closed subsets of
Ln+' in the Euclidean topology. For this class of space-times, Cheng and Yau
(1976) showed that the condition of constant mean curvature Ho implies that
S is complete in the induced Riemannian metric. (A complete treatment of the
issue of the achronality en route to the proof of completeness is given in Harris
(1988a, pp. 112, 118). Also, Harris (1988a, p. 118) showed that the hypothesis
of bounded principal curvatures may be substituted for constant mean curva-
ture.) Further, Cheng and Yau (1976) showed that constant mean curvature
Ho implies that the length of the second fundamental form is bounded by
n /No[ and also that S has nonpositive Ricci curvature. In particular, in the
case of a maximal (i.e., Ho = 0) spacelike hypersurface, this estimate shows
that the second fundamental form is trivial. Hence these results of Cheng and
Yau (1976) combine with earlier work of Calabi (1968) for n = 3 to yield the
result that the only maximal spacelike hypersurface which is a closed subset
of Minkowski space is a linear hyperplane. Nishikawa (1984) investigated the
more general case of a locally symmetric target manifold satisfying the timelike
convergence condition and a condition that the sectional curvature of all non-
degenerate two-planes spanned by a pair of spacelike vectors be nonnegative.
Nishikawa showed that a complete maximal spacelike hypersurface in such a
target space-time would be totally geodesic.
In the case of nonzero constant mean curvature, Goddard (1977b) car-
ried out perturbation calculations for hyperboloids in Minkowski space (and
also for appropriate submanifolds of de Sitter space-time) which suggested
that perhaps all entire, constant mean curvature, spacelike hypersurfaces of
Minkowski space should be hyperboloids. This issue was thoroughly inves-
tigated by Treibergs (1982), who took the starting point that in the case of
positive mean curvature, an entire spacelike hypersurface could be realized as
214 6 COMPLETENESS AND EXTENDIBILITY 6.4 IDEAL BOUNDARIES 215
the graph of a convex function. Now for an entire convex function f whose
graph is a spacelike hypersurface, Treibergs (1982, p. 51) defines the projective
boundary values o f f at infinity, Vj, as
f (7.~1 Vj(x) = lim -. r++m T
Treibergs defines two entire, constant mean curvature, spacelike hypersurfaces
to be equivalent if they have the same projective boundary values a t infinity.
Treibergs then proves that the set of such equivalence classes coincides with
convex homogeneous functions whose gradient has norm one whenever defined.
Treibergs further shows that constant mean curvature spacelike hypersurfaces
with given projective boundary data a t infinity are highly nonunique, because
arbitrary finite perturbations of the given light cone (at infinity) may be made
producing different f (x), hence different spacelike hypersurfaces, each strongly
asymptotic to its own perturbed light cone, yet all having the same projective
boundary behavior V(x) a t infinity. Thus the geometry for entire, constant
(but nonzero) mean curvature, spacelike hypersurfaces turns out to be more
complicated than the maximal (No = 0) case.
6.4 Ideal Boundaries
In this section we give brief descriptions of the b-boundary and the causal
boundary for a space-time. Further details may be found in Hawking and Ellis
(1973, Sections 6.8 and 8.3) or Dodson (1978).
The b-boundary of a space-time (M,g) will be denoted by dbM. This
boundary is formed by defining a certain positive definite metric on the bundle
of linear frames L ( M ) over M , taking the Cauchy completion of L(M), and
then using the newly formed ideal points of L(M) to obtain ideal points of
M . The b-boundary is particularly useful in telling whether or not some
points have been removed from the space-time. Somewhat unfortunately, the
b-boundary often consists of just a single point [cf. Bosshard (1976), Johnson
(1977)l. This boundary is not invariant under conformal changes and also
is not directly related to the causal structure of (M,g). A discussion of the
merits and demerits of the b-boundary and geodesic incompleteness may be
found in the review article of Tipler, Clarke, and Ellis (1980).
Recall that a curve y : [O,a) --t M is said to be b-incomplete if it has
finite generalized affine parameter (cf. Section 6.2). .4ny curve y : [O,a) --, M
which is both b-incomplete and inextendible to t = a defines a point of dbM
corresponding to y(a). In Minkowski space-time, generalized affine parameter
values along a curve can be made to correspond to Euclidean arc length. Thus
Minkowski space-time has an empty b-boundary, and each b-incomplete curve
in Minkowski space-time has an endpoint in the space-time.
The causal boundary of a space-time (M,g) will be denoted by dcM. This
boundary is constructed using the causal structure of the space-time. Thus it
is invariant under conformal changes. We will only be interested in considering
this boundary for strongly causal space-times.
The causal boundary is formed using indecomposable past (respectively, fu-
ture) sets which do not correspond to the past (respectively, future) of any
point of M . A past (respectively, future) set A is a subset of M such that
I-(A) C A (respectively, I+(A) A). The open past (respectively, future)
sets are characterized by I-(A) = A (respectively, If (A) = A). An indecom-
posable past set (IP) is an open past set that cannot be written as a union
of two proper subsets both of which are open past sets. An indecomposable
future set (IF) is defined dually.
A terminal indecomposable past set (TIP) is a subset A of M such that
(1) A is an indecomposable past set, and
(2) A is not the chronological past of any point p E M.
A terminal indecomposable future set (TIF) is defined dually. The causal
boundary bcM is formed using TIP's and TIF's after making certain identifica-
tions which will be described below [cf. Hawking and Ellis (1973, pp. 218-221)].
These identifications allow the topology of M to be extended to M * = MU&M
in such a way that the causal completion of M is Hausdorff, provided that
(M, g) satisfies certain more restrictive causality conditions such as stable
causality.
The use of TIP's and TIF's to represent ideal points of the causal boundary
of (M, g) is illustrated in Figure 6.4.
216 6 COMPLETENESS AND EXTENDIBILITY
- D
FIGURE 6.4. The ideal point p in 8,M is represented by the termi-
nal indecomposable past set A, and the ideal point ?j is represented
by the terminal indecomposable future set B. The point 5: is repre-
sented by both the set C, which is a TIP, and the set D, which is a
TIF.
We now show that a TIP may be represented as the chronological past of a
future inextendible timelike curve. This result is due to Geroch, Kronheimer,
and Penrose (1972, p. 551).
Proposition 6.14. A subset W of the strongly causal space-time (M, g) is
a TIP iff there exists a future directed and future inextendible timelike curve
y such that W = I-(?).
Proof. Assume there is a future inextendible timelike curve y with W =
I - (7 ) . Using the strong causality of (M,g) , it follows that if W is an IP, then
W is a TIP. To show W is an IP, assume that W = Ul U U2 for nonempty
6.4 IDEAL BOUNDARIES 217
open past sets Ul and U2 such that neither is a subset of the other. Choose
rl E U1 - U2 and 7-2 E U2 - Ul . There must exist points r: t y such that
r , E I - ( r : ) for i = 1,2 because Ul U U2 = I - (7 ) . However, whichever Ui contains the futuremost of r; and r; must then contain all four of r l , 7-2, T: ,
and T;. This contradicts either the definition of rl or of r2.
On the other hand, assume that W is a TIP. If p is any point of W , then
W = [ W n I + ( p ) ] u ( W - I+(p)] and thus W = I - ( W n I f ( p ) ) u I - ( W - I+ ( p ) )
since W is a past set. Since W is an IP, either W = I - ( W n I+(p)) or
W = I - ( W - I+(p)). Consequently, as p $ I-(W - I+(p)) , we have W =
I - ( W n I+(p)). Thus, given any q # p in W , there must be some point r in
W which is in the chronological future of both p and q. Inductively, for each
finite subset of W , there exists some point of W in the chronological future
of each point of the subset. Now choose a sequence of points {p,) which
forms a countable dense subset of W . We will define a second sequence {q,)
inductively. Let qo be a point of W in the chronological future of po. If qi for
i = 1,2 , . . . , k - 1 has been defined, then choose q k to be a point of W in the
chronological future of pk and of q, for i = 1,2,. . . , k - 1.
Finally let y be any future directed timelike curve which begins a t qo and
connects each qi to the next qi+1. Clearly, each p, lies in I - ( y ) and I-(?)
W . Using the openness of W and the denseness of the sequence {p,), it follows
that W = I - ( y ) as required. 0
In space-times which are not strongly causal, there may exist future directed
and future inextendible timelike curves y such that I - ( y ) is the chronological
past of some point [i.e., I - ( 7 ) is not a TIP]. Consider, for example, the cylinder
W1 x S 1 with the flat metric ds2 = dt dB and the usual time orientation with
the future corresponding to increasing t. The lower half of the cylinder W =
{ ( t , 8 ) : t < 0) is an IP which can be represented as I - ( y ) for a future directed
and future inextendible timelike curve y. However, W is not a TIP since W
can be represented as the chronological past of any point on the circle t = 0.
By restricting our attention to strongly causal space-times, the IP's which are
not TIP'S are in one-to-one correspondence with the points of M. The dual
statement holds for IF'S which are not TIF1s.
218 6 COMPLETENESS AND EXTENDIBILITY
We now define M (respectively, M) to be the collection of all IP's (respec-
tively, IF'S). Fhrthermore, let ~d = M U M / N, where for each p E M
the element I-(p) of M is identified with the element I+(p) of M. The map
I+ : M -+ MI given by p -+ I+(p) then identifies M with a subset of Ms.
Using this identification, the set Mu corresponds to M together with all TIP'S
and TIF's.
In order to define a topology on Mu, first define for any A E M the sets ~ i n t and Aext by
= {V E M : v n A # 0 )
and
AeXt = {V E M : V = I - (W) implies I+(w) 9 A).
Similar definitions of BInt and BeXt are made for any B E M. A subbasis
for a topology on Mfl IS then given by all sets of the form Alnt, AeXt, BRnt,
and Bext. The sets Alnt and BInt are analogues of sets of the form I+(p) and -
I-(p), respectively. The sets A"* and BeXt are analogues of M - I+(p) and -
M - I - (p), respectively.
Now in Geroch, Kronheimer, and Penrose (1972), it is proposed to obtain
a Hausdorff space M * = M U a,M from Mu, with the topology given as
above, by identifying the smallest number of points of MI necessary to obtain
a Hausdorff space M*. Equivalently, it is proposed that M * = M U 8,M
should be taken to be the quotient M I I R ~ , where Rh is the intersection of all
equivalence relations R on MI such that M ~ / R is Hausdorff.
Unfortunately, Szabados (1988) and Rube (1988, 1990) independently ob-
served that the assumption of strong causality for the given space-time (M, g)
was not sufficient to ensure that the minimal equivalence relation Rh exists,
and hence the Geroch-PenroseKronheimer choice of topology may not be
used to induce a Hausdorff causal completion M * for a general strongly causal
space-time. Szabados points out two difficulties that may arise for general
strongly causal space-times if the topology given via {Atnt, Aext, BInt, Bext)
is chosen. First, inner points corresponding to the embedding of M into M@
via the map I+ and pre-boundary points are in general only TI-separated, but
not Tz-separated, with this choice of topology. Second, an example is given to
6.5 LOCAL EXTENSIONS 219
show, for a nonspacelike curve y with endpoint q in M , that the induced curve
I+ o y does not necessarily have a unique endpoint in M. Both Rube (1988)
and Szabados (1988) suggest stronger causality conditions on the underlying
space-time (M, g) which will ensure that the equivalence relation Rh proposed
as above will exist, and hence this construction will produce a Hausdorff causal
boundary d,M for M identified with its image in M * under the mapping If.
The simplest such condition to impose is that (M,g) be stably causal.
6.5 Local Extensions
In this section, extendibility and inextendibility of Lorentzian manifolds are
defined. Also, two types of local extendibility are discussed. Most of the results
of this section hold for Lorentzian manifolds which are not time orientable as
well as for space-times.
Definition 6.15. (Extension) An extension of a Lorentzian manifold
(M,g) is a Lorentzian manifold (MIlg') together with an isometry f : M -+
M' which maps M onto a proper open subset of MI. An analytic extension
of (M, g) is an extension f : (M, g) -+ (MI, g') such that both Lorentzian
manifolds are analytic and the map f : M -+ M' is analytic. If (M,g) has no
extensions, it is said to be inextendible.
Suppose that the Lorentzian manifold (M, g) has an extension f : (M, g ) - (M',gl). Since M' is connected and f ( M ) is assumed t o be open in MI, it
follows that
- where f ( M ) denotes the closure of f (M) in M'. Because d(f (M)) # 0 and
the isometry f : M -+ M' maps geodesics in M into geodesics in M' lying
in f (M), it is easily seen that (M,g) cannot be timelike, null, or spacelike
geodesically complete. Recalling that b-completeness and b.a. completeness
both imply timelike geodesic completeness (cf. Section 6.2), we thus have the
following criteria for Lorentzian manifolds to be inextendible.
220 6 COMPLETENESS AND EXTENDIBILITY
Proposition 6.16. A Lorentzian manifold ( M , g ) is inextendible if i t is
complete in any of the following ways:
(1 ) b-complete;
(2) b.a. complete;
(3 ) timelike geodesically complete;
(4) null geodesically complete; or
(5 ) spacelike geodesically complete.
We now define two types of local extensions [cf. Clarke (1973, p. 207), Beem
(1980), Hawking and Ellis (1973, p. 59)].
Definition 6.17. (Local Eztension) Let ( M , g) be a Lorentzian manifold.
(1) Suppose y : [0, a ) --t M is a b-incomplete curve which is not extendible to
t = a in M. A local b-boundary extension about 7 is an open neighborhood
U C_ M of 7 and an extension (U', g ' ) of (U, gl,) such that the image of y in
U' is C0 extendible beyond t = a.
(2 ) A local extension of ( M , g ) is a connected open subset U of M having
noncompact closure in M and an extension (U', g') of (U, gl,) such that the
image of U has compact closure in U'.
Remark 6.18. This definition of local extension differs from the corr*
sponding definition of local extension in Hawking and Ellis (1973, p. 59) in
that U is required to be connected in Definition 6.17-(2) but not in Hawking
and Ellis (cf. Figures 6.5 and 6.6).
We now investigate the relationships between these two types of local exten-
sions. An arbitrary space-time may contain a b-incomplete curve 7 : 10, a ) -+
M which is not extendible to t = a , yet y[O, a ) has compact closure in M.
However, Schmidt has shown that such space-times contain compactly im-
prisoned inextendible null geodesics [cf. Schmidt (1973), Hawking and Ellis
(1973, p. 280)]. On the other hand, if ( M , g ) contains no imprisoned non-
spacelike curves, and ( M , g ) has a local b-boundary extension about y , we
now show this same extension yields a local extension.
6.5 LOCAL EXTENSIONS 221
FIGURE 6.5. Let y : [O, a ) -+ M be a b-incomplete curve which
is not extendible to t = a in the space-time (M,g) . Assume that
there is an isometry f : (U, glu) -+ (U',gl) which takes y to a curve
f o y having an endpoint p in U'. Then f o y may be continu-
ously extended beyond t = a. Thus ( M , g) has a local b-boundary
extension about r.
Lemma 6.19. If ( M , g) is a space-time with no imprisoned nonspacelike
curves and if ( M , g) has a local b-boundary extension about y, then (M, g)
has a local extension.
Proof. Suppose that f : (U, g l l l ) + (U', g') is a local b-boundary extension
about y. Then f o y : 1 0 , ~ ) 4 U' is extendible, and f o y ( t ) converges to some
p E U' as t --+ a. Let W' be an open neighborhood of f j in U' with compact
closure in U'. Choose to E [O,a) such that f o y ( t ) E W' for all to 5 t < a.
Set Vl = f-'(W'), and let V be the component of Vl in U which contains
the noncompact set y I [to,a). Since U is open in M, the set V is a connected
open set in M with noncompact closure in M . Also f ( V ) has compact closure
in U' since f ( V ) C W' Thus f Jv . (V, glv) -+ (U', g ' ) 1s a local extension of
(M,g) . 0
6 COMPLETENESS AND EXTENDIBILITY
FIGURE 6.6. Let U be a connected open subset of M having non-
compact closure in M . A local extension is defined to be an isom-
etry f : (U, glu) -+ (U', g') such that f (U) has compact closure in
U'. Minkowski space-time shows that even real analytic b-complete
space-times may admit analytic local extensions (cf. Example 6.21).
Thus a space-time may admit local extensions but not admit local
b-boundary extensions.
We now show that both types of local inextendibility imply global inex-
tendibility.
Proposition 6.20. If the Lorentzian manifold (M, g) has no local exten-
sions of either of the two types in Definition 6.1 7, then (M,g) is inextendible.
Proof. Suppose (M, g) has an extension F : (M, g) + (M1,g'). Let p E
B(F(M)), and choose a geodesic a : [O, I] --t (M1,g') with o(0) E F(M)
and a(1) = p. Since F ( M ) is open in M' and 5 4 F(M) , there exists some
to E (0, I] such that u(t) E F (M) for all 0 < t < to but a(to) 4 F(M). Then
the curve 7 = F-' 0 a][o,to) : [0, to) -+ M is b-incomplete, inextendible to
6.5 LOCAL EXTENSIONS 223
t = to in M , and has noncompact closure in M. Taking U = M , U' = MI, and
f = F in (1) of Definition 6.17, it follows that (M,g) has a local b-boundary
extension about y. Taking W to be any open subset about p with compact
closure in M', U' = M', and U to be the component of F- ' (W) containing y,
we obtain a local extension Flu : (U, glU) -t (M', 9'). 0
The next example shows that Minkowski space-time has local extensions.
Since Minkowski space-time is b-complete, this example shows that even
though b-completeness is an obstruction to global extensions, it is not an
obstruction to local extensions (cf. Proposition 6.16). This example is unusual
in that it does not correspond to a local extension of M over a point of either
the b-boundary a b h l or the causal boundary d,M. I t is a local extension of a
set which extends to i0 (cf. Figure 5.4).
Example 6.21. Let ( M = Rn,g) be n-dimensional Minkowski space-time
and let M' = W x Tn-l , where Tn-' = {(82,83,. . . ,On) : 0 5 8, < 1) is the
(n - 1)-dimensional torus (using the usual identifications). We may define a
Lorentzian metric g' for M' by g' = ( d ~ ' ) ~ = -dt2 + d822 $ . . . + don2. Then
( M , g) is the universal Lorentzian covering space of (M ' , g') with covering map
f : (M, g) -+ (M', g') given by
f (xl, . . . , xn) = (XI, x2(mod 1), xa(mod I) , . . . , xn(mod 1)).
Fix ,B > 0 and consider the curve y : [I, m) 4 M given by
Then f o y : [I, ca) -+ M' is a spiral which is asymptotic to the circle t = B3 =
. . = 8, = 0 in M'. Let U be an open tubular neighborhood about y in M
such that U is contained in some open set {(XI, . . . , x,) E Rn : 0 < XI < a ) for
some fixed a > 1 and such that f 1, : U -t M' is a homeomorphism onto its
image (cf. Figure 6.7). Intuitively, the set U must be chosen to be thinner as
s -+ c=a in order to satisfy the requirement xl > 0 for (XI, . . . , x,) E U. While
U does not have compact closure in Minkowski space-time, f ( U ) does have
compact closure in M ' since f (U) is contained in the compact set [O, a] x Tn-l.
6 COMPLETENESS AND EXTENDIBILITY
FIGURE 6.7. Minkowski space-time has analytic local extensions.
Let M = Wn be given the usual Minkowskian metric g, and let Tn-I
be the (n - 1)-dimensional torus with the usual positive definite
flat metric h. Let M' = W x Tn-' be given the Lorentzian product
metric g' = -dt2 8 h. Then (M,g) is the universal covering space
of (MI, g'), and the quotient map f : M -+ M' is locally isometric.
Choose U to be an open set in M about y(s) = (s-P, s, . . . , 0),
7 : [I, m) -+ M, such that f ] U is one-to-one and f ( U ) has compact
closure in Mi. Then f 1~ : (U,gu) -+ (M1,g') is an analytic local
extension of Minkowski space, but this extension is not across points
of dcM and not across points of dbM.
6.6 SINGULARITIES 225
Thus f : (U, glU) --, (M', g') is an analytic local extension of Minkowski space-
time. Notice that if y~ : [0, a) --+ U is any curve with noncompact closure in
M , then f o yl cannot be extended to t = a in MI.
Let d M denote an ideal boundary of M (i.e., d M represents either dbM
or d,M). A point q E d M is said to be a regular boundary point of M if
there exists a global extension (MI, g') of (M, g) such that q may be naturally
identified with a point of MI. A regular boundary point may thus be regarded
as being a removable singularity of M.
Let y : 10, a) -+ M be an inextendible curve such that y(a) corresponds
to an ideal point of M. The curve y is said to define a curvature singularity
[cf. Ellis and Schmidt (1977, p. 916)] if some component of Ra6cd;el,,, , ,ek is not
C0 on [0, a] when measured in a parallelly propagated orthonormal basis along
y. A curvature singularity is an obstruction to a local b-boundary extension
about y because if there is a local b-boundary extension about y, then the
curvature tensor and all of its derivatives measured in a parallelly propagated
orthonormal basis must be continuous and hence converge to well-defined limits
as t -+ a-. A related but somewhat different notion is that of strung curvature
singularity which may be defined using expansion 8 along null geodesics. The
notion of strong curvature singularity has been useful in connection with cosmic
censorship [cf. Kr6lak (1992), Kr6lak and Rudnicki (1993)l.
A b-boundary point q E dbM which is neither a regular boundary point
nor a curvature singularity is called a quasi-regular singularity. Clarke (1973,
p. 208) has proven that if y : [0, a) -+ M is an inextendible b-complete
curve which correspo,lds to a quasi-regular singularity, then there is a local
b-boundary extension about y. This shows that curvature singularities are
the only real obstructions to local b-boundary extensions.
In general, it can be quite difficult to decide if a given space-time has local
extensions of some type. However, for analytical local b-boundary extensions
of analytic space-times, the situation is somewhat simpler (cf. Theorem 6.23).
226 6 COMPLETENESS AND EXTENDIBILITY
For the proof of Theorem 6.23, it is useful to prove the following proposi-
tion about real analytic space-times and local isometries. Recall that a local
isometry F : M -+ M' is a map such that for each p E M , there exists an open
neighborhood U(p) of p on which F is an isometry. Thus local isometries are
local diffeomorphisms but need not be globally one-to-one.
Proposition 6.22. Let (M, g) and (Ml,gl) be real analytic space-times
of the same dimension and suppose that F : M -+ MI is a real analytic map.
If M contains an open set U such that Flu : U + Ml is an isometry, then F
is a local isometry.
Proof. Let W = {m E M : F,v # 0 for all v # 0 in T,M), which is an
open subset of M by the inverse function theorem. Since Flu is an isometry,
U is contained in W. Fix any p E U, and let V be the path connected
component of W containing p. We will establish the proposition by showing
first that Ffv is a local isometry and second that V = M.
Let q be any point of V. Choose a curve y : [0,1] -+ V with $0) = p and
$1) = q. By the usual compactness arguments, we may cover y[O, I] with a
finite chain of coordinate charts (Ul, (U2, +2), . . . , (Uk, &) such that each
U, is simply connected, Flu, : U, -+ MI is an analytic diffeomorphism, p E
Ul C u n v , q E UI;, and U,nU,+l # 0 for each i with 1 I i I k-1. Since UI C UnV, we have g = (Flul)*gl on Ul. Thus g = (FIUl)*gl on UlnU2. Since Uln
U2 is an open subset of U2 and F is a real analytic diffeomorphism of U2 onto its
image, it follows that g = ( FIU2)*gl on U2. Continuing inductively, we obtain
g = (FIUk)*gl on Uk, whence F is an isometry in the open neighborhood Uk
of q. Thus FIv : V -+ MI is a local isometry.
It remains to show that V = M. Suppose V # M. Choose any point
rl E M - V. Let yl : [O, 11 -+ M be a smooth curve with y(0) = p and
y(1) = rl. There is a smallest to E [O, 1) such that T = y(to) E M - V. Then
F restricted to the neighborhood V of yl{O, to) is a local isometry. It suffices
to show that r E V to obtain the desired contradiction. Since r E M - V,
there exists a tangent vector x # 0 in T,M with F,x = 0. Let X be the
unique parallel field along 7 with X(to) = x. Then F, o X is a parallel field
along F o yl : [O, to) -+ MI since F is a local isometry in a neighborhood of
6.6 SINGULARITIES 227
71 : [O,to) -+ M. But since F is smooth, F,x = F,X(to) = lim,,5 F,X(t).
Because F, o X is a parallel vector field for all t with 0 < t < to, it follows
that lim,-,o F,X(t) # 0. Hence Fax # 0. Thus F is nonsingular a t the point
r, whence r E V in contradiction. 0
We are now ready to turn to the proof of Theorem 6.23 on local b-boundary
extensions of real analytic space-times.
Theorem 6.23. Suppose (M, g) is an analytic space-time with no impris-
oned nonspacelike curves, which has an analytic local b-boundary extension
about y : [O,a) + M. Then there are timelike, null, and spacelike geodesics
of finite &ne parameter which are inextendible in one direction and which do
not correspond to curvature singularities.
The proof of Theorem 6.23 will involve two lemmas.
Lemma 6.24. Suppose (M,g) is an analytic space-time with no impris-
oned nonspacelike curves, which has an analytic local b-boundary extension
about y : [O, a ) -+ M. Then (M, g) has an incomplete geodesic.
Proof. Let f : (U, glU) -+ (U1,g') be an analytic extension about y. We
may assume U contains the image of y. Also, f o y is extendible in U'. Thus
f o y(t) -+ p E U' as t -+ a-. Let W' be a neighborhood of p such that
W' is a convex normal neighborhood of each of its points. Then exp;l :
W' -+ T,U' is a diffeomorphism for each fixed x E TV'. Assume to is chosen
with f 0 y(t) E W' for all to < t < a. Set q = y(t0) and T = f(q). Then
H = exp, of;' o exP;' : W' -+ M is analytic and is a t least defined near r .
The map H takes geodesics starting a t r to geodesics starting a t q, and H
preserves lengths along these geodesics. In fact, H agrees with f-' near r .
The map H need not be one-to-one since expq is not necessarily one-to-one.
Because the domain of exp, is a union of line segments starting at the origin
of T,M, the domain V' of H must be some subset of W' which is a union of
geodesic segments starting a t r. Hence the set V' fails to be all of W' only
when expq : T,M -t M is defined on a proper subset of TqM which does not
include all of the image f;' o exp;l(W1). Thus if we show V' # W', there is
some incomplete, inextendible geodesic starting at q. But the analytic maps
6.6 SINGULARITIES 228 6 COMPLETENESS AND EXTENDIBILITY 229
H and f -' must agree on the component off (U) n V' which contains r. This
implies V' # W'. Otherwise, H and f -' would agree on f (U) n W' and hence
on a neighborhood of f o ?[to, a). This yields H o f o y = y for to 5 t < a and
implies y is extendible in M across the point H(p), in contradiction.
We will continue with the same notation in the next lemma. f - Lemma 6.25. The map H : V' + M is a local isometry.
Proof. The space-time (M, g) is analytic, (U', g') is analytic, and H is an-
alytic. Furthermore, H agrees with the isometry f -l near r , and H is defined
on an arcwise connected set V'. Thus Proposition 6.22 implies H is a local
isometry. O
We are now ready to complete the
FIGURE 6.8. In the proof of Theorem 6.23, the map f : (U, glu) -+ Proof of Theorem 6.23. There are three cases to consider corresponding
(U', g') is an isometry which is an analytic local b-boundary exten- to incomplete timelike, null, and spacelike geodesics. We only give the proof
sion about y. The point p E U' is the endpoint of f o y in U'. In for the timelike case. Let U, U', f , etc., be as in Lemmas 6.24 and 6.25.
this figure, f (9) = r, and all points of f o y between r. and p are in Assume without loss of generality that there is some point x E W' such that
the chronological future of x. in a chronological ordering on W', we have x << p and x << f o y(t) for all
to 5 t < a (cf. Figure 6.8).
If x 4 V', let a: be the geodesic segment in W' from r to x. Then H takes
an V' to an inextendible, incomplete, timelike geodesic starting at q. Lemma
6.25 implies that this geodesic does not correspond to a curvature singularity. Hence the analyticity in the hypothesis of Theorem 6.23 cannot be replaced
If x E V', let y = H(x) and define H' = exp, OH,= o exp;' : W' + M. by a Cw assumption.
The map H' is defined on some subset V" of W'. It is a local isometry for the Corollary 6.27. Let (M, g) be an analytic space-time with no imprisoned
same reasons that H is a local isometry, and H' agrees with both H and f -' nonspacelike curves such that each timelike geodesic y : [0, a) --+ M which is
near r. The set V" cannot contain all of f o y for y on to 5 t < a since this inextendible to t = a is either complete (i.e., a = CQ) in the indicated direction
would yield an endpoint H1(p) of y in M. Using x << f o y(t) for to 5 t < a, or else corresponds to a curvature singularity. Then (M, g) has no analytic
we conclude that there is an inextendible incomplete timelike geodesic starting local b-boundary extensions.
at y in M which does not correspond to a curvature singularity. 13
Extensive work has been done on classifying singularities by various differ-
Remark 6.26. There are examples of Cw space-times which are both ent methodologies. Indeed, the monograph by Clarke (1993) may be consulted
geodesically complete and locally b-extendible [cf. Beem (1976c, p. 506)]. to obtain much more detailed information than that which is provided in this
230 6 COMPLETENESS AND EXTENDIBILITY
chapter. Our treatment here will be limited to a discussion of the "abstract
boundary" &(M) and its application to the classification of singularities as
given in Scott and Szekeres (1994) and Fama and Scott (1994). (In Clarke
(1993), a somewhat different "A-boundary" obtained by taking the closures of
an atlas of normal coordinate charts is discussed.) To form the a-boundary,
equivalence classes of boundary points of a given smooth manifold under all
possible open embeddings are considered. The a-boundary thus has no depen-
dence on any choice of affine connection or semi-Riemannian metric. When
the manifold is further endowed with an extra structure, like an affine con-
nection or a semi-Riemannian metric, then abstract boundary points may be
classified as regular boundary points, points a t infinity, unapproachable points,
or singularities, with the classification having some dependence upon a par-
ticular curve family C selected, which is associated to the added structure for
M . The abstract boundary also has the feature that if a closed region is re-
moved from a singularity-free semi-Riemannian manifold, then the resulting
semi-Ftiemannian manifold is still singularity-free since only regular boundary
points are introduced by the excision of the closed set.
We will begin by discussing the construction of the a-boundary for a given
smooth manifold M of dimension n. Since no metric is yet involved, we will
adopt the terminology of Scott and Szekeres (1994) and define an enveloped
manifold (MI MI, f l ) to be a pair of (connected) smooth manifolds M , MI of
the same dimension together with a smooth embedding fl : M + MI. Since
both manifolds have the same dimension, f l (M) is an open submanifold of
MI. (A very special type of envelopment would then be provided by a global
extension of a given space-time [cf. Definition 6.151.)
Definition 6.28. (Boundary Point, Boundary Set) A boundary point p
of an enveloped manifold (M, MI, f 1) is a point p f MI - fl (M) such that
every open neighborhood U of p in Ml has non-empty intersection with f l(M),
i.e., p belongs to the topological boundary d(fl(M)) = f l (M) - f l(M). A
boundary set B contained in MI - f l (M) is any set of boundary points for the
enveloped manifold (M, MI, fl). We will use the notation (M, MI, f l , B ) for
an enveloped manifold with distinguished boundary set B.
6.6 SINGULARITIES 231
The set of enveloped manifolds and resulting boundary sets is much too
large an object to use to form a useful boundary. Thus the following relation
is introduced between enveloped manifolds and boundary sets.
Definition 6.29. (Covering Boundary Sets) Let (M, MI , f l , B1) and (M, M2, f2, B2) be two envelopments for M with boundary sets B1 and B2,
respectively. Then the boundary set Bl is said to cover the boundary set B2
if for every open neighborhood Ul of B1 in MI, there exists an open neighbor-
hood U2 of Bz in M2 such that
(6.2) fl (u2 n f 2 ( ~ ) ) c u,.
This requirement may be conveniently checked in terms of sequences as
follows.
Proposition 6.30. The boundary set B1 covers the boundary set B2 iff
for every sequence {pk) in hf such that the sequence { f2(pk)) has a limit point
in B2, the sequence (fl (pk)) is required to have a limit point in B1.
Since the requirement (6.2) is to some degree only a topological restriction,
examples may be given of different envelopments of M = Rn - ((0)) for which
one envelopment produces the boundary set B1 = ((0)) consisting of precisely
one point, but a second envelopment produces the boundary set B2 = Sn-l.
and even more remarkably, B1 covers B2 and B2 covers B1. Thus a single
point which is the boundary set of a given envelopment may cover znfinzte sets
of points which are contained in a boundary set for a second envelopment.
Definition 6.31. (Equivalent Boundary Sets) Let (M, M I , f l , B1) and
(M, M2, f2, B2) be two enveloped manifolds with boundary sets B1 and B2, respectively. Then B1 is said to be equivalent to B2 iff B1 covers B2 and Bz
covers B1.
Denote this equivalence relation on the class of boundary sets for M by
B1 - B2. An abstract boundary set [B] is then an equivalence class of bound-
ary sets. In Fama and Scott (1994), topological properties of boundary set
equivalence are studied. It is noted that if a boundary set B of a given envel-
opment is compact, then all boundary sets B' of all other envelopments which
232 6 COMPLETENESS AND EXTENDIBILITY
are equivalent to B must also be compact. In particular, all boundary sets
B equivalent to a single boundary point [ p ] are compact. However, closed-
ness, connectedness, and simple connectedness of boundary sets are unfortu-
nately not invariant under boundary set equivalence. Fama and Scott (1994)
give a detailed discussion of "topological neighborhood properties," including
the connected neighborhood property and the simply connected neighborhood
property, which are preserved by boundary set equivalence.
With all of these preliminaries settled, we may now define the a-boundary
as in Scott and Szekeres (1994).
Definition 6.32. (The a-boundary aa(M)) If p is a boundary point of an
enveloped manifold (M, MI, f l) , then the equivalence class [p ] of the boundary
set {p) under the equivalence relation - of Definition 6.31 is called an abstract
boundary point of M. The set aa(M) of all such abstract boundary points for
all envelopments of the given manifold M is called the abstract boundary or
the a-boundary.
More symbolically, one has that
&(M) = ( I p ] : p is in f l (M) - f l ( M ) for some envelopment (M, Mi , fl)).
In the beginning of this section, the concept of regular boundary point was
defined for space-times in terms of global extensions. A similar definition is
given in Scott and Szekeres (1994) for general semi-Riemannian manifolds.
Now let (M, g) be a semi-Riemannian manifold with a metric tensor g of class
Ck. Let (M,g, MI, f l ) be an envelopment of M . The induced metric tensor
(fcl)* g on f l (M) will be denoted by g when there is no risk of ambiguity.
Definition 6.33. (C1-Extension of (M,g)) A Cl-extension (1 5 1 5 k)
of a Ck semi-=emannian manifold (M, g) is an envelopment of (M, g) by a C1
semi-Remannian manifold (Ml,gl), fi : M -+ M I , such that
This will be denoted by (M, g, MI, gl, fi).
With this definition in hand, the concept of a C1-regular boundary point
may be formulated independent of any choice of curve family for (M, g).
6.6 SINGULARITIES 233
Definition 6.34. (C1-Regular Boundary Point) A boundary point p
of an envelopment (M, g, MI, gl, f 1) is C1-regular for g if there exists a C'
semi-Riemannian manifold (M2, gz) such that f l (M) U {p) C Mz C MI and
(M, g, Mz,gz, f i ) is a CLextension of (M, g).
Further analysis of the boundary points as "singular boundary points" or
"points a t infinity" is possible only if a certain family of curves C with the
bounded parameter property has been specified, and indeed it is natural that
this classification should depend on the choice of the particular curve family
selected. Thus we assume that the smooth manifold M is furnished with a
family C of parametrized curves satisfying the bounded parameter property.
Here a parametrized curve will be taken to mean a C' map y : [a, b) -+ M
with a < b 5 +m and with yl(t) # 0 for all t in [a, b).
Definition 6.35. (Bounded Parameter Property) A family C of param-
etrized curves in M is said to satisfy the bounded parameter property if the
members of C satisfy the following:
(1) Through any point p of M passes a t least one curve y of the family C;
(2) If y is a curve in C, then so is every subcurve of y; and
(3) For any pair of curves yl and 7 2 in C which are obtained from each
other by a monotone increasing C1 change of parameter, either the
parameter on both curves is bounded, or it is unbounded on both
curves.
Examples of suitable families of curves C in our context include: (1) the
family of all geodesics with affine parameter in a manifold M with affine
connection; (2) all C' curves parametrized by "generalized affine parameter"
(cf. Definition 6.8) in an affine manifold; and (3) all future timelike geodesics
parametrized by arc length in a space-time.
As an alternative to Proposition 6.30, boundary sets may be studied using
parametrized curves.
Definition 6.36. (Limit Point, Endpoint of Curve) Let y : [a, b) -+ M
be a parametrized curve. Then
(1) A point p in M is a limit point of the curve y : [a, b) 4 M if there
234 6 COMPLETENESS AND EXTENDIBILITY
exists an increasing sequence ti -4 b such that $ti) -4 p.
(2) A point p in M is an endpoint of the curve y if y(t) -+ p as t + b-
(For Hausdorff manifolds, curve endpoints are unique.)
Definition 6.37. (Approaching a Boundary Set) Let y : [a, b) - M be
a parametrized curve and (M, MI, fl , B ) a boundary set in an envelopment of
M .
(1) The parametrized curve y : [a, b) -t M approaches the boundary set B
if the curve f l o y has a limit point lying in B.
(2) The parametrized curve y : [a, b) -+ M has its endpoint in B if the
curve f o y : [a, b) -+ MI has its endpoint in B.
The analogue of Proposition 6.30 above in this setting is then
Proposition 6.38. If a boundary set B1 covers a boundary set B2, then
every curve y : [a, b) 4 M which approaches B2 also approaches B1.
Now we restrict our attention to a semi-Riemannian manifold (M,g) for
which a family C of curves for M with the bounded parameter property has
been selected, and we use the notation (M, g, C) to denote this selection.
Definition 6.39. (Approachable Boundary Point, C-Boundary Point) If
(M,AJ1, f l ) is an envelopment of (M,g,C), then a boundary point p of this
envelopment is approachable, or a C-boundary point, if p is a limit point of
some curve 7 : [a, b) -+ M of the family C. Boundary points which are not
C-boundary points will be called unapproachable.
As a result of Proposition 6.38, this definition passes to the a-boundary:
Definition 6.40. (Approachable a-Boundary Point, Abstract C-Boundary
Point) An abstract boundary point [p] is an abstract C-boundary point, or
approachable, if p is a C-boundary point. Similarly, an abstract boundary point
[ p ] is unapproachable if p is not a C-boundary point.
With the family C specified, the points a t infinity for (M, g, C) may now be
detected by determining whether they may be reached along any curve in C
a t a finite value of the given parameter.
6.6 SINGULARITIES 235
Definition 6.41. (C1-Point at Infinity for (M, g, C)) Given (M, g, C), a
boundary point p of an envelopment (M, g, C, MI , f l ) is said to be a C1-point
at infinity for C if
(1) p is not a C1-regular boundary point (recall Definition 6.34),
(2) p is a C-boundary point, and
(3) no curve in C approaches p with bounded parameter.
Since the family C is assumed to satisfy the bounded parameter property,
the concept of a point a t infinity is Independent of the choice of parametrization
for the curves from C which approach p. Condition (3) says more explicitly
that for no interval [a, b) with b finite is there a curve y : [a, b) -, (M,g) in
the family C and an increasing sequence of real numbers t, -, b such that
A tricky aspect of a point p at infinity for (M,g,C) is that it is possible
that p is covered by a boundary set B of another embedding consisting only of
regular or unapproachable boundary points. In this case, Scott and Szekeres
(1994, p. 238) term the point p at infinity for (M, g,C) a removable point at
infinity. If no such covering by a boundary set of another embedding exists,
then p is termed an essential point at infinity. It may be shown that the
concept of being an essential point a t infinity passes to the abstract boundary,
i.e., these points cannot be transformed away by a change of coordinates. Thus
an abstract boundary point [p ] is termed an abstract point at infinity if it has
a representative in some envelopment which is an essential point at infinity
for that envelopment. Essential points at infinity may cover regular boundary
points of another embedding. Hence, the following final dichotomy is used for
essential points a t infinity: an essential boundary point a t infinity which covers
a regular boundary point is termed a mixed point at infinity. Otherwise, p is
termed a pure point at infinity.
So far the two categories of regular points and points at infinity have been
defined. The final category to be treated is that of singular boundary points.
Here a more subtle viewpoint is taken than that of Definition 6.3 above in
which "geodesic singularity" is defined. According to this previous definition)
236 6 COMPLETENESS AND EXTENDIBILITY
if one takes, for example, two points p and q in Minkowski space with p << q
and puts M = 14(p) n If (q) with the induced metric from the inclusion of M
in Minkowski space, then M is a singular space-time. Indeed, every geodesic
of (M,g) is incomplete. Yet the given space-time has a global embedding
in Minkowski space such that after this enlargement, every geodesic becomes
complete. Thus this example should perhaps not really be regarded as a sin-
gular space-time, Definition 6.3 notwithstanding.
Definition 6.42. ((7'-Singular Point) A boundary point p of an envel-
opment (M,g,C, MI, f l ) is said to be C1-singular if
(1) p is not a C1-regular boundary point,
(2) p is a C-boundary point, and
(3) there exists a curve in the family C which approaches p with bounded
parameter.
Thus a singular boundary point for the envelopment is a C-boundary point
which is not C1-regular and is not a point a t infinity. Again, in this case,
a finer subclassification is made. If p is covered by a non-singular boundary
set of a second envelopment, then p is termed a removable singularity. If
not, then p is an essential singularity, and a further classification is made as
follows: if p covers some regular boundary points or points a t infinity of another
embedding, then p is commonly called a mixed or directional singularity. If
no such covering behavior is exhibited, then p is said to be a pure singularity.
Then an abstract boundary point [ p ] in d,(M) may be termed an abstract
singularity if it has a representative which is an essential singularity.
We conclude this section with considerations akin to the more simple con-
cepts of Section 6.2 and "geodesic singularity" of Definition 6.3. First, the
notion of geodesic completeness for a space-time may be extended to that of
C-completeness for the manifold M with distinguished curve family C.
Definition 6.43. (C-Completeness) Given a manifold M with a curve
family C satisfying the bounded parameter property, M is said to be C-complete
if every curve y : [a, b) -+ M in C with bounded parameter, i.e., b < +co, has
an endpoint in M.
6.6 SINGULARITIES 237
Definition 6.44. (C1-Singularity) A semi-Ftiemannian manifold (M, g)
with a distinguished class C of curves satisfying the bounded parameter p r o p
erty has a C'-singularity if there exists an envelopment of M having an es-
sential C1-singularity p, i.e., p is a C1-singular point for some envelopment
of (M,g,C) which is not covered by a CL-non-singular boundary set B of
any other envelopment for (M,g,C). Conversely, (hf,g, C) is said to be C1-
singularity free if it has no C1-singularities, i.e., for every envelopment of M ,
the boundary points are either C"nonsingular (C1-regular boundary points,
C'-points at infinity, or unapproachable boundary points), or C'-removable
singularities.
According to Scott and Szekeres (1994)) any theory of singularities ought
to pass over the following hurdle.
Theorem 6.45. Every compact semi-ftiemannian manifold (M, g, C), with
any family of curves C satisfying the bounded parameter property, is singularity
free.
Proof. A compact manifold M has no non-trivial envelopments (M, MI , f l) ,
for MI is required to be connected yet would contain f l (M) as a compact
open subset. Thus f l (M) would be both open and closed in MI, whence
MI = f l(M). Since M has no envelopments, b,(M) is empty and hence can
contain no singular points.
Theorem 6.46. If (M, g) is a semi-Riemannian manifold which is C-com-
plete for a curve family C satisfying the bounded parameter property, then
(M, g, C) is singularity-free.
Proof. Suppose that (M, g, C) contains a singularity. Then there exists an
envelopment (M, g, C, MI , f i) which has a C-boundary point p E M1 - f l(M).
Hence, there exists a curve y : [a, b) -+ M in C which has p as a limit point
and also b < +m. Now by the assumed C-completeness, y has an endpoint
q in M . But endpoints are unique limit points of curves so that of necessity
p = f l (q), which is impossible since p $ fl(M). O
In Section 6.5, Proposition 6.16, it was noted that various types of complete-
ness such as b-completeness, b.a. completeness, timelike geodesic complete
238 6 COMPLETENESS AND EXTENDIBILITY
ness, null geodesic completeness, and spacelike geodesic completeness preclude
global extendibility. In the present context, the analogous statement is the fol-
lowing.
Theorem 6.47. If the semi-Riemannian manifold (M, g) is geodesically
complete, then (M, g) has no regular boundary points.
Complete discussions of all of these boundary points classifications together
with extensive examples, including Taub-NUT space-time as simplified by
Misner (1967) and the Curzon space-time, may be found in Scott and Szekeres
(1994) and Fama and Scott (1994).
CHAPTER 7
STABILITY OF COMPLETENESS AND INCOMPLETENESS
In proving singularity theorems in general relativity, it is important to use
hypotheses that hold not just for the given "background" Lorentzian metric g
for M but in addition for all metrics gl for M sufficiently close to g. Not only
does the imprecision of astronomical measurements mean tha t the Lorentzian
metric of the universe cannot be determined exactly, but also cosmological
assumptions like the spatial homogeneity of the universe hold only approxi-
mately. Nevertheless, if an incompleteness theorem can b e obtained for the
idealized model (M, g) using hypotheses valid for all metrics gl for M in an
open neighborhood of g, then all space-times (M, gl) with gl sufficiently close
to g will also be incomplete. Hence if the model is believed to be sufficiently
accurate, conclusions valid for the model are also valid for t h e actual universe.
Recall that Lor(M) denotes the space of all Lorentzian metrics for a gzven
manzfold M and that Con(M) denotes the quotient space formed by identifyzng
all pozntwise globally conformal metrics gl = Rg2 for M. where R : M +
(0, CQ) is smooth. Let T : Lor(M) + Con(M) denote the natural projection
map which assigns to each Lorentzian metric g for M the set ~ ( g ) = g of
all Lorentzian metrics for M pointwise globally conformal to g. Given g E
Con(M), set C(M, g) = T - ' ( ~ ) C Lor(M). It is customary in general relativity
to say that a curvature or causality condition for a space-time (M,g) is Cr
stable in Lor(M) [respectively, Con(M)], if the validity of the condition for
(M, g) implies the validity of the condition for all gl in a Cr-open neighborhood
of g in Lor(M) [respectively, Con(M)]. More generally, a stable condztion for
a set of metrics zs one which holds on an open subset of such metrics.
After the singularity theorems described in Chapter 8 of Hawking and Ellis
(1973) were obtained, it was of interest to study the Cr stability of conditions
238 6 COMPLETENESS AND EXTENDIBILITY
ness, null geodesic completeness, and spacelike geodesic completeness preclude
global extendibility. In the present context, the analogous statement is the fol-
lowing.
Theorem 6.47. If the semi-Riemannian manifold (M, g) is geodesically
complete, then (M, g) has no regular boundary points.
Complete discussions of all of these boundary points classifications together
with extensive examples, including Taub-NUT space-time as simplified by
Misner (1967) and the Curzon space-time, may be found in Scott and Szekeres
(1994) and Fama and Scott (1994).
CHAPTER 7
STABILITY OF COMPLETENESS AND INCOMPLETENESS
In proving singularity theorems in general relativity, it is important to use
hypotheses that hold not just for the given "background" Lorentzian metric g
for M but in addition for all metrics gl for M sufficiently close to g. Not only
does the imprecision of astronomical measurements mean that the Lorentzian
metric of the universe cannot be determined exactly, but also cosmological
assumptions like the spatial homogeneity of the universe hold only approxi-
mately. Nevertheless, if an incompleteness theorem can be obtained for the
idealized model (M, g) using hypotheses valid for all metrics gl for M in an
open neighborhood of g, then all space-times (M, gl) with gl sufficiently close
to g will also be incomplete. Hence if the model is believed to be sufficiently
accurate, conclusions valid for the model are also valid for the actual universe.
Recall that Lor(M) denotes the space of all Lorentzian metrics for a given
manifold M and that Con(M) denotes the quotient space formed by identifying
all pointwise globally conformal metrics gl = Rg2 for M, where R : M +
(0,w) is smooth. Let T : Lor(M) + Con(M) denote the natural projection
map which assigns to each Lorentzian metric g for M the set ~ ( g ) = 6 of
all Lorentzian metrics for M pointwise globally conformal to g. Given g E
Con(M), set C(M, g) = T - ' ( ~ ) C Lor(M). It is customary in general relativity
to say that a curvature or causality condition for a space-time (M, g) is Cr
stable in Lor(M) [respectively, Con(M)], if the validity of the condition for
(M, g) implies the validity of the condition for all gl in a Cr-open neighborhood
of g in Lor(M) [respectively, Con(M)]. More generally, a stable condition for
a set of metrics is one which holds on an open subset of such metrics.
After the singularity theorems described in Chapter 8 of Hawking and Ellis
(1973) were obtained, it was of interest to study the Cr stability of conditions
240 7 STABILITY OF COMPLETENESS AND INCOMPLETENESS
such as the existence of closed trapped surfaces, positive nonspacelike Rcci
curvature, and geodesic completeness, which played such a key role in these
singularity theorems. Geroch (1970a) established the stability of global hy-
perbolicity in the interval topology on Con(M). Then Lerner (1973) made a
thorough study of the stability in Lor(M) and Con(M) of causality and cur-
vature conditions useful in general relativity. In particular, Lerner noted that
the interval and quotient topologies for Con(M) coincide. Hence Geroch's sta-
bility result for global hyperbolicity holds for Con(M) in the quotient topology
and thus automatically holds in Lor(M). In Section 7.1 we define the fine Cr
topologies and the interval topology for Con(M). We then review stability
properties of Lor(M) and Con(M) which were established by Geroch (1970a)
and Lerner (1973). We give examples of Williams (1984) to show that both ge-
odesic completeness and geodesic incompleteness may fail to be stable. These
two properties are C0 stable for definite spaces, but for all signatures (s,r)
with s 2 1 and r > 1 one may construct examples for which these properties
are unstable.
In Section 7.2, using the "Euclidean norm"
induced on ( T M J U , E ) by a coordinate chart (U, x) for M and standard esti-
mates from the theory of systems of ordinary differential equations in Rn, we
obtain estimates for the behavior of geodesics in (U, x) under C1 metric pertur-
bations. In Section 7.3 we apply these estimates to coordinate charts adapted
to the product structure M = (a, b) x f H of a Robertson-Walker space-time
(cf. Definition 5.10) to study the stability of geodesic incompleteness for such
space-times. We show (Theorem 7.19) that if ((a, b) x j H, g) is a Robertson-
Walker space-time with a > -m, then there is a fine C0 neighborhood U(g)
of g in Lor(M) such that all timelike geodesics of each space-time (M, gl) are
past incomplete for all gl E U(g). If we assume b < oo as well, we may obtain
(Theorem 7.20) both future and past incompleteness of all timelike geodesics
for all gl E U(g). A similar result (Theorem 7.23) may be established for null
geodesic incompleteness using the C' topology on Lor(M). Combining these
7.1 STABLE PROPERTIES OF Lor(M) AND Con(M) 241
results yields the C1 stabzhty of past nonspacelzke geodeszc zncompleteness for
Robertson- Walker space-tames.
In the last sectlon of this chapter we consider space-times which need not
have any symmetry properties. We show (Theorem 7.30) that nonzmpnson-
ment zs a suficient condztzon for the C' stabzlzty of geodeszc zncompleteness
[cf. Beem (1994)l. Sufficient condit~ons for the stability of geodesic complete-
ness involve pseudoconvexity as well as nonimprisonment. Here a space is
said to have a pseudoconvex class of geodesics if for each compact subset K
there is a larger compact subset H such that any geodesic segment of the
class with endpoints in K lies entirely in H. Nonzmpnsonment and pseu-
doconvexzty of nonspacelzke geodesics taken together are suficzent for the C'
stabilzty of nonspacelzke geodeszc completeness. This result (Theorem 7.35) im-
plies that nonspacelike geodesic completeness is stable for globally hyperbolic
space-times.
At the end of Section 3.6 we discuss the relationship between the choice
of warping function f : (a, b) + (0, oo) and the nonspacelike geodesic incom-
pleteness of a given Lorentzian warped product space-time (a, b) x j H.
7.1 Stable Properties of Lor(M) and Con(M)
An equivalence relation C may be placed on the space Lor(M) of Lorentzian
metrics for M by defining gl,g2 E Lor(M) to be equivalent if there exists
a smooth conformal factor R : M -+ ( 0 , ~ ) such that gl = Rg2. As in
Chapter 1, we will denote the equivalence class of g in Lor(M) by C(M, g). The
quotient space Lor(M)/C of equivalence classes will be denoted by Con(M).
: There is then a natural projection map T : Lor(M) -+ Con(M) given by t 1 ~ ( 9 ) = C(M, g).
The fine C0 topology (cf. Section 3.2) on Lor(M) induces a quotient topology
on Con(M) as usual. A subset A of Con(M) is defined to be open in this
topology if the inverse image ?-'(A) is open in the fine C0 topology on Lor(M).
Con(M) may also be given the interval topology [cf. Geroch (1970a, p. 447)).
Recall from our discussion of stable causality in Section 3.2 that a partial
ordering may be defined on Lor(M) by defining gl < g2 if gl(v, v) 5 0 implies
242 7 STABILITY OF COMPLETENESS AND INCOMPLETENESS
g2(u, v) < 0 for all v # 0 in TM. It may then be checked that gl , g2 E Lor(M)
satisfy gl < g2 if and only if g; < g; for all g; E C(M,gl) and g$ E C(M, gz).
Thus the partial ordering < for Lor(M) projects to a partial ordering on
Con(M) which will also be denoted by <. A subbasis for the internal topology
on Con(M) is then given by all sets of the form
where gl and g2 are arbitrary Lorentzian metrics for M with gl < g2. The
quotient and interval topologies agree on Con(M) [cf. Lerner (1973, p. 23)).
Thus intuitively, two conformal classes C(M,gl) and C(M,g2) are close in
either of these topologies on Con(M) if and only if at all points p of M the
metrics gl and g2 have light cones which are close in T,M. A property defined on Lor(M) which holds on a Cr-open subset of Lor(M)
is said to be C' stable. Also, a property defined on Lor(M) which is invariant
under the conformal relation C is said to be confomally stable if it holds
for an open set of equivalence classes in the quotient (or interval) topology on
Con(M). The continuity of the projection map r : Lor(M) -+ Con(M) implies
that any conformally stable property defined on Lor(M) is also C0 stable on
Lor(M). Furthermore, since the fine Cr topology is strictly finer than the fine
CS topology on Lor(M) for r > s , any conformally stable property defined on
Lor(M) is also Cr stable for all r 2 0.
Example 7.1. (Stable Causality) Stable causality is conformally stable
and hence also Cr stable for all r 2 0. Indeed, a metric g E Lor(M) may be
defined to be stably causal if the property of causality is C0 stable in Lor(M)
at g.
A second example of a conformally stable property is furnished by a result
of Geroch (1970a, p. 448).
Theorem 7.2. Global hyperbolicity is conformally stable and hence Cr
stable in Lor(M) for all T 2 0.
It may also be shown that if S is a smooth Cauchy surface for (M, g), there
exists a CO neighborhood U of g in Lor(M) such that if gl E U, then S is a
Cauchy hypersurface for (M,gl) [cf. Geroch (1970a, p. 448)].
7.1 STABLE PROPERTIES OF Lor(M) AND Con(M) 243
The Ricci curvature involves the first two partials of the metric tensor but
110 higher derivatives. Using this fact, Lerner (1973) established the following
result.
Proposition 7.3. If (M,g) is a Lorentzian manifold such that g(v, v) 5 0
and v # 0 in T M imply Ric(g)(v, v) > 0, then there is a fine C2 neighborhood
U(g) of g in Lor(M) such that for all gl E U(g), the relations gl(v, v) 5 0 and
v # 0 in T M imply Ric(gl)(v, u) > 0.
It is well known that compact positive definite Riemannian manifolds are
always complete. In contrast, compact space-times need not be complete
[cf. Fierz and Jost (1965)]. Examples of Williams (1984) show that both
geodesic completeness and geodesic incompleteness are, in general, unstable
properties for space-times. In fact, the examples show that both of these
properties may fail to be stable for compact space-times as well as non-compact
space-times. These instabilities are related to questions involving sprays and
the Levi-Civita map [cf. Del Riego and Dodson (1988)j.
An inextendible closed geodesic c is one which repeatedly retraces the same
image. For spacelike and timelike geodesics this implies the geodesic is com-
plete since these geodesics have constant nonzero speed and thus increase in
affine parameter by the same amount for each circuit of the image. However,
there are closed null geodesics which are inextendible and incomplete. This
incompleteness results from the fact that the tangent vector to a null geodesic
may fail to return to itself each time the geodesic traverses one circuit of the
image. For closed null geodesics, the tangent vector may return to a scalar
multiple of itself where the multiple is different from one. In this case, the
closed geodesic fails to be a periodic map and the domain is an open subset
of W which is bounded either above or below. More precisely, we have the
following result.
Lemma 7.4. Let (M,g) be a semi-Riemannian manifold with an inex-
tendible null geodesic P : (a, b) + M satisfying P(0) = P(1) and p ( 0 ) =
XP/(O). If X = 1, then ,D is complete. If 0 < X < 1, then P is incomplete with
a > -m and b = +m. If 1 < A, then ,D is incomplete with a = -co and
b < +m.
244 7 STABILITY OF COMPLETENESS AND INCOMPLETENESS
Proof. If X = 1, then P[O, l ] is a closed geodesic which is periodic (i.e.,
p(t + 1) = P(t) for all t), and clearly the domain of ,B is all of R.
If 1 < A, then the speed of /3 increases by a factor of X each time around the
image in the positive direction. In particular, if y(s) = P(Xs), then y(Xml) =
P(1) = P(0) = y(0) and y'(0) = Xfl(0) = fl(1). Thus y(t) = P(1 + t), and
the first circuit of y (i.e., 0 5 t 5 X-') corresponds to the second circuit of ,O
(i.e., 1 < t 5 1 + A-I). Thus a countably infinite number of circuits of P in
the positive direction starting with t = 0 increase the &ne parameter of by
It follows that the domain of P has a finite upper bound, and one finds
b = (1 - < +m. Of course, traversing the geodesic P in the negative
direction starting at t = 0 changes the affine parameter by x:=l An = co,
which yields a = -co.
If 0 < X < 1, then similar arguments show a > -m and b = +m.
We will now consider two examples of Williams (1984). These examples
are for metrics of the form dx dy + f (x)dy2 on the torus. Let S1 x S1 =
M = {(x, y) 10 5 x 5 27r, 0 5 y < 27r) with the usual identifications. Let
f : W1 -+ W1 be a periodic function with period 2a. One may then easily
calculate the Christoffel symbols for the metric g = dxdy + f (x)dy2.
Example 7.5. (Instability of Geodesic Completeness) Let g = dxdy + f (x)dy2. If f is identically zero, one has a flat metric g = dxdy on the
torus M which is geodesically complete. In particular, the closed null geo-
desic corresponding to x = 0 is complete. On the other hand, using fn(x) =
( l ln ) sin(x) one may directly calculate that the closed null geodesic x = 0 of
gn = dx dy + fn(x)dy2 is not complete [cf. Williams (1984)l. This incomplete-
ness results from the fact that the tangent vector to this null geodesic of g,
fails to return to itself each time the geodesic traverses the circle corresponding
to x = 0. Each time around the circle, the tangent vector returns to a scalar
multiple of itself where the multiple is different from one. Lemma 7.4 yields
that this null geodesic is incomplete for each g,. Since each Cr neighborhood
7.1 STABLE PROPERTIES OF Lor(M) AND Con(M) 245
of the original flat metric g has metrics g, for large n , it follows that geodesic
completeness is not stable at the Lorentzian metric g.
This last example may be generalized [cf. Beem and Ehrlich (1987, p. 328)].
If (M, g) is a geodesically complete semi-Remannian manifold which contains
a closed null geodesic, then each CT neighborhood of g contains incomplete
metrics. On the other hand, it has recently been shown that any Lorentzian
metric on a compact manifold which has zero curvature, i.e., R = 0, is geodesi-
cally complete [cf. Carrikre (1989), Yurtsever (1992)l. Hence no deformation
of the flat metric g = dxdy on the torus through flat metrics can produce
geodesic incompleteness, as in Example 7.5.
Example 7.6. (Instability of Geodesic Incompleteness) Williams modi-
fied his previous example somewhat to demonstrate the instability of geodesic
incompleteness. Using f (x) = 1 - cosx and xo with 0 < xo < 27r, one may
show that the null geodesic starting at ($0, yo) with dyldx # 0 at xo will
be incomplete for the metric g = dxdy + f (x)dy2. Williams shows this by
solving for x = x(t) and then obtaining an integral formula for y = y(x)
involving integrating the reciprocal of f(x). However, he also finds that
using f,(x) = 1 - cosx + ( l ln ) one obtains geodesically complete metrics
gn = dxdy + fn(x)dy2 which are arbitrarily close to the incomplete metric g.
Examples 7.5 and 7.6 both involve two-dimensional Lorentzian manifolds.
However, the examples may be slightly modified to give examples which show
geodesic completeness and incompleteness are not necessarily stable for any
metric signature (s, r ) with both s and r positive.
If (MI, gl) and (M2,gz) are semi-Riemannian manifolds, the product mani-
fold MI x M2 may be given the usual product metnc g = gl @g2, and with this
metric the geodesics of the product are of the form (yl(t),y2(t)) where each
factor y,(t) is either a geodesic of (Mz,g,) or else is a constant map. Clearly,
the product MI x M2 is geodesically complete if and only if each (M,,gz) is
geodesically complete.
The semi-Euclidean space of signature (s, r) is the product manifold MI x
M2 where Ml = W8 with g1 = C:=l -dxz2 and M2 = Rr with gz = Cr=l dxt2.
246 7 STABILITY OF COMPLETENESS AND INCOMPLETENESS
By taking the first manifold (Ml,gl) to be the manifold (M,g) of Exam-
ple 7.5 (respectively, Example 7.6) and taking the second manifold (M2,g2)
to be the semi-Euclidean space of signature ( r - 1,s - I ) , one may obtain
an example MI x M2 which shows that geodesic completeness (respectively,
incompleteness) may fail to be stable for signature (s, r ) whenever neither s
nor T is zero.
The situation for positive definite (i.e., s = 0) and negative definite (i.e.,
T = 0) signatures is quite different. For these signatures, both geodesic com-
pleteness and geodesic incompleteness are C0 stable. For example, if (M, g) is
positive definite, then the set U(g) consisting of all positive definite metrics h
on M with 1 h(v,v) < 4 - < - 4 g(v1v)
for all nontrivial vectors v is a Co-open subset of the collection Riem(M) of
all Riemannian metrics on M. Clearly, U(g) contains the metric g. If h is a
fixed metric in U(g), then any given curve has an h-length Lh and a g-length
L, with 1 -L , 5 Lh 5 2Lg. 2
This inequality shows that either g and h are both complete or else both are
incomplete. It follows that geodesic completeness and geodesic incompleteness
are C0 stable for positive definite spaces. The same reasoning applies in the
negative definite case, and consequently one obtains the following result.
Proposition 7.7. Let (M, g) be either a positive definite or negative def-
inite semi-Riemannian manifold. If (M, g) is geodesically complete, then for
each r 2 0 there is a Cr neighborhood U(g) with each gl E U(g) complete. If
(M, g) is geodesically incomplete, then for each r 2 0 there is a Cr neighbor-
hood U(g) with each gl E U(g) incomplete.
We summarize this last result and the examples of Williams in the following
remark.
Remark 7.8. Geodesic completeness and geodesic incompleteness are sta-
ble properties in the Whitney topology for both positive definite and nega-
tive definite semi-Riemannian manifolds. For all other metric signatures (s, T),
7.2 THE C1 TOPOLOGY AND GEODESIC SYSTEMS 247
there are examples to show that both geodesic completeness and geodesic in-
completeness may fail to be stable for the Whitney Ck topologies for k > 0.
In this chapter the manifold M is always fixed. However, one-parameter
families (MA, g ~ ) of manifolds and Lorentzian metrics have been considered in
general relativity [cf. Geroch (1969)].
7.2 The C' Topology a n d Geodesic Systems
If (M, g) is an arbitrary Lorentzian manifold, then metrics in Lor(M) which
are close to g in the fine C1 topology have geodesic systems which are close to
the geodesic system of g. The purpose of this section is to give a more analytic
formulation of this concept, which is needed for our investigation of the C1
stability of null geodesic incompleteness for Robertson-Walker space-times in
Section 7.3.
We begin by recalling a well-known estimate from the theory of ordinary
differential equations Icf. Birkhoff and Rota (1969, p. 155)]. We will al-
ways use /1x1I2 to denote the Euclidean norm [ELl xi2] of the point x =
(51, 2 2 , . . . ,x,) E Wrn.
Proposition 7.9. Suppose that f = ( f l , . . . , f,) and h = (h l , . . . , h,) are
continuous functions defined on a common domain D R x Wm, and suppose
that f satisfies the Lipschitz condition
Ilf (5-1x1 - f(~15)112 I Lllx - 4 1 2
for all (s, x), ( s , f ) E D.
Let x(s) = (xl(s), . . . , xm(s)) and y(s) = (y~(s ) , . . . , ym(s)) be solutions for
0 5 s 5 b of the differential equations
dx - = f (8, x) and - d~ = h(s, y), ds ds
respectively. Then if 11 f (s, x) - h(s, x) ] I 2 5 e for all (s, x) E D with 0 5 s 5 b,
the following inequality holds for all 0 5 s 5 b:
246 7 STABILITY O F COMPLETENESS AND INCOMPLETENESS
By taking the first manifold (M1,gl) to be the manifold (M,g) of Exam-
ple 7.5 (respectively, Example 7.6) and taking the second manifold (Mz,g2)
to be the semi-Euclidean space of signature (T - 1, s - I), one may obtain
an example MI x M2 which shows that geodesic completeness (respectively,
incompleteness) may fail to be stable for signature ( s , ~ ) whenever neither s
nor T is zero.
The situation for positive definite (i.e., s = 0) and negative definite (i.e.,
T = 0) signatures is quite different. For these signatures, both geodesic com-
pleteness and geodesic incompleteness are C0 stable. For example, if (M, g) is
positive definite, then the set U(g) consisting of all positive definite metrics h
on M with
L, with
If (M, g) is an arbitrary Lorentzian manifold, then metrics in Lor(M) which
are close to g in the fine C' topology have geodesic systems which are close to
the geodesic system of g. The purpose of this section is to give a more analytic
formulation of this concept, which is needed for our investigation of the C1
stability of null geodesic incompleteness for Robertson-Walker space-tlmes in
for all nontrivial vectors v is a Co-open subset of the collection Kem(M) of
all Riemannian metrics on M. Clearly, U(g) contains the metric g. If h is We begin by recalling a well-known estimate from the theory of ordinary
fixed metric in U ( g ) , then any given curve has an h-length LI, and a g-lengt fferential equations [cf. Birkhoff and Rota (1969, p. 155)]. We will al-
ays use IlzIlz to denote the Euclidean norm [C:, xZ2] of the point x = 1 -L, 5 Lh 1 2Lg. 2
This inequality shows that either g and h are both complete or else bot roposition 7.9. Suppose that f = ( f l , . . . , f,) and h = (h l , . . . , h,) are
incomplete. It follows that geodesic completeness and geodesic incomple
are CO stable for positive definite
negative definite case, and consequently one obtains the following result Ilf ( s > x) - f (.,z)llz 5 Lllx - z112
Proposition 7.7. Let (M, g)
inite ,cjemi-fi~annian manifold.
& r 2 0 thexe i s a CY neighborhood . , y,(s)) be solutions for
.\M. 3) & *xm*e- -
k,GT,i *edLz?-s'- J -111-
7.2 T H E C' TOPOLOGY AND GEODESIC SYSTEMS 247
there are examples to show that both geodesic completeness and geodesic in-
completeness may fail to be stable for the Whitney C"opologies for k > 0.
In this chapter the manifold M is always fixed. However, one-parameter
families ( M A , gx) of manifolds and Lorentzian metrics have been considered in
general relativity [cf. Geroch (1969)l.
7.2 T h e C' Topology a n d Geodesic Systems
248 7 STABILITY OF COMPLETENESS AND INCOMPLETENESS
Now let M be a smooth manifold, and let (U, x) be any coordinate chart
for M. We may obtain an associated coordinate chart
z = ( ~ 1 , ~ 2 , . . - ,xn,xn+lr. - . 1 ~ 2 n )
for TMlu as follows. Let d/dxl, . . . , d/dxn be the basis vector fields defined
on U bv the local coordinates x = (xi,. . . ,xn). Given v E TqM for q E n we may write v = zEl ai $1". Then E(v) is defined to be Z(v)
(xl(q), x2(q), . . . , xn(q),al, a2,. . . ,an). These coordinate charts may then be
used to define Euclidean coordinate distances on U and TMIU. Explicitly,
given p, q E U and v, w f TMIU, set
and
respectively. Also if T 2 0 is given, we will use the notation IJgl -g2\lr,v < 6
gl,g2 E Lor(M) and a positive constant 6 > 0 to mean that calculating wi
the local coordinates (U,x), all the corresponding entries of the two metri
tensors and all their corresponding partial derivatives up to order r are 6clos
at each point of U.
We will denote the Christoffel symbols of the second kind for gl,g2 E
Lor(M) by rJk(gl) and rJk(g2): respectively. Then for a = 1,2, the geodesic
equations in the coordinate chart (U, x) for (M, g,) are given by
for 1 5 i, j, k 5 n, where we employ the Einstein summation convention
throughout this chapter.
7.2 THE C1 TOPOLOGY AND GEODESIC SYSTEMS 249
We will use the notation expq[g,] for the exponential map at q E (M,g,),
a = 1,2. If v E TMIU, then s --+ expq[ga](sv) is the solution of (7.1) in U
with initial conditions (q,v) for (M,g,). In order to apply Proposition 7.9
to these exponential maps, we identify TMlu with a subset of W2" using the
coordinate chart (TMIu , Z) and define f (s, X ) = f (X) and h(s, X ) = h(X)
fi+n(X) = -rjk(gl)~j+n~k+n,
and = -rjk(92)xj+nxk+n
for 15 i , j , k 5 n a n d X = (xl ,..., x2,) E WZn.
Lemma 7.10. Let (U, x) be a local coordinate chart for the n-manifold
M. Let (p, v) E TMIu, and assume that cl(s) = exp,[gl](sv) lies in U for all
0 < s 5 b. Given E > 0, there exists a constant 6 > 0 such that JJv - wl12 < 6
and llgl-g2111,u < 6implythatca(s) = expq[g2](sw) liesinU ford10 5 s < b,
and moreover,
Ibj 0 CI)'(S) - (53 O CZ)'(S)I < E
for all 15 j 5 n a n d 0 5 s 5 b.
Proof. Let f (X) and h(X) be defined as above. Then X(s) = (cl (s), cll(s))
and Y(s) = (cz(s), czl(s)) are solutions to the differential equations dX/ds =
f(Xf and dY/ds = h(Y), respectively. Choose Do to be an open set in TMlu
about the image of the curve X(s) such that DO is compact. Then there exists
a constant L such that f satisfies a Lipschitz condition 11 f (X) - f ( x ) ] / 2 5
L1IX - Xllz on Do.
We may make the term jlX(0) - Y(0)IJ2 in the estimate of Proposition
7.9 as small as required by making Ilu - wll2 small. Furthermore, since the
Chriitoffel symbols depend only on the coefficients of the metric tensor and
on their first partial derivatives, we may make 11 f (X) - h(X)ll2 as small as we
250 7 STABILITY OF COMPLETENESS AND INCOMPLETENESS
wish on Do by requiring that IJgl - g2jll,u be small. Hence for a sufficiently
small 6 > 0, Proposition 7.9 may be applied to guarantee that c2(s) E U for all
0 5 s 5 b and also to yield the estimate IIX(s) - Y(s)l12 < E for all 0 5 s 5 b.
Consequently, Ixi(s) - yi(s)I < E for all 1 5 i 5 2n and 0 5 s < b. In view of
(7.1), this establishes the desired inequalities. 0
The following slightly more technical lemma, which is needed in Section
7.3, follows directly from Lemma 7.10 by using the triangle inequality and the
continuity of the geodesic solution X ( s ) = (cl(s), cl' (s)).
Lemma 7.11. Let (U, x) be a local coordinate chart on the n-manifold
M. Suppose that cl(v) = exp,[gl](sv) lies in U for all 0 < s 5 b. Let
E > 0 and s l with 0 < s l < b be given. Then there exists a constant 6 > 0
such that if ilv - wllz < 6, 1/91 - 92111,~ < 6, and ]SO - s l J < 6, the geodesic
c2(s) = exp,[g2](sw) lies in U for all 0 5 s < b, and moreover,
and
for all 1 5 j 5 n. Furthermore, if (xl 0 cl)'(sl) # 0, then the constant 6 > 0 may be chosen
such that
7.3 Stability of Geodesic Incompleteness for Robertson-Walker
Space-times
In this section we investigate the stability in the space of Lorentzian rnetrics
of the nonspacelike geodesic incompleteness of Robertson-Walker space-times
M = (a, b) x f H (cf. Definition 5.10). It turns out, however, that the proof of
the C0 stability of timelike geodesic incompleteness uses only the homogeneity
of the Riemannian factor (H, h) and not the isotropy of (H, h). Accordingly,
7.3 GEODESIC INCOMPLETENESS 251
we will formulate the results in the first portion of this section for the larger
class of Lorentzian warped products M = (a, b) x f H with -00 5 a < b 5 t o o
and (H, h) a homogeneous Riemannian manifold. We will let xl = t denote
the usual coordinate on (a , b) throughout this section.
In order to study the geodesic incompleteness of such space-times under
metric perturbations, it is helpful to use coordinates adapted to the product
structure. Fix p = ( t l ,h l ) E (a,b) x H. Since the submanifold {tl) x H
of M is spacelike, the Lorentzian metric g for M restricts to a positive defi-
nite inner product on the tangent space to this submanifold at p. Identifying
{tl) x H and H, we may use an orthonormal basis for the tangent space to
i t l ) x H at p to define Riemannian normal coordinates xz, . . . , x, for H in a
neighborhood V of hl. Then (XI, x2,. . . , x,) defines a coordinate system for
M on (a, b) x V. By construction, g has the form diag{-1, + I , . . . , +1) at p
in these coordinates. Because the submanifold {tl) x H is not necessarily to-
tally geodesic iff is nonconstant, these coordinates are not necessarily normal
coordinates. Nonetheless, the coordinates (XI, x2, . . . , x,) are well adapted to
the product structure since the level sets xl(t) = X are just {A) x V.
We will say that coordinates (XI, . . . , x,) constructed as above are adapted
at p E M and call such coordinates adapted coordinates. It will also be useful
to define adapted normal neighborhoods.
Definition 7.12. (Adapted Normal Neighborhood) An arbitrary con- vex normal neighborhood U of (M,g) with compact closure 77 is said to
be an adapted normal neighborhood if U is covered by adapted coordinates
(xl,x2,. . . ,x,) which are adapted at some point of U such that the following
hold:
(1) At every point of U , the components gij of the metric tensor g ex-
pressed in the given coordinates (XI, xz, . . . , x,) differ from the matrix
diag(-1, +I,. . . , +1) by at most 112.
(2) The metric g satisfies g <U 71, where 71 is the Minkowskian metric
ds2 = -2dx12 +dxz2 + ... + dxn2 for U (see the definition of stably
causal in Section 3.2 for the notation g <u ql).
252 7 STABILITY OF COMPLETENESS AND INCOMPLETENESS
Thus on the neighborhood U of Definition 7.12 the Lorentzian metric g may
be expressed as
where the functions kiJ : U -+ IR satisfy Ikajl I 1/2 for all 1 5 i, j I n. For use in the sequel, we need to establish the existence o f countable chains
{Uk) o f adapted normal neighborhoods covering future directed, past inex-
tendible, timelike geodesics o f the form c(t) = ( t , yo). Since in Definition a 7.14 and in Lemma 7.15 it is possible that a = -co, we adopt the following
convention throughout this section.
Convention 7.13. Let wo denote any fixed interior point o f the interval
( a , b).
Now we make the following definition.
Definition 7.14. (Admissible Chain) Let M = (a , b) x f H denote a Lor-
entzian warped product with metric g = -dt2 @ fh. Fix any yo E H and let
c : (a , wo] -+ ( M , 3 ) be the future directed, past inextendible, timelike geodesic
given by c( t ) = ( t , yo). A countable covering {Uk)p=l o f c by open sets and
a strictly monotone decreasing sequence {tk)& with tl = wo and tk -+ a+
as k -+ co is said to be an admissible chain for c : (a,wo] -+ ( M , g ) i f the
following two conditions hold:
(1) Each Uk is an adapted normal neighborhood containing c(tk) = (tk, yo)
which is adapted at some point of c.
(2) For each k , every future directed and past inextendible nonspaceliie
curve u ( t ) = ( t , a z ( t ) ) with u ( t k ) = ( t k , yo) remains in Uk for all t with
tk+l I t I t k .
Any future directed nonspacelike curve u in ( M , g ) may be given a par-
ametrization of the form u ( t ) = (t, ul ( t ) ) . Thus condition (2) applies to all
future directed nonspacelike curves issuing from (tk, y o ) W e now show that
admissible chains exist.
Lemma 7.15. Let M = (a,b) x f H with a 2 -m and 3 = -dt2 $ fh be a Lorentzian warped product. For any yo E H , the timelike geodesic
7.3 GEODESIC INCOMPLETENESS 253
c : (a,wo] -4 ( M , i j ) given by c(t) = (t, yo) has a covering by an admissible
chain.
Proof. W e will say that {Uk), { tk) is an admissible chain for c I (9, wo], 9 > a, i f {Uk), { t k ) satisfy the properties o f Definition 7.14 except that tk -+ Bt
as k + oo instead of t k --+ a+ as k -t co. Set
T = inf{8 E [a, wo] : there is an admissible chain
{Uk), I t k ) for c 1 (el, wo] for all 0' 9).
W e must show that T = a.
By taking an adapted normal neighborhood centered at c(wo), it is easily
seen that T < WO. Suppose that T > a. Let U be any adapted normal neigh-
borhood adapted at the point (T , yo) E M . Choose T > T such that all future
directed nonspacelike curves u ( t ) = (t, u l ( t ) ) originating at (r , yo) lie in U for
all T - E I t I T , where E > 0. There exists an admissible chain {U,), {t,) for - C ( ( T + 7 ) / 2 , W O ] with tm < r for some m. Define U, = = U. Extending - the finite chain {U1,U2,. . . , Urn-1, Urn, Um+l), { t l , t2 , . . . , t m - l l t m l ~ - 6) to
an infinite admissible chain yields the required contradiction. 0
W e now show that the subset o f Uk for which property (2) o f Definition 7.14
holds may be extended from the point ( t k , yo) to a neighborhood I t k ) x Vk(yo)
in { t k ) x H . The notation Ilg - glllo,pk < 6 has been introduced in Section
7.2.
Lemma 7.16. Let {Uk) , { tk) be an admissible chain for the timelike ge-
odesic c ( t ) = (t, yo), c : (a,wo] -+ ( M , g ) . For each k , there is a neigh-
borhood Vk(y0) of yo in H such that any future directed nonspacelike curve
a ( t ) = ( t , a l ( t ) ) with ~ ( t k ) E { tk ) x &(yo) remains in Uk for d l t with
tk+l 5 t 5 t k . Furthermore, Vk(yo) and 6 > 0 may be chosen such that
i f gl E Lor(M) and I(g - glllo,uk < 6, then the following two conditions
are satisfied. If y ( t ) = (t,-yl(t)) is any nonspacelike curve of ( M , g l ) with
~ ( t k ) E { tk ) x V ~ ( Y O ) , then
(1 ) y remains in Uk for tk+l 5 t 5 t k ; and
(2) The gl-length o f yl [ t k+~, tk] is at most &n(tk - tk+l).
254 7 STABILITY OF COMPLETENESS AND INCOMPLETENESS
Proof. First recall that z : M = (a, b) x H --+ W given by ~ ( t , h ) = t serves
as a global time function for M . In particular, the vector field V z satisfies
g ( V z , V z ) < 0 at all points of M . Define g E Lor(M) by
I t follows that g < $ on M so that U2 = { i 2 E Con(M) : g2 < ~ ( 9 ) ) is an
open neighborhood o f C ( M , g ) in Con(M). Let Ul = r-l(U2). Then Ul is
a Co-open neighborhood of g in Lor(M) such that i f gl E Ul , the projection
map z : M --+ W is a global time function for ( M , gl). Hence the hypersurfaces
{ t ) x H , t E (a , b), remain spacelike in ( M l g l ) . Thus any nonspacelike curve
y of ( M , gl ) , gl E Ul , may be parametrized as y ( t ) = ( t , y l ( t ) ) . Thus the
lemma will apply to any nonspacelike curve of ( M , gl ) originating at any point
o f { t k ) x Vk(yO) provided g1 E Ul is sufficiently close to g on Uk.
Let ( x l , . . . , x,) denote the given adapted coordinate system for the adapted
normal neighborhood Uk. In view of condition (2) o f Definition 7.12, we may
find 61 > 0 such that llg - g1110,~, < 61 implies gl < 72 on Uk, where 772 is
the Lorentzian metric on Uk given in the adapted local coordinates by 772 =
-3 dx12 + dx22 + - . . + dxn2. Secondly, since Co-close metrics have close light
cones, it follows by a compactness argument that there exist a neighborhood
Vk(yo) o f yo in H and a constant 62 > 0 such that i f gl E Lor(M) satisfies
llg-glllo,vk < 62 and y ( t ) = ( t , y l ( t ) ) is any future directed nonspacelike curve
of ( M , g l ) with y(tk) E { tk ) x V ~ ( Y O ) , then y( t ) E Uk for tk+l i t I tk.
I t remains to establish the length estimate (2). Set 6 = min{61,62, 1/21.
Suppose that g' E Lor(M) satisfies jig'-gllo,uk < 6, and let y ( t ) = (t, y l ( t ) ) be
any nonspacelike curve o f ( M , g') with y (tk) E { tk) x Vk (yo) , y : [ t k + l , tk] -+ M .
Let L ( y ) denote the length o f y in ( M , g'). Thus
Fkom Definition 7.12 and the choice of the 6's, we have lgi31 5 (1+ 1/2)+1/2 =
2 and Iyil(t)l 5 & for all 1 2 i, j 5 n. Thus, as required,
7.3 GEODESIC INCOMPLETENESS 255
Assuming now that the Riemannian factor (H, h ) o f the Lorentzian warped
product is homogeneous, we may extend Lemma 7.16 from Uk to [tk+l, t k ] x H.
W e will use the notation lgl - glo < 6, as defined in Section 3.2.
Lemma 7.17. Let ( M , g ) be a Lorentzian warped product with ( H , h )
homogeneous, and let {Uk), { tk) be an admissible chain for c( t ) = ( t , yo),
c : (a , wo] -+ M. For each k , there is a continuous function bk : [tk+l, tk] x H -+
(0, co) such that i f gl € Lor(M) and lg - gllo < 6k on [&+I, tk] x H , then any
nonspacelike curve y ( t ) = ( t , y l ( t ) ) , y : [tk+l,tk] -' ( M , g l ) , joining any point
of {tk+l) x H to any point o f { t k ) x H , has length at most &n(tk - tk i l ) .
Proof. Fix any k > 0. Let 6 > 0 be the constant given by Lemma 7.16
such that i f gl E Lor(M) satisfies Ilgl - gllo,pk < 6, then any nonspacelike
curve y ( t ) = ( t , y l ( t ) ) in ( M , g l ) with y(tk) E { tk ) x &(yo) remains in Uk for
tk+l I t i t k and has length at most &n (tk - &+I) . Also let ( x l , . . . , x,)
denote the given adapted coordinates for Uk.
W e may find isometries {$,)El in I ( H ) such that i f yi = $i(yo) and
V k ( ~ i ) = $i(Vk(yo)), then the sets {l/k(yi))& together with Vk(y0) form a
locally finite covering of H. Let @, : M -+ M be the isometry given by
@i(t , h) = ( t , +i(h)), and set Ui = @i(Uk) for each i. Then the sets {Gi)
cover [tk+l, t k ] X H and ( X I , x2 0 ah1, . . . , x, 0 @, l ) form adapted local co-
ordinates for Ui for each i. Since everything is constructed with isometries,
the constant 6 > 0 that works in Lemma 7.16 for Uk and c( t ) = ( t , yo) works
equally well for each 6i and @i 0 c, provided that the adapted coordinates
( ~ 1 ~ x 2 0 @ i l l . . . , x , 0 a;') are used for Ui. ~f we let bk : [tk+l, tk] x H -+ M
be any continuous function such that ilgl - gllo < hk on [tk+1, t k ] x H implies
that llgl - g110,5i < 6 for each i, then the lemma is immediate from Lemma
7.16 Cl
W e are now ready to prove the C0 stability of timelike geodesic incomplete-
ness for Lorentzian warped products M = (a , b) x f H with a > -m and (H. h )
homogeneous.
Theorem 7.18. Let ( M , g ) be a warped product space-time of the form
M = (a, b) x f H with a > -03, g = -dt2 @ fh, and (H , h ) a homogeneous
256 7 STABILITY OF COMPLETENESS AND INCOMPLETENESS 7.3 GEODESIC INCOMPLETENESS 257
Riemannian manifold. Then there exists a fine CO neighborhood U ( g ) of g 'I'heorem 7-19. Let ( M , g) be a Robertson-Walker space-time o f the form
in Lor(M) o f globally hyperbolic metrics such that all timelike geodesics of M = (a , b) x f H with a > -W. Then there exists a fine C0 neighborhood U ( g )
( M , gl ) are past incomplete for each gl E U(g) . of g in Lor(M) o f globally hyperbolic metrics such that all timelike geodesics
of ( M , g l ) are past incomplete for each gl E U(g). Proof. Fix any yo E M and let c : (a,wo] 4 M be the past inextendible
future directed geodesic given by c(t) = (t, yo). Let {Uk), { tk) be an admissible I f we change the time function on ( M , 9 ) to T I : M -+ R defined by ( t , h) =
chain for c, guaranteed by Lemma 7.15. Also choose 6k : [&+I, t k ] x H 4 (0, a) -t, and apply Lemmas 7.16 and 7.17 to the resulting space-time, we obtain
for each tk according to Lemma 7.17. Let 6 : M -+ (0, CO) be a continuous the exact analogue o f these lemmas for the future directed timelike geodesic c :
function such that 6(q) I &(q) for all q E [tk+i,tk] X H and each k > 0. ["o, b) + ( M , 9 ) given by c(t) = (t, yo) in the given space-time. Hence i f ( M , g )
Set v ~ ( ~ ) = {g l E Lor(M) : lgl - glo < 6). Since global hyperbolicity is a is a Lorentzian warped product M = (a, b) x f H with ( H , h ) homogeneous and
Co-open condition, we may also assume that all metrics in q ( g ) are globally b < oo, the same proof as for Theorem 7.18 yields the C0 stability o f the future
hyperbolic. timelike geodesic incompleteness. Combining this remark with Theorem 7.18
By the first paragraph o f the proof of Lemma 7.16, we may choose a CO then yields the following result.
V2(g) o f g in Lor(M) such that for all gl E V Z ( ~ ) , each hyper- Thorem 7-20. Let ( M , g ) be a Lorentzian warped product o f the form
surface { t ) x H , t E (a, b), is spacelike in (M,gl) . Then every nons~acelike M = (a , b) X f H , g = -dt2 @ f h , with both a and b finite and ( H , h ) ho-
curve y : (a, p) -+ ( M , g ~ ) may be parametrized in the form y( t ) = ( t , yl(t)) . mogeneous. Then there is a fine C0 neighborhood U ( g ) o f g in Lor(M) of
Hence Lemma 7.16 may be applied to all inextendible nons~acelike geodesics globally hyperbolic metrics such that all timelike geodesics o f ( M , gl ) for each
in ( M , g l ) with gl E Vz(g). 91 f U(g) are both past incomplete and future incomplete.
Now U ( g ) = Vl(g)nVz(g) is a fine neighborhood o f in the CO topology. It is interesting to note that while the finiteness of a and b is essential to
Let gl E U(g] , and let y : ( a , p ) -t ( M , gl) be any future directed inextendible the proof o f Theorem 7.20, the proof is independent of the particular choice o f timelike geodesic. W e may assume that { t l ) x H is a Cauchy surface
warping function f : (a, b) -t (0, CO). While the homogeneity o f the Riemann- ( M , g l ) by the arguments o f Geroch (1970a, p. 448), and hence there exists an ian factor ( H , h ) is also used in the proof o f Theorem 7.20, no other geometric so E ( a , p) such that y(s0) E { t ~ } x H. In passing f k m {tk+i) X H t o { tk) r topological property o f ( H , h ) is needed. the gl length of y is at most &n ( t k - &+I), applying Lemma 7.17 for each In general relativity and cosmology, closed big bang models for the uni- k. Summing up these estimates, it follows that the gl length o f (a, so] is at
se are considered [cf. Hawking and Ellis (1973, Section 5.3)]. These models most f i n (tl - a) . Since yl (a, so] is a past inextendible timelike geodesic Robertson-Walker space-times for which b - a < oo and H is compact. finite gl length, it follows that y is past incomplete in ( M , g l ) . 0
ce Theorem 7.20 implies, in particular, the 60 stability o f timelike geodesic
Lerner raised the following question (1973, p. 35) about the Roberts mpleteness for these models.
Walker big bang models ( M , g): under small C2 perturbations o f the me now turn t o the proof o f the C1 stability o f null geodesic incompleteness
does each nonspacelike geodesic remain incomplete? Since the Ftiemannl Robertson-Walker space-times. Taking M = ( 0 , l ) x f W with f ( t ) =
factor ( H , h) of a Robertson-Walker spacetime is homogeneous, we obt )-2 and 3 = -dt2 @ fdx2, it may be checked using the results o f Section
the following corollary t o Theorem 7.18 which settles affirmatively for time that the curve 7 : (-m,O) -t ( M , g ) given by ~ ( t ) = (et, e2t ) is a past
geodesics the question raised by Lerner (1973, P. 35). plete null geodesic. Thus by choosing the warping function suitably, it
258 7 STABILITY OF COMPLETENESS AND INCOMPLETENESS
is possible to construct Robertson-Walker space-times with a > -m which
are past null geodesically complete. Thus unlike the proof o f stability for
timelike geodesic incompleteness, it is necessary to assume that ( M , g) contains
a past incomplete (respectively, past and future incomplete) null geodesic to
obtain the null analogue o f Theorem 7.19 (respectively, Theorem 7.20). Not
surprisingly, the proof of the C 1 stability of null geodesic incompleteness is
more complicated than for the timelike case since affine parameters must be
used instead of Lorentzian arc length to establish null incompleteness. Also
for the proof o f Lemma 7.22, we need the isotropy as well as the homogeneity
o f ( H , h). Thus we will assume that M = (a , b) x f H is a Robertson-Walker
space-time in the rest o f this section.
Let (V, x l , . . . , x,) denote an adapted normal neighborhood of ( M , g) with
adapted coordinates ( x l , . . . , x,). For the proof o f Lemma 7.22, it is neces-
sary t o define a distance between compact subsets o f vectors that are null for
different Lorentzian metrics for M and are attached at different points o f V .
Recall from Section 7.2 that local coordinates ( x l , . . . , x,) for V give rise to
local coordinates Z = ( x l , . . . ,x,, x,+l,. . . , x2,) for T V = TMIV. Thus given
any q E V , gl E Lor(M), and cu > 0, we may define
S(q,a ,gl ) = { v E T,M : gl(v,v) = 0 and z,+l(v) = -a).
Then S(q, a, gl) is a compact subset o f T,M for any cr > 0 and gl E Lor(M).
Given p,q f V , gl,g2 E Lor(M), and cul,az > 0, define the Hawdorff distance
between S (p , ~ 1 , gl) and S(q, a2,g2) by
The continuity o f the components o f the metric tensor g as functions g,? :
V x V -+ W and the closeness o f light cones for Lorentzian metrics close in the
C0 topology imply the continuity o f this distance in p, a , and g [cf. Busemann
(1955, pp. 11-12)].
7.3 GEODESIC INCOMPLETENESS 259
Lemma 7.21. Let V be an adapted normal neighborhood adapted at p E
(M,g) . Given a > 0 and E > 0, there exists 6 > 0 such that Ilp - ql/2 < 6,
g1 E Lor(M) with Ilg - g1110,v < 6, and la1 - a / < 6 together imply that
dist(S(91 cul,gl), S(P, a , 9 ) ) < c.
Now let ( M , g) be a Robertson-Walker space-time (a , b) x f H which is
past null incomplete. Thus some past directed, past inextendible null geodesic
c : [O,A) -+ (M,g) is past incomplete (i.e., A < ca). Since ( H , h) is isotropic
and spatially homogeneous, and since isometries map geodesics to geodesics,
it follows that all null geodesics are past incomplete. W e now fix through the
proof o f Theorem 7.23 this past inextendible, past incomplete, null geodesic
c : [0, A ) --+ ( M , g ) with the given parametrization.
Let (wo, yo) = c(0) E M = (a, b) x f H. With this choice o f wo, apply
Lemma 7.15 to the future directed timelike geodesic t -+ ( t , yo), t 5 wo, to
get an admissible chain {Uk), { tk) for this timelike geodesic. Using this choice
of { tk) , we may find sk with 0 = sl < s:, < . . . < sk < ... < A such that
c ( s ~ ) f { tk ) x H for each k. Set Ask = sk+l - sk. AS above, let xl : M -+ W denote the projection map x l ( t , h ) = t on the first factor o f M = (a , b) x f H.
Notice that i f (V, X I , x2,.. . ,x,) is any adapted coordinate chart, then the
coordinate function xl : V -r R coincides with this projection map. I f y is any
smooth curve o f M which intersects each hypersurface { t ) x H o f M exactly
once and y ( s ) E { t ) x H , we will say that l ( x lo y)'(s)l is the $1 speed o f y at
{ t ) x H. In particular, we will denote by ak = / ( X I o c)'(.qk)l the x l speed o f
the fixed null geodesic c : [0, A ) -+ ( M , g) at { t k ) x H for each k .
Lemma 7.22. Let E > 0 be given. Then for each k > 0 , there exists a
continuous function 6k : [tk+l, tk] X H -' ( 0 , m ) with the following properties.
Let gl E Lor(M) with lg - g1]1 < 6k on [tk+l,tk] x H , and let y : [O,B) --+ M
be any past directed, past inextendible, null geodesic with $0) E i t k ) x H
and with x1 speed o f cuk at { tk) x H. Then y reaches { & + I ) x H with an
ncrease in f ine parameter o f at most 2Ask, and moreover, the x1 speed 9 of
y at {tk+l) x H satisfies the estimate
260 7 STABILITY OF COMPLETENESS AND INCOMPLETENESS
Proof. Let c : [0, A) -+ (M, g) be the given past incomplete null geodesic as
above. Fix k > 0. By the spatial homogeneity of Robertson-Walker space-
times, we may find an isometry $ € I ( H ) such that $ = id x 4 € I (M,g)
satisfies +(c(sk)) = (tk, YO) with yo as above. Since k is fixed during the
course of this proof, we may set p = (tk, yo) without danger of confusion. Put
cl(s) = $ o c(s + sk). Then cl is a past inextendible, past incomplete, null
geodesic of (M,g) with cl(0) f {tk) x H, cl(Ask) E {tk+l) x H , and cl(s) =
exp,[g](sv) for v = $,(cl(sk)). Choose b > 0 with Ask < b < 2Ask such that
cl(s) E Uk for all s with 0 5 s < b. Since b > Ask, we have cl(b) € {t) x H
for some t < tk+l. Hence (xi o cl) (Ask) - (xi 0 cl)(b) = tk+i - t > 0. Set
€1 = min{c, tk+l - (xi 0 cl)(b)) > 0.
Now let gl E Lor(M), and let q E Uk n ({tk) x H). Suppose that y : [0, B) --+
(M, gl) is any past directed, past inextendible, null gl geodesic with y(0) = q
and with xl speed a k a t q. Then w = yl(0) f TqM satisfies gl(w, w) = 0 and
xn+,(w) < 0. Moreover, y(s) = expq[gl](sw). Applying Lemmas 7.10 and
7.11 to cl and c2 = y with the constant €1 as above, we may find a constant
60 > 0 with 0 < 60 < Ask such that Ilv - wll2 < 60, 119 - gllll,uk < 60, and
Iso - Ask\ < 60 imply that
(I) I(XI 0 c ~ ) ( s ) - (21 0 ca)(~)I < €1 < tk+l - (XI 0 ~ l ) ( b )
for all s with 0 5 s 5 b and
(2) 1 - €1 < 1 ("I O C2)'(s0) 1 < 1 + €1. (21 O cl)'(Ask)
Setting s = b in (I), we obtain ](XI ocl)(b) -(xi oc2)(b)l < tk+l- (xi ocl)(b)
from which (xl o c2)(b) < tk+l. Hence there exists an s1 with 0 < s' < b su
that (xl o c2)(s1) = tk+l. But then s' < b < 2Ask, which shows that th
increase in affine parameter of c2 in passing from {tk) x H to {tk+l) x H
less than 2Ask provided that 60 is chosen as above.
For any geodesic q ( s ) = exp[gl](sw) with gl and w 60-close to g an
as above, let s' denote the value of the f i n e parameter s of c2 such t
x1 o c2(s1) = tk+l. AS 60 -+ 0, the corresponding value of s' must appro
Ask by Lemma 7.10. Thus by continuity we may choose b1 with 0 < 61 <
7.3 GEODESIC INCOMPLETENESS 261
such that for any geodesic c2(s) = exp[gl](sw) with gl and w both 61-close to
g and v, we have 21 0 c~(s ' ) = tk+l for some st f [Ask - bo, Ask + bo]. Hence
as 61 < 60, we may apply estimate (2) above with so = s' to obtain
We now need to extend these estimates from a neighborhood of v E T,M
to a neighborhood of S(p, a k , g). TO this end, note that since (a, b) x f H =
M is a warped product of H and the one-dimensional factor (a, b) with f :
(a, b) -, R, it follows that I (H) acts transitively on S(p, ak,g). Thus given
any t € S(p, ak,g), we may apply the previous arguments using the same
admissible chain {Uk), {tk) to find a constant 61(t) > 0 such that if w E T M
satisfies Ilw - 4 2 < 61(z), .rr(w) E Uk n ({tk) x HI, llgl - 9lll,uk < bl(z), and
4 s ) = exp[gl](sw) has xl speed a k at {tk) x H , then c2 has an increase
in affine parameter of at most 2Ask in passing from {tk) x H to {tk+l) x H
and satisfies estimate (7.4). Using the compactness of S(p, ak,g), we may
choose null vectors vl, v2, . . . , v, E S(p, ak, g) such that S(p, a k , g) is covered
by the sets {w f S(p,ak,g) : I I w - vrnI12 < 61(vrn)) for m = 1,2,. . . , j. Set
b2 = mini61 (v,) : 1 5 m 5 j}. By Lemma 7.21 we may find a constant b3 with
0 < 63 < 62 such that if I ~ P - 4\12 < 63, llgi - 9111,~~ < 63, and w E S(P, ak, gi),
then Ilw - vrnl12 < 61(vm) for some m. Hence b3 has the following properties.
If y : [0, B) -+ (M, gl) is any past inextendible, past directed, null geodesic of
(M,gl) such that Ilgl-gl(l,vk < 63, ~ ( 0 ) ({tk)xH)n{q f Uk : II~--9112 < 63) where p = (tk, yo), and y has xl speed a k at y(O), then the conclusions of the
theorem apply to y. Since I (H) acts transitively on H , we may extend this
result from ({tk) x H ) n {q f Uk : Ilp - 9112 < 63) to all of {tk) x H just as
in the proof of Lemma 7.17. The function 6k : [tk+l,tk] x H -+ (0, a) may be
constructed exactly as in Lemma 7.17. 0
With Lemma 7.22 in hand, we are now ready to prove the C1 stability of
t null geodesic incompleteness for Robertson-Walker space-times. Since
bertson-Walker space-times are isotropic and spatially homogeneous, past
incompleteness of one inextendible null geodesic implies past incompleteness
of all null geodesics. Thus the stability theorem may be formulated as follows.
262 7 STABILITY OF COMPLETENESS AND INCOMPLETENESS
T h e o r e m 7.23. Let ( M , g) be a Robertson-Walker space-time containing
an inextendible null geodesic which is past incomplete. Then there is a fine C'
neighborhood U(g) o f g in Lor(M) o f globally hyperbolic metrics such that all
null geodesics of ( M , gl ) are past incomplete for each gl f U(g).
Proof. Let M = (a, b) x f H and let c : [0, A ) + ( M , g ) be the given inex-
tendible past incomplete null geodesic. Wi th wo = xl(c(O)), let {Uk), { t k ) ,
{ s k ) , and { a k ) be chosen as in the paragraph preceding Lemma 7.22. Let
{Pk) be a sequence of real numbers with 0 < Pk < 1 for each k such that
112 < n L l ( 1 - P k ) < 1. Thus for each m 2 1, we have
For each k 2 1, we apply Lemma 7.22 with E = Pk to obtain a continuous
function bk : i tkf l , tk] X H -' (0, m) with the properties o f Lemma 7.22.
Choose a continuous function 6 : M + (0, m) such that for each q in M we
have 6(q) < bk(q) for ail k with q in the domain of 6k. Let Ul(g) = {gl E
Lor(M) : lgl - gjl < 6 ) . Also choose a C1-open neighborhood Uz(g) o f g in
Lor(M) such that all metrics in U2(g) are globally hyperbolic and such that
each hypersurface { t ) x H is spacelike in (M,g l ) for all t E (a , b) and any
gl E U2(g) (cf . Lemma 7.16). Set U(g) = Ul(g) n Uz(g).
Now suppose that gl E U(g) and that y : [0, B ) + M is any past directed
and past inextendible null geodesic o f ( M , gl). Reparametrizing y i f necessary,
we may assume that x l (y (0 ) ) = tk for some k 2 1 and that y has xl speed
C Y ~ at { t k ) x H. By Lemma 7.22, y changes in d n e parameter by at most
2Ask in passing from {tk) x H to x H. In order to apply Lemma
7.22 to y as y passes from {tk+l) x H to {tk+2) x H , it may be necessary
to reparametrize y at {tk+l) x H to have xl speed ak+l at {tk+l) x H.
Nonetheless, i f Bk+' denotes the xl speed of y at {tk+l) x H , we have 1 - Pk < lek+l/~~k+ll < 1 +Pk from Lemma 7.22. Thus the xl speed of y at {&+I) x H
cannot be less than ( 1 - r C j k ) ~ k + ~ . Hence the affine parameter o f y increases
in passing from {tk+l) x H to {tk+2) x H by at most 2(1 - Pk)-lAsk+l.
Arguing inductively, it may be seen that y increases in affine parameter by at
most AS^+^ nfzA(l- ~ k + % ) - ' in passing from {tk+l) x H t o (tk+~+l) x H.
7.4 SUFFICIENT CONDITIONS FOR STABILITY 263
Using inequality (7.5), we thus have that y increases in affine parameter by at
most 4ASk+, in passing from {tk+l) x H to {tk+l+l) x H. Since ELl Ask = A ,
it follows that the total afFine length B o f y is less than 4A. Since 4 A < m, it
follows that y is past incomplete as required. [3
By reversing the time orientation, we may obtain the analogue o f The-
orem 7.23 for Robertson-Walker space-times having future incomplete null
geodesics. Thus Theorem 7.23 implies the following result.
Theorem 7.24. Let ( M , g) be a Robertson-Walker s p a c e t ~ m e containing
an inextendible null geodesic which is both past and future incomplete. Then
there is a fine C' neighborhood U(g) of g in Lor(M) o f globally hyperbolic
metrics such that all null geodesics of ( M , gl) are past and future incomplete
for each gl E U(g) .
W e now obtain two stability theorems for nonspacelike geodesic incomplete-
ness by combining Theorems 7.19 and 7.23 and by combining Theorems 7.20
and 7.24. The first o f these theorems applies to all big bang models, and the
second theorem applies to the closed big bang models.
T h e o r e m 7.25. Let ( M , g) be a Robertson-Walker space-time of the form
(a, b) x f H , where a > -m. Assume that ( M , g) contains a past incomplete
and past inextendible null geodesic. Then there is a fine C' neighborhood
U(g) o f g in Lor(M) o f globally hyperbolic metrics such that all nonspacelike
geodesics o f ( M , g l ) are past incomplete for each gl E U(g) .
Theorem 7.26. Let ( M , g) be a Robertson-Walker space-time o f the form
(a, b) X / H , where both a and b are finite. Assume that ( M , g ) contains an
inextendible null geodesic which is both past and future incomplete. Then
there is a fine C' neighborhood U(g) o f g in Lor(M) o f globally hyperbolic
metrics such that all nonspacelike geodesics o f ( M , g l ) are both past and future
incomplete for each gl E U(g) .
k 7.4 Sufficient Conditions for Stability
In this section we show that nonimprisonment is a sufficient condition for the
C' stability o f incompleteness [cf. Beem (1994)l. Sufficient conditions for the
264 7 STABILITY O F COMPLETENESS AND INCOMPLETENESS
stability o f completeness involve both nonimprisonment and pseudoconvexity.
W e first state two lemmas which follow from standard results (cf . Lemmas 7.10
and 7.11).
Lemma 7.27. Let ( M , g) be a given semi-Riemannian manifold, fix a ge-
odesic y : (a, b) -+ M of ( M , g ) , and let W = W ( y I Itl,t2]) be a neighborhood
o f y I [ t l , t z ] . There is a neighborhood V o f y1(tl) in T M and a constant E > 0
such that i f llg - gllll,w < E and i f c is a geodesic of gl with c l ( t l ) E V , then
the domain of c includes the value t 2 and c( t ) E W for all tl < t 5 t2.
Lemma 7.28. Let ( M , g ) be a given semi-Riemannian manifold, fix a ge-
odesic y : (a, b) -+ M of ( M , g ) , and let W = W ( y I [tl , t2]) be a neighborhood
o f y 1 [ t l , t2] . Given any neighborhood Vl o f y l ( t l ) in T M , there is a constant
E > 0 and a neighborhood fi of y1(t2) in T M such that i f llg - glll1,w < 6 and
i f c is a geodesic of gl with c1(t2) f V2, then cl(t l) E Vl and c(t) E W for all
tl < t < t2. Furthermore, i f y is timelike (respectively, spacelike), then V2 and
E > 0 may be chosen such that each v E 7 2 o f each such metric gl is timelike
(respectively, spacelike). I f y is null, then E > 0 may be chosen such that each
such metric gl has some null vectors in V2.
W e now define partial imprisonment and imprisonment.
Definition 7.29. (Imprisonment and Partial Imprisonment) Let y :
(a , b) -+ M be an endless (i.e., inextendible) geodesic.
(1) The geodesic y is partially imprisoned as t -+ b i f there is a compact
set K M and a sequence { t j ) with t , -+ b- such that y ( t j ) E K for
all j .
(2) The geodesic y is imprisoned i f there is a compact set K such that the
entire image of y is contained in K.
In other words, y is partially imprisoned in K as t 4 b i f either y ( t ) E K
for all t sufficiently near b or i f y leaves and returns to K an infinite number of
times as t + b-. An imprisoned geodesic is clearly partially imprisoned, but for
general space-times one may have some partially imprisoned geodesics which
fail t o be imprisoned. The stability o f incompleteness result (Theorem 7.30)
will hold for geodesics which fail to be partially imprisoned in the direction
7.4 SUFFICIENT CONDITIONS FOR STABILITY 265
of incompleteness. In contrast, the stability o f (nonspacelike) completeness
(Theorem 7.35) result will require the nonimprisonment o f all nonspacelike
geodesics as well as the pseudoconvexity of nonspacelike geodesics.
Let M be given an auxiliary, positive definite, and complete Riemannian
metric. This metric has a complete distance function do : M x M -+ W, and the
Hopf-Rinow Theorem [cf. Hicks (1965)l guarantees that the compact subsets
o f M are exactly the subsets which are closed and bounded with respect to
do. The proof of Theorem 7.30 [cf. Beem (1994)l will use a sequence of n-
dimensional annuli {W,) [i.e., sets bounded by spherical shells with respect
to the distance do which are centered at a fixed point y(to)]. The approach
involves requiring the perturbed metric gl to be sufficiently close to the original
metric g on the sequence {W,). For a fixed gl, one constructs a sequence o f
geodesics cj o f gl such that a given c, is either tangent or almost tangent to
the original y at a certain point y ( t j ) o f W j . A limit geodesic o f the sequence
will be the desired geodesic c o f gl.
Theorem 7.30. Let ( M , g) be a semi-Riemannian manifold. Assume that
( M , g ) has an endless geodesic y : (a, b) -+ M such that y is incomplete in
the forward direction (i.e., b # oo). I f y is not partiaI1y imprisoned in any
compact set as t -+ b, then there is a C 1 neighborhood U ( g ) o f g such that
each gl in U(g) has at least one incomplete geodesic c. Furthermore, i f y is
timelike (respectively, null, spacelike) then c may also be taken as timelike
(respectively, null, spacelike).
Proof. Choose some to in the interval (a, b). We will construct sequences
t,, D,, and L,. Let tl and Dl be chosen with Dl > 1 and t l > to such
that do(y( to) ,y( t l ) ) = Dl and do(y(to), y ( t ) ) > Dl for all tl < t < b. In
other words, y 1 [to, b) leaves the closed ball o f radius Dl for the last time at
t l . The existence o f t l follows using the fact that y is not partially imprisoned
as t -+ b. Set Lo = 0 and L1 = 1 + sup{do(y(to), y ( t ) ) 1 to < t < t l ) . Notice
that on the interval [to, t l ] the geodesic y remains within the closed ball o f
radius L1 - 1 about y(t0). I f t l , . . . , t,-1, Dl , . . . , D,-1, and Lo, L1, . . . , L,-1
have been defined, then t, and D, are defined by letting D, > L, - 1 i- 1 and
requiring both do(y(to), y(t,)) = D, and &($to), ? ( t ) ) > D, for all t, < t < b.
266 7 STABILITY O F COMPLETENESS AND INCOMPLETENESS
Existence again follows using the nonimprisonment hypotheses. Define L j =
1 + sup{do(y(to),r(t)) It0 < t < tj). Note that limy(tj) does not exist as
j -+ co, and hence t j -+ b as j -+ w . Set Wl = {x E M I &(y(to), x) < L1),
W2 = {X E M I &(y(to), X) < L2), and Wj = {X E M I Lj-2 < do(y(to), X) < Lj) for j > 2. The sets Wj are n-dimensional annuli with respect to the
distance do. We now define a sequence of positive numbers {cj) and a sequence
of open sets {V,) of TM. Let Vo be an open neighborhood of yl(to) in TM with
compact closure, and assume that the closure of Vo does not contain any trivial
vectors. If g(yl(to), yl(to)) > 0 [respectively, g(yl(to), yl(to)) < 01, we assume
without loss of generality that all v E 70 satisfy g(v, v) > 0 [respectively,
g(v, v) < 01. Assume that &, Vl, . . . , T/;-l have been defined. Use Lemma 7.28
to obtain an open set V, = T/;(yl(tj)) of yl(tj) in T M and a positive number
cj such that if Ilg-gljll,p < cj where P = Wj, and if c is a geodesic of gl with
cl(tj) E Vj, then cl(tj-1) E y-l and c(t) E Wj for all tj-l 5 t 5 tj. Lemma
7.28 implies that if the original geodesic y is timelike (respectively, spacelike),
then we may assume each v in Vj is timelike (respectively, spacelike) for gl. If
y is null, then we may assume that gl has some null vectors in V,. Lemma 7.27
implies we may assume that if c is a geodesic of such a gl with cl(tj) E vj, then the domain of c contains tj+l. We may also assume without loss of
generality that the diameter with respect to & of K(V,) is less than 112 and
cj+l < cj for all j . Notice that points of M are in a t most two consecutive
sets of the sequence Wj. It follows that there is a continuous positive-valued
function E : M -+ (0 ,w) with ~ ( x ) < ej for all j with x 6 Wj. Let gl satisfy
119 - gllll,M < ~ ( x ) . We will construct a sequence of geodesics of gl. Assume
first that the original geodesic y is either timelike or spacelike. For each j,
let cj(t) be the geodesic of gl which satisfies cjl(tj) = yl(tj). Notice that if
y is timelike (respectively, spacelike), then each cj is timelike (respectively,
spacelike). If the original geodesic y is null, then we choose c j such that
cjl(tj) E Vj and cjl(tj) is null. The above construction yields that cjt(tk) E Vk
for all 0 5 k 5 j. Since Vo has compact closure and this closure does not
contain any trivial vectors, one obtains a nontrivial vector v in To and a
subsequence { m ) of {j) such that cml(to) -+ v. The constructions of Vo and
7.4 SUFFICIENT CONDITIONS FOR STABILITY 267
the sequence {cj) yield that v is timelike (respectively, null or spacelike) for
gl if the original geodesic y is timelike (respectively, null or spacelike) for g.
Let c be the endless (i.e., inextendible) geodesic of gl with cl(to) = v. Note
that each geodesic c, satisfies h ( t j ) E .rr(V,) for all 0 5 j 5 m. Also, for
each t in the domain of c, cm(t) converges to c(t) and cml(t) converges to cl(t)
as m -+ w . By construction, cl(tj) f 6 implies c(tj+l) exists. Furthermore,
ckt(tj+1) f V,+I for all k > j + 1 yields cl(tj+l) 6 vj+l. Thus, c is defined for
all values of t j , and &(y(to), c(tj)) -+ oo as j -+ w . It is easily shown that the
limit c is not partially imprisoned in any compact set as t + b. It only remains
to show that c cannot have any domain values above b (i.e., c(b) does not exist).
Assume the domain of c contains b. Since c is continuous, the set c([to, b])
would be compact in contradiction to the condition do(yo(to), c(tj)) -+ co. 0
An interesting aspect of the construction used in the above proof is that the
final geodesic c of the metric gl has the same value b for the least upper bound
of its domain as the original geodesic y of the metric g. In essence, this is
due to cjl(tj) E T/, implying cjl(to) E Vo and to the fact that do(cj(tj),cj(to))
diverges to infinity. Notice that the methods used in the proof of Theorem
7.30 show that if gl is another semi-Riemannian metric on M and if gl is close
to g in the C' sense on a neighborhood of y , then gl has a corresponding
geodesic which is incomplete. In other words, the proof of Theorem 7.30 only
really requires that g and gl be metrics which are C1 close near y. Thus,
if (M, g) has a finite number N of incomplete geodesics and none of these
are partially imprisoned in any compact set, then one may construct a C1
neighborhood U(g) of g such that each gl in U(g) has at least N incomplete
geodesics. On the other hand, one may construct space-times which have all
geodesics complete except for a single geodesic which is incomplete and which
is not partially imprisoned [cf. Beem (1976c)j. Thus, the existence of a single
incomplete geodesic does not imply that there is more than one such geodesic.
Corollary 7.31. If (M,g) is a strongly causal space-time which is causally
geodesically incomplete, then there is a C1 neighborhood U(g) of g such that
each gl in U(g) is nonspacelike geodesically incomplete.
268 7 STABILITY OF COMPLETENESS AND INCOMPLETENESS
Proof. This follows from Theorem 7.30 using the fact that for strongly
causal space-times, no causal geodesic is future or past partially imprisoned
in any compact set. 0
It is known that no null geodesic in a twc-dimensional Lorentzian manifold
has conjugate points. However, a chronological space-time in which all null
geodesics have conjugate points is strongly causal and has dimension at least
three. Thus, we obtain the following corollary.
Corollary 7.32. Let (M, g) be a chronological space-time of dimension at
least three which is causally geodesically incomplete. If (M, g) has conjugate
points along all null geodesics, then there is a C1 neighborhood U(g) of g such
that each gl in U(g) is nonspacelike geodesically incomplete.
Let (a, b) x f H be given the metric g = -dt2 + f (t)dcr2. If either a or b is
finite, then (a, b) x f H is timelike geodesically incomplete. Note that (a, b) x H
is necessarily stably causal since f (t , y) = t is a time function. Using Corollary
7.31 and the fact that stably causal space-times are strongly causal, we obtain
Corollary 7.33. Assume (H, h) is a positive definite Riemannian manifold,
and let the warped product (a, b) x f H be given the Lorentzian metric g =
-dt2 @ fh . If either a # -ca o rb # +ca, then there is a fine C1 neighborhood
U(g) of g such that each metric gl in this neighborhood is timelike geodesically
incomplete.
This last corollary applies to a much more general class of spacetimes
than the Robertson-Walker space-times because we have not made symmetry
assumptions on (H, h). Of course, for the special case of Robertson-Walker
space-times we have already established the stronger conclusion that there is
a C0 neighborhood U(g) with each gl in U(g) having all timelike geodesics
incomplete (cf. Theorem 7.18).
If K is a subset of Wn, then the conuez hull K H of K is the union of all
Euclidean line segments with both endpoints in K. It is well known that the
convex hull of a compact set is again a compact set. Pseudoconvexity is a gen-
eralization of this property to manifolds. In Definition 7.34 below, we assume
the class of geodesic segments under consideration to be nonspacelike. One
7.4 SUFFICIENT CONDITIONS FOR STABILITY 269
can, of course, consider this property for other classes such as the class of all
null geodesic segments, all spacelike segments, etc. Furthermore, pseudocon-
vexity and disprisonment have also been applied to sprays [cf. Del Riego and
Parker (1995)l.
Definition 7.34. (Nonspacelike Pseudoconuexity) We say (M, g) has a
pseudoconuex nonspacelike geodesic system if for each compact subset K of M ,
there is a second compact set H such that each nonspacelike geodesic segment
with both endpoints in K lies in H.
The following [cf. Beem and Ehrlich (1987, p. 324)] gives sufficient condi-
tions for the stability of nonspacelike completeness.
Theorem 7.35. Let (M,g) be a Lorentzian manifold which has no im-
prisoned nonspacelike geodesics and which has a pseudoconvex nonspacelike
geodesic system. If (M, g) is nonspacelike geodesically complete, then there is a
C1 neighborhood U(g) ofg in Lor(M) such that each gl f U(g) is nonspacelike
complete.
Pseudoconvexity is a type of "internal" completeness condition in somewhat
the same sense that global hyperbolicity is such a condition. The next propo-
sition shows that nonspacelike pseudoconvexity is a generalization of global
hyperbolicity. An example of a nonspacelike pseudoconvex space-time which
fails to be globally hyperbolic is given by the open strip {( t , x) : 0 < x < 1) in
the Minkowski plane.
Proposition 7.36. If (M,g) is a globally hyperbolic space-time, then
(M, g) has a pseudoconvex nonspacelike geodesic system.
Proof. Let K be a compact subset of M. For each p E K choose points q
and r with q in the chronological past I-(p) of p and r in the chronological
future If (p) of p. Then U(p) = I+(q) n I-(T) is an open set containing p.
Since (M, g) is globally hyperbolic, the set U(p) has compact closure given by
J + ( q ) n J-(T). Cover the compact set K with a finite number of open set
U(p,) = I+(q,) n I-(r,) for i = 1 ,2 , . . . , k, and let H be the union of the k2
compact sets of the form J+(q,) n J-(r,) for 1 5 i, j 5 k. It is easily seen that
270 7 STABILITY OF COMPLETENESS AND INCOMPLETENESS
all nonspacelike geodesic segments with both endpoints in K must lie in the
compact set H.
Since globally hyperbolic space-times are strongly causal, they have no im-
prisoned nonspacelike geodesics. Consequently, Theorem 7.35 and Proposition
7.36 yield the following corollary.
Corollary 7.37. Let (M, g) be a globally hyperbolic space-time. If (M, g)
is nonspacelike geodesically complete, then there is a C1 neighborhood U(g)
of g in Lor(M) such that each gl E U(g) is nonspacelike complete.
Minkowski space-time is globally hyperbolic and geodesically complete.
Furthermore, its (entire) geodesic system is pseudoconvex. In other words,
given any compact set K , there is a larger compact set H such that any ge-
odesic segment (timelike, null, or spacelike) with endpoints in K must lie in
H. In fact, one may take H to be the usual convex hull of K. The next
proposition gives the C1 stability of global hyperbolicity, nonimprisonment,
completeness, and inextendibility for Minkowski space-time. The C1 stability
of global hyperbolicity follows from the stronger C0 stability result of Geroch
(1970a). The stability of nonimprisonment and of geodesic completeness (
all geodesics) follows from Beem and Ehrlich (1987, pp. 324-325). See a
Beem and Parker (1985, p. 18). The stability of inextendibility follows fro
the stability of completeness using Proposition 6.16 which guarantees th
geodesically complete space-times are inextendible.
Proposition 7.38. There is a C1 neighborhood U(7) of n-dimension
Minkowski space-time (M, 7 ) such that for each metric gl in this neighbor-
hood:
(1) (M, gl) is globally hyperbolic;
(2) No geodesic of (M, gl) is imprisoned;
(3) (M, gl) is geodesically complete; and
(4) (M, gl) is an inextendible space-time.
CHAPTER 8
MAXIMAL GEODESICS AND CAUSALLY
DISCONNECTED S P A C S T I M E S
Many basic properties of complete, noncompact Riemannian manifolds stem
from the principle that a limit curve of a sequence of minimal geodesics is
itself a minimal geodesic. After the correct formulation of completeness had
been given by Hopf and Rinow (1931), Rinow (1932) and Myers (1935) were
able to establish the existence of a geodesic ray issuing from every point of
a complete noncompact Riemannian manifold using this principle. Here a
geodesic y : [0, co) --t (N,go) is said to be a ray if y realizes the Riemannian
distance between every pair of its points. Rinow and Myers constructed the
desired geodesic ray as follows. Since (N, go) is complete and noncompact,
there exists an infinite sequence (p,) of points in N such that for every point
p E N, do(p,p,) -+ CXI as n -+ m. Let y, be a minimal (i.e., distance realizing)
unit speed geodesic segment from p = y,(O) to p,. This segment exists by
the completeness of (N,go). If v E T,N is any accumulation point of the
sequence {ynl(0)) of unit tangent vectors in TpN, then y(t) = exp, tv is the
required geodesic ray. Intuitively, y is a ray since it is a limit curve of some
subsequence of the minimal geodesic segments (7,). The existence of geodesic
rays through every point has been an essential tool in the structure theory of
both positively curved [cf. Cheeger and Gromoll (1971, 1972)] and negatively
curved [cf. Eberlein and O'Neill (1973)l complete noncompact Riemannian
manifolds.
A second application of this basic principle of constructing geodesics as
limits of minimal geodesic segments is a concrete geometric realization for
complete Riemannian manifolds of the theory of ends for noncompact Haus-
dorff topological spaces [cf. Cohn-Vossen (1936)l. An infinite sequence (p,)
of points in a manifold is said to diverge to infinity if, given any compact sub-
272 8 MAXIMAL GEODESICS, DISCONNECTED SPACE-TIMES 8.1 ALMOST MAXIMAL CURVES AND MAXIMAL GEODESICS 273
set K , only finitely many members of the sequence are contained in K. If a disconnecting two divergent sequences is said to be causally disconnected. It
complete Riemannian manifold (N, go) has more than one end, there exists a is evident from the definition that causal disconnection is a global conformal
compact subset K of N and sequences {p,) and {q,) which diverge to infinity invariant of C(M, g). Applying the principle of Section 8.1, we show that if the
such that 0 < &(p,, q,) --+ oo and every curve from p, to q, meets K for each strongly causal space-time (M,g) is causally disconnected by the compact set
n. Let -y, be a minimal (i.e., distance realizing) geodesic segment from p, to K, then (M, g) contains a nonspacelike geodesic line y : (a, b) -4 M which in-
q,. Since each y, meets K , a limit geodesic y : R -+ M may be constructed. tersects K. That is, d(y(s), y(t)) = L(y I [s, t]) for all s , t with a < s t < b.
Moreover, y is minimal as a limit of a sequence of minimal curves. Then This result is essential to the proof of the singularity theorem 6.3 in Beem
"y(-m)" corresponds to the end of N represented by { p , ) and "y(+m)" to and Ehrlich (1979a), as will be seen in Chapter 12. We conclude this chapter
the end represented by (9,). In particular, a complete Riemannian manifold by studying conditions on the global geodesic structure of a given space-time
with more than one end contains a line, i.e., a geodesic y : (-co, +m) N (M,g) which imply that (M,g) is causally disconnected. In particular, we
that is distance realizing between any two of its points. show that all two-dimensional globally hyperbolic space-times are causally
disconnected. Also, it follows from one of these conditions and the existence
Motivated by these Riemannian constructions, we study similar existence of nonspacelike geodesic lines in strongly causal, causally disconnected space- theorems for geodesic rays and lines in strongly causal space-times. From times that a strongly causal space-time containing no future directed null the viewpoint of general relativity, it is desirable to have constructions that geodesic rays contains a timelike geodesic line.
are valid not only for globally hyperbolic subsets of space-times, but also for
strongly causal space-times. However, if we only assume strong causality, it 8.1 Almost Maximal Curves and Maximal Geodesics not true in general that causally related points may be joined by maximal g e
odesic segments. Thus a slightly weaker principle for construction of maximal The purpose of this section is to show how geodesics may be constructed
geodesics is needed for Lorentzian manifolds than for complete Riemannian as limits of "almost maximal" curves in strongly causal spacetimes. In both
manifolds. Namely, in strongly causal space-times, limit curves of sequences o constructions, the upper semicontinuity of Lorentzian arc length in the C0
"almost maximal" curves are maximal and hence are also geodesics. In Secti topology on curves for strongly causal space-times and the lower semicontinu-
8.1 we give two methods for constructing families of almost maximal curv ity of Lorentzian distance play important roles. The strong causality of (M, g )
whose limit curves in strongly causal space-times are maximal geodesics. T is used in our approach here so that convergence in the limit curve sense and in
strong causality is needed to ensure the upper semicontinuity of arc length the CO topology on curves are closely related (cf. Proposition 3.34). The first
the C0 topology on curves and also so that Proposition 3.34 may be appli nstruction may be applied to pairs of chronologically related points p, q with
In Section 8.2 we apply this construction to prove the existence of past and (p, q) < a. While this approach is therefore sufficient to show the existence
ture directed nonspacelike geodesic rays issuing from every point of a stron of nonspacelike geodesic rays in globally hyperbolic space-times [cf. Beem and
causal space-time. In Section 8.3 we study the class of causally disconnec Ehrlich ( 1 9 7 9 ~ ~ Theorem 4.2)], it is not valid for points at infinite distance.
space-times. Here a space-time is said to be causally disconnected by a cordingly, for use in Sections 8.2 and 8.3, we give a second construction
pact set K if there are two infinite sequences {pn) and {qn) , both divergi ich may be used in arbitrary strongly causal space-times. In Section 14.1 a
infinity, such that p, < q,, p, # qn, and all nonspacelike curves from Pn ghtly different approach to constructing maximal segments from sequences of
meet K for each n. A space-time (M, g) admitting such a compact K caus onspacelike almost maximal curves is presented [cf. Galloway (1986a)J. Here,
274 8 MAXIMAL GEODESICS, DISCONNECTED SPACE-TIMES
the use of uniform convergence in a unit speed reparametrization with respect
to an auxiliary complete Riemannian metric is employed to dispense with the
global requirement of strong causality which we assume in this chapter.
Let (M, g) be an arbitrary space-time and suppose that p and q are distinct
points of M with p < q. If d(p,q) = 0, then letting y be any future directed
nonspacelike curve from p to q, we have L(y) 5 d(p, q) = 0. Hence L(y) =
d(p, q) and 7 may be reparametrized to a maximal null geodesic segment from
p to q by Theorem 4.13. Thus suppose that p << q, or equivalently, that
d(p, q) > 0. If d(p, q) < co as well, then by Definition 4.1 there exists a future
directed nonspacelike curve y from p to q with
for any E > 0. Of course, inequality (8.1) is only a restriction on L(y) provided
E < d(p, q). In this case, we will call y an almost maximal curve.
We note the following elementary consequence of the reverse triangle in-
equality.
Remark 8.1. Let y : [O, 11 -+ M be a future directed nonspacelike curve
from p to q, p # q, with
Then for any s < t in [O,l], we have
Proof. Assume that L ( y I [s, t ]) < d(y(s), y(t)) - c for some s < t in [O,l].
Then
in contradiction. 0
8.1 ALMOST MAXIMAL CURVES AND MAXIMAL GEODESICS 275
We are now ready to give an example of the principle that for strongly
causal space-times, limits of almost maximal curves are maximal geodesics.
Strong causality is used here (with all curves given a "Lorentzian parametriza-
tion") since convergence in the limit curve sense and in the C0 topology are
closely related for strongly causal space-times but not for arbitrary space-
times (cf. Section 14.1).
Proposit ion 8.2. Let (M,g) be a strongly causal space-time. Suppose
that p, -+ p and q, -+ q where p, < q, for each n and 0 < d (p , q) < m. Let
y, : [a, b ] -+ M be a future directed nonspacelike curve from p , to q, with
where 6 , --, 0 as n -+ m. If y : [a, b] --+ M is a limit curve of the sequence
{m) with y(a) = p and y(b) = q, then L(y) = d(p,q). Thus y may be
reparametrized to be a smooth maximal geodesic from p to q.
Proof. First, y is nonspacelike by Lemma 3.29. Second, by Proposition
3.34, a subsequence {y,) of {y,) converges to y in the C0 topology on curves.
By the upper semicontinuity of arc length in this topology for strongly causal
space-times (cf. Remark 3.35), we then have
k using the lower semicontinuity of Lorentzian distance (Lemma 4.4). But by
definition of distance, d(p,q) 2 L(y). Thus L(y) = d(p, q) and the last state-
[ ment of the proposition follows from Theorem 4.13. El
We now consider a second method for constructing maximal geodesics in
strongly causal space-times (M,g) which may be applied to points at infinite
Lorentzian distance. For this purpose, we fix throughout the rest of Chapter
8 an arbitrary point po E M and a complete (positive definite) Riemannian
metric h for the paracompact manifold M. Let do : M x M -+ R denote the
276 8 MAXIMAL GEODESICS, DISCONNECTED SPACE-TIMES
Riemannian distance function induced on M by h. For all positive integers n ,
the sets - B n = {m E M : do(po,m) F n)
are compact by the Hopf-Rinow Theorem. Thus the sets {B, : n > 0) form a
compact exhaustion of M by connected sets. For each n, let
denote the Lorentzian distance function induced on B, by the inclusion B, (M, g). That is, given p f B,, set d [Bn](p, q) = 0 if q $! J + ( ~ , R), and for
4 E J+(~ ,B, ) , let d [Bn](p,q) be the supremum of lengths of future directed
nonspacelike curves from p to q which are contained in B,. It is then immediate
that d q) 5 d(p, q) for all p, q E B,. However, d [B,] is not in general
the restriction of the given Lorentzian distance function d of (MI g) to the set - B, x B,. Nonetheless, for strongly causal space-times these two distances
coincide "in the limit."
Lemma 8.3. Let (M, g) be strongly causal. Then for all p, q E M , we have
d(p, 9) = lim d [Bn](p, 9).
Proof. Since d [Bn](p,q) 5 d(p,q), the desired equality is obvious in the
event that d(p, q) = 0. Thus suppose that d(p, q) > 0. By definition of
Lorentzian distance, we may find a sequence {yk) of future directed nonspace-
like curves from p to q such that L(yk) -+ d(p, q) as k -+ m. (If d(p, q) = m,
choose (yk) such that L(yk) > k for each k.) Since the image of yk in M is
compact and the Riemannian distance function do : M x M -+ W is continuous
and finite-valued, there exists an n(k) > 0 for each k such that yk C Bj for all j 2 n(k). Thus d(p, q) = lim L(yk) 5 limd [Bn](p, q). Hence as
d [Bn](p, q) 5 d(p, q) for each n, the lemma is established.
It will be convenient to introduce the following notational convention for
use throughout the rest of this chapter.
Notational Convention 8.4. Let y be a future directed nonspacelike
curve in a causal space-time. Suppose p = ~ ( s ) and q = y(t) with s < t and
p # q. We will let y[p, q ] denote the restriction of 7 to the interval [s, t] .
8.1 ALMOST MAXIMAL CURVES AND MAXIMAL GEODESICS 277
For strongly causal space-times, the Lorentzian distance function d and the
d [Bn]'s are related by the following lower semicontinuity.
Lemma 8.5. Let (M,g) be strongly causal. If p, -+ p and q, -+ q, then
d ( ~ , 9) < lim inf d [Bn] (pn qn) .
Proof. If d(p, q) = 0, there is nothing to prove. Thus we first assume that
0 < d(p, q) < a. Let 6 > 0 be given. By definition of Lorentzian distance and
standard results from elementary causality theory [cf. Penrose (1972, pp. 15-
16)], a timelike curve y from p to q may be found with d(p, q) - 6 < L(y) 5 d(p, q). Since y is timelike and L(y) > d(p,q) - E , we may find TI, rz E y with
d(p, q) - 6 < L(y[rl,rz]) and p << TI << r2 << q. Since I - (TI) and I+(r2) are
open and pn -+ p, qn -+ q, we have p, << r l << rg << qn for all n sufficiently
large. Also y C Bn, p, E J-(rl, B,), and q, E J+(rz, B,) for all n sufficiently
large. Consequently, d(p, q) - 6 < L(y[rl, m]) 5 d [Z,](p,, q,) for all large n.
Since e > 0 was arbitrary, we thus have d(p, q) 5 lim inf d [Bn](pn, q,) in
the case that 0 < d(p, q) < m . Assume finally that d(p, q) = co. Choosing
timelike curves yk from p to q with L(yk) > k for each k, we have for each
k that d [B,](p,, q,) 1 k - 6 for all n sufficiently large as above. Hence
limd [B,] (pn, qn) = co as required. 13
Since we are assuming that (M,g) is strongly causal but not necessarily
globally hyperbolic, it is possible that the Lorentzian distance function d :
M x M --t W U {m) assumes the value +w. Nonetheless, for any given Bn the distance function d [B,] : Bn x B, -+ R U {m) is finite-valued. This is
a consequence of the compactness of the Bn and the compactness of certain
subspaces of nonspacelike curves in the C0 topology on curves [if. Penrose
(1972, p. 50, Theorem 6.5)]. Moreover, this compactness also implies the
existence of curves realizing the d [B,] distance for points p, q E Bn with
E J+(P, Bn).
Lemma 8.6. Let (M,g) be a strongly causal space-time and let n > 0 be
arbitrary. If q E J+(~,B,) , then d [Bn](p,q) < XI and there exists a future
directed nonspacelike curve y in B, joining p to q which satisfies L(y) =
d[B,l(p,q).
278 8 MAXIMAL GEODESICS, DISCONNECTED SPACE-TIMES
Proof. By definition o f the distance d [B,], i f d [Bn] (p ,q ) = 0 and q E
J+(~,B,) , then there exists a future directed nonspacelike curve y in B, from p to q with L ( y ) 5 d [Bn](p,q) = 0. Hence L ( y ) = d [Bn](p,q) as required. Thus we may suppose that d q) > 0. Again by definition
o f d [ & I , we may find a sequence {yk ) o f future directed nonspacelike curves
from p to q with L(yk) -t d [ R ] ( p l q ) . ( I f d [B,](p, q) = oo, choose yk with
L(yk) > k for each k.) Since 3, is compact and ( M , g ) is strongly causal,
there exists a future directed nonspacelike curve y in B, joining p to q with
the property that a subsequence {y,} o f { y k ) converges t o y in the C O topology
on curves by Theorem 6.5 o f Penrose (1972, pp. 50-51). But then using the
upper semicontinuity o f arc length in the C0 topology on curves, we have
d [B,](p, q) = lim L(y,) 5 L ( y ) which implies the finiteness o f d [B,](p, q).
Since L ( y ) <_ d [ R ] ( p , q ) from the definition, we also have d (B,](p,q) = L(y)
as required.
Now let p, q E M with p < q, p # q, be arbitrary. Choose any nonspace-
like curve yo from p to q. Since the image of yo is compact in M and the
Riemannian distance function is continuous, we may find an N > 0 such that
yo is contained in BN. Hence q E J+(p ,Bn) for all n 2 N . Thus using
Lemma 8.6 we may find a future directed nonspacelike curve y, from p to q
with L(y,) = d [B,](p, q) for each n 2 N . For C0 limit curves o f the sequence
{y,), we then have the following analogue o f Proposition 8.2.
Proposition 8.7. Let ( M I g) be strongly causal and let p, q f M be dis-
tinct points with p < q. For all n > 0 sufficiently large, let yn be a future
directed nonspacelike curve from p to q in B, with L(y,) = d [Bn](p, q). If -y
is a nonspacelike curve from p to q such that {y,} converges to y in the C0
topology on curves, then L ( y ) = d(p, q) and hence y may be reparametrized
to a maximal geodesic segment from p to q.
Proof. Using Lemma 8.5 and the upper semicontinuity of arc length in the
C 0 topology on curves in strongly causal space-times, we have
8.2 GEODESIC RAYS IN CAUSAL SPACE-TIMES 279
d(p, q) L lim inf d [Bn] ( P , q)
= lim inf L(yn)
L lim sup L(yn) L L ( y )
L d ( ~ ! q)
as required. C3
Now let p, q be distinct points o f an arbitrary strongly causal space-time
with p < q and let a sequence {y,} of nonspacelike curves from p to q be
chosen as in Proposition 8.7. While a limit curve y for the sequence {y,) with
y(0) = p may always be extracted by Proposition 3.31, we have no guarantee
that y reaches q unless ( M , g ) is globally hyperbolic. Indeed, i f d(p, q) = co,
then there is no maximal geodesic from p to q, and hence no limit curve y o f
the sequence (7,) with y(0) = p passes through q. Thus the hypothesis that y
joins p to q in Proposition 8.7 together with the conclusion o f Proposition 8.7
implies that d(p, q) < co. On the other hand, the condition that d(p, q) < co does not imply that any limit curve y o f (7,) with y(0) = p reaches q when
( M , g ) is not globally hyperbolic. Examples may easily be constructed by
deleting points from Minkowski space-time.
8.2 Nonspacelike Geodesic Rays i n Strongly Causal Space-times
The purpose o f this section is to establish the existence o f past and future
directed nonspacelike geodesic rays issuing from every point o f a strongly causal
space-time ( M , g).
Definit ion 8.8. (Future and Past Directed Nonspacelike Geodesic Rays)
A future directed (respectively, past directed) nonspacelike geodesic ray is a
future (respectively, past) directed, future (respectively, past) inextendible,
nonspacelike geodesic y : [O,a) -t ( M , g ) for which d(y(O), y ( t ) ) = L ( y 1 [0, t ] )
(respectively, d (y ( t ) , y (0)) = L ( y I [0, t ] ) ) for all t with 0 5 t < a.
The reverse triangle inequality then implies that a nonspacelike geodesic
ray is maximal between any pair o f its points.
Using Lemmas 8.5 and 8.6 we first prove a proposition that will be needed
not only for the proof o f the existence o f nonspacelike geodesic rays, but also
280 8 MAXIMAL GEODESICS, DISCONNECTED SPACE-TIMES
for the proof of the existence of nonspacelike geodesic lines in strongly causal,
causally disconnected spacetimes in Section 8.3. Let and d [B,] : x - B, -+ R be constructed as in Section 8.1.
Proposition 8.9. Let (M, g) be a strongly causal space-time and let K be
any compact subset of M. Suppose that p and q are distinct points of M such
that p < q and every future directed nonspacelike curve from p to q meets K.
Then at least one of the following holds:
(1) There exists a future directed maximal nonspacelike geodesic segment
from p to q which intersects K.
(2) There exists a future directed maximal nonspacelike geodesic which
starts at p, intersects K, and is future inextendible.
(3) There exists a future directed maximal nonspacelike geodesic which
ends at q, intersects K , and is past inextendible.
(4) There exists a maximal nonspacelike geodesic which intersects K and
is both past and future inextendible.
Proof. Let yo be any future directed nonspacelike curve in M from p to q.
Since K U yo is compact, there exists an N > 0 such that K U yo is contained
in Bn for all n 2 N. Hence q E J+(P,B,) for all n 2 N. Thus by Lemma
8.6, for each n >_ N there exists a future directed nonspacelike curve y, in Bn joining p to q with L(yn) = d [B,](p, q). By hypothesis, each yn intersects
K in some point r,. Since K is compact, there exists a point T E K and a
subsequence {r,) of {r,) such that r, -+ r as m + m. Extend each curve
ym to a past and future inextendible nonspacelike curve which we will still
denote by ym. By Proposition 3.31, there exists an inextendible nonspacelike
limit curve y for the subsequence (7,) such that y contains T. Relabeling if
necessary, we may assume that {y,) distinguishes y.
Now the limit curve y may contain both p and q, only p, only q, or neither
p nor q. These four cases give rise respectively to the four cases (1) to (4)
of the proposition. Since the proofs are similar, we will only give the proof
for the second case. Thus we assume that y : (a, b) -+ M contains p = y(to)
but not q. We must show that y 1 [to, b) is maximal. To this end, let x be an
arbitrary point of y 1 [to, b). Since 1%) distinguishes y, we may find points
8.2 GEODESIC RAYS IN CAUSAL SPACE-TIMES 281
x, E y, with x, -+ x as m -+ w. Passing to a subsequence {yk) of {y,)
if necessary, we may assume by Proposition 3.34 that yk[p,xk] converges to
y[p, x] in the C0 topology on curves (recall Notational Convention 8.4). Since
y[p, x] is closed in M and q # y, there exists an open set V containing y[p, x]
with q # V . Since yk[p,xk] -+ y[p,x] in the C0 topology on curves, there
exists an Nl > O such that yk[p,xk] 2 V for all k 2 Nl. Hence q @ yk[p,xk]
for all k 2 NI . Thus yk[p,xk] E yk[p, q] for all k 2 N1 which implies that
L(yk[p,xk]) = d [Bk] (P ,~k) for all k > N1. By Lemma 8.5 and the upper
semicontinuity of arc length in the C0 topology on curves for strongly causal
spacetimes, we have
d(p, x) < lim inf d [Bk](p, xk)
= lim inf L(yk[p, ~ k ] )
5 f i m s u ~ L ( y k \ ~ , xk]) 5 L(y[p, XI). Since L(y[p, x]) < d(p,x) by definition of Lorentzian distance, we thus have
d(p, x) = L(y[p, XI) as required.
For globally hyperbolic space-times, case (1) of Proposition 8.9 always ap-
plies because J+(p) n J-(q) is compact and no inextendible nonspacelike curve
is past or future imprisoned in a compact set. However, space-tirnes which
are strongly causal but not globally hyperbolic and which have chronologically
related points p << q to which exactly one of cases (2) to (4) applies may be
constructed by deleting points from Minkowski space-time.
With Proposition 8.9 in hand, we are now ready to prove the existence of
past and future directed nonspacelike geodesic rays issuing from each point
of a strongly causal space-time. By the usual duality, it suffices to show the
existence of a future directed ray at each point.
Theorem 8.10. Let (M, g) be a strongly causal space-time and let p E M
be arbitrary. Then there exists a future directed nonspacelike geodesic ray
y : [O,a) -+ M with y(0) = p, i.e., d(p,y(t)) = L(y 1 [ O , t ] ) for all t with
O l t < a .
Proof. Let c : [0, b) --+ M be a future directed, future inextendible, timelike
curve with c(0) = p. Since (M,g) is strongly causal, c cannot be future
282 8 MAXIMAL GEODESICS, DISCONNECTED SPACE-TIMES
imprisoned in any compact set (cf. Proposition 3.13). Thus there is a sequence
{t,) with t , --+ b such that do(p,c(tn)) --+ co as n co. Set q, = c(t,) for
each n.
We now apply Proposition 8.9 to each pair p,qn with K = {p). Thus
for each n, either (1) there is a maximal future directed nonspacelike geodesic
segment from p to q,, or (2) there is a future directed, future-inextendible, non-
spacelike geodesic ray starting at p. If case (2) occurs for some n, we are done.
Thus assume that for each n there is a maximal future directed nonspacelike
geodesic segment y, from p to q,. Extend each y, to a future directed, future
inextendible, nonspacelike curve, still denoted by y,. By Proposition 3.31, the
sequence (7,) has a future directed, future inextendible, nonspacelike limit
curve y : [O,a) + M with y(0) = p. Relabeling the q, if necessary, we may
suppose that the sequence {y,) itself distinguishes y.
It remains to show that if x E 7 with p # x is arbitrary, then L(y[p, x]) =
d(p, x). Since {y,) distinguishes y, we may choose x, E y, for each n such
that x, -+ x as n + co. Choose a subsequence (7,) of {y,) by Proposition
3.34 such that {~m[p,xm]) converges to y[p,x] in the C0 topology on curves.
Hence there exists an N > 0 such that y,[p,x,] C BN for all m 2 N.
Since BN is compact and do(p, q,) + co, there exists an Nl 2 N such that
q, $ BN for all m 2 Nl. Hence L(ymip, x,]) = d(p, x,) = d [B,](p, 2,) for
all m 2 N1. Using Lemma 8.5 and the upper semicontinuity of Lorentzian arc
length in the C0 topology on curves, we then obtain
d(p, x) 5 lim inf d x,) = lim inf L(y,[p, x,])
5 limsupL(ym[p, xm1) 5 L(y[p,xI)
whence d(p, x) = L(y[p, x]) as in Proposition 8.9. 0
8.3 Causally Disconnected Space-times a n d Nonspacelike
Geodesic Lines
In this section we define and study the class of causally disconnected space-
times. Our definition of this class of spacetimes is motivated by the geometric
realization, discussed in the introduction to this chapter, of the ends of a
8.3 DISCONNECTED SPACE-TIMES, NONSPACELIKE LINES 283
noncompact complete Riemannian manifold by geodesic lines [cf. Freudenthal
(1931) for the original definition of ends of a noncompact Hausdorff topological
space]. Recall that an infinite sequence in a noncompact topological space is
said to diverge to infinity if given any compact subset C, only finitely many
elements of the sequence are contained in C.
Definition 8.11. (Causally Disconnected Space-time) A space-time is
said to be causally disconnected by a compact set K if there exist two infinite
sequences {p,) and {q,) diverging to infinity such that for each n, p, < q,,
p, # q,, and all future directed nonspacelike curves from p, to q, meet K. A
space-time (M, g) that is causally disconnected by some compact K is said to
be causally disconnected.
Note first that if k # n, then pk is not necessarily causally related to q,
or p,. Also the compact set K may be quite different from a Cauchy surface
(cf. Theorem 3.17) and non-globally hyperbolic, strongly causal space-times
may be causally disconnected even though they contain no Cauchy surfaces.
An example is provided by a Reissner-Nordstrom space-time with e2 = m2
(cf. Figure 8.1).
It is immediate from Definition 8.11 that if ( M , g ) is causally disconnected
and gl E C(M, 9) is arbitrary, then (M,gl) is causally disconnected. Thus
causal disconnection is a global conformal invariant.
We have previously used a more restrictive version of the concept of causal
disconnection in which we assumed in addition to the conditions of Definition
8.11 that 0 < d(p,, q,) < co for each n [cf. Beem and Ehrlich (1979a, p. 171,
1979c)l. With this additional condition our previous definition was, in general,
conformally invariant only for the class of globally hyperbolic space-times.
We now give the following definition.
Definition 8.12. (Nonspacelike Geodesic Line) Let (M,g) be an ar-
bitrary space-time. A past and future inextendible, future directed, non-
spacelike geodesic y : (a, b) 4 M is said to be a nonspacelike geodesic line if
L(y ] IS, t ] ) = d(-y(s),r(t)) for all s, t with a < s < t < b.
284 8 MAXIMAL GEODESICS, DISCONNECTED SPACE-TIMES
FIGURE 8.1. A Penrose diagram for a Reissner-Nordstrom space-
time with e2 = m2 containing a causally disconnected set K and
associated divergent sequences { p , ) and {q,) is shown. Thii s p a c e
time contains no Cauchy surfaces because it is not globally hyper-
bolic.
8.3 DISCONNECTED SPACE-TIMES, NONSPACELIKE LINES 285
We now establish the existence of nonspacelike geodesic lines for strongly
causal, causally disconnected space-times. This result will be an important
ingredient in the proof of singularity theorems for causally disconnected s p a c e
times in Chapter 12.
Theorem 8.13. Let ( M , g ) denote a strongly causal space-time which is
causally disconnected by a compact set K. Then M contains a nonspacelike
geodesic line y : (a, b) -+ M which intersects K.
Proof. Let K, {p,), and {q,) be as in Definition 8.11. Applying Proposition
8.9 to K , p,, and q, for each n, we obtain a future directed nonspacelike
geodesic 7, intersecting K at some point r, and satisfying at least one of the
cases (1) to (4) of Proposition 8.9. If case (4) holds for any y,, then we are
done. Thus assume that no y, satisfies case (4). Hence at least one of cases (I),
(2), or (3) holds for infinitely many n. Since the proofs are similar, we will only
give the proof assuming that case (2) holds for infinitely many n. Passing to
a subsequence if necessary, we may suppose that condition (2) holds for all n.
Since K is compact, there exists a subsequence {r,) of {r,) such that r, -+ r
as m --+ m. Extend each y, past p, to get a nonspacelike curve, still denoted
by y,, that is past as well as future inextendible for each m. By Proposition
3.31, the sequence {y,) has a future directed, past and future inextendible,
nonspacelike limit curve y with r f y. Relabeling if necessary, we may assume
that {y,) distinguishes y. We will prove that y is the required nonspacelike
line. To prove this, it suffices to show that if z , y f y are distinct points with
x Q r 6 y, then L(y[x, y]) = d(x, y). Since {y,) distinguishes y, we may find
points x,, y, E ym such that x, + x and y, --+ y as m --+ m. Passing to
a subsequence {yk} of (7,) if necessary, we may suppose by Proposition 3.34
that yk [xk, yk] converges to y [x, y] in the fl topology on curves. Since y [x, y]
is compact in M , there exists an N > 0 such that y[x, y] C Int(BN). By
definition of the C0 topology on curves, there is then an Nl 2 N such that
yk[zk, yk] C Int(BN) for all k 2 Nl. Since {pk) diverges to infinity and BN is
compact, there is an Nz > Nl such that pk $ BN for all k 2 Nz. Consequently,
xk comes after pk on yk for all k 2 N2 so that yk[xk,yk] is maximal for all
286 8 MAXIMAL GEODESICS, DISCONNECTED SPACE-TIMES
k > N2. We thus have
d(x, y) 5 lim inf d(xk, yk) = lim inf L(yk[zk, yk])
5 l i m s u ~ L ( ~ k [ ~ k , ~ k ] ) 5 L(Y[x, Y]) < ~ ( x , Y ) .
Hence d(x, y) = L(y[x, y]) as required. 0
We now give several criteria, expressed in terms of the global geodesic struc-
ture, for globally hyperbolic space-times and for strongly causal space-times
to be causally disconnected. In particular, we are able to show that all two-
dimensional globally hyperbolic spacetimes are causally disconnected. Also
one of our criteria (Proposition 8.18) together with Theorem 8.13 implies that
if a strongly causal space-time (M,g) has no null geodesic rays, then (M, g)
contains a timelike geodesic line.
Recall that an inextendible null geodesic y : (a, b) -+ (M, g) is said to be a
null geodesic line if d(y(s),y(t)) = 0 for all s, t with a < s 5 t < b.
Proposition 8.14. Let (M,g) be strongly causal. If (M, g) contains a null
geodesic line, then (M, g) is causally disconnected.
Proof. Let c : (a, b) --+ (M, g) be the given null geodesic line. Choose d with
a < d < band put K = {c(d)). Choose sequences {s,), {t,) with s, < d < t, and s, -+ a, t, + b. Put p, = c(s,) and q, = c(t,). Since c is both future and
past inextendible and (M, g) is strongly causal, both {p,) and {q,) diverge to
infinity by Proposition 3.13. Now because c is a maximal null geodesic, any
future directed nonspacelike curve a from p, to q, is a reparametrization of
cl [s,, t,], whence a meets K as required. (First, a may be reparametrized
to be a smooth future directed null geodesic segment a : [O, 11 --+ M with
a(0) = p,, a(1) = q, by Theorem 4.13. If a l ( l ) # Xcl(tn) for some X > 0,
then q,+~ E I+(pn). But this contradicts d(p,, qn+l) = 0 since c is maximal.
Hence, a'(1) = Xc1(tn) for some X > 0. But then since (M, g) is causal, a must
simply be a reparametrization of c 1 [s,, t,].) D
Proposition 8.14 implies that Minkowski space-time, de Sitter space-time,
and the Friedmann cosmological models are all causally disconnected. Ako,
the Einstein static universe (cf. Example 5.11) shows that there are globally
8.3 DISCONNECTED SPACSTIMES, NONSPACELIKE LINES 287
hyperbolic, causally disconnected spacetimes which do not have null geodesic
lines. Thus the existence of a null geodesic line is not a necessary condition
for a globally hyperbolic space-time to be causally disconnected.
Evidently, the strong causality in Proposition 8.14 may be replaced by any
other nonimprisonment condition which guarantees that both ends of c diverge
to infinity, together with the requirement that (M, g) be causal.
In the next proposition we will give a sufficient condition for a strongly
causal space-time (M,g) to be causally disconnected. For the proof of this
result (Proposition 8.18), it is necessary to recall some additional concepts
from elementary causality theory. A subset S of (M, g) is said to be achronal
if no two polnts of S are chronologically related. Given a closed subset S of (M,g), the future Cauchy development or domazn of dependence D f (S)
of S is defined as the set of all points q such that every past inextendible
nonspacelike curve from q intersects S. The future Cauchy honzon H+(S) is
given by Hf (S) = D+(S) - I-(D+(S)). The future honsmos E+(S) of S is
defined to be E+(S) = J+(S) - I+(S). An achronal set S is said to be future
trapped if E+(S) is compact. Details about these concepts may be found in
Hawking and Ellis (1973, pp. 201, 202, 184, and 267, respectively).
For the proof of Proposition 8.18 we also need to use a result first obtained
in Hawking and Penrose (1970, p. 537, Lemma 2.12). This result is presented
somewhat differently during the course of the proof of Theorem 2 in the text of
Hawking and Ellis (1973, p. 266). In the proof of this theorem, it is assumed
that dim M > 3 and that (M,g) has everywhere nonnegative nonspacelike
Ricci curvatures and satisfies the generic condition [conditions (1) and (2) of
Theorem 21. However, it may be seen that in the proof of Lemma 8.2.1 and
the following corollary in Hawking and Ellis (1973, pp. 267-269), it is only
necessary to assume that (M, g) is strongly causal to obtain our Lemma 8.15
and Corollary 8.16. We now state these two results for completeness.
Lemma 8.15. Let A be a closed subset of the strongly causal space-time
(M,g). Then H+(Ef (A)) is noncompact or empty.
From this lemma, one obtains as in Hawking and Penrose (1970, p. 537) or
Hawking and Ellis (1973, pp. 268-269) the following corollary.
288 8 MAXIMAL GEODESICS, DISCONNECTED SPACE-TIMES
Corollary 8.16. Let (M,g) be strongly causd. If S Ss future trapped in
(M,g), i.e., E+(S) is compact, then there is a future inextendible timelike
curve y contained in D+(E+(S)).
It will also be convenient to prove the following lemma for the proof of
Proposition 8.18.
Lemma 8.17. Let (M, g) be strongly causal. If E+(p) is noncompact, then
E+(p) contains an infinite sequence {q,) which diverges to infinity.
Proof. If E+(p) is closed, this is immediate since a closed and noncompact
subset of M must be unbounded with respect to do. Thus assume that E+(p)
is not closed. Then there exists an infinite sequence {x,) C E+(p) such that
xn -t x 4 E+(p) as n -4 co. Since x, E E+(p), we have d(p, x,) = 0 and hence
as x, E J+(p), there exists a maximal future directed null geodesic segment
yn from p to x, for each n. Extend each /, beyond x, to a future inextendible
nonspacelike curve still denoted by y,. By Proposition 3.31, the sequence {y,)
has a future inextendible, future directed, nonspacelike limit curve y : [O, a) +
M with y(0) = p. We may assume that the sequence {y,) itself distinguishes
y. If x E y, then x E J+(p). Since d(p,x) 5 liminfd(p,z,) = 0, we then
have x E J+(p) - I+(p) = E+(p), in contradiction to the assumption that
x 4 E+(p). Thus x f y. We now show that y[O, a) is contained in E+(p). To
this end, let z E y be arbitrary. Since {y,) distinguishes y, we may find z, E y,
such that z, -t z as n -+ a. By Proposition 3.34, there is a subsequence {yk)
of {?,) such that 7k[P, zk] converges to yip, z] in the c0 topology on curves.
Since x 4 7, we may find an open set U containing 7[p, z] such that x f n. Since xk -t x, it follows that zk comes before xk on yk for all k sufficiently
large. Thus y[p, zk] is maximal and d(p, zk) = 0 for all k sufficiently large.
Hence d(p, z) 5 liminf d(p, zk) = 0. Since z was arbitrary, we have thus shown
that d(p, z) = 0 for all z E y. Since y is a nonspacelike curve, 7 is then a
maximal, future directed, future inextendible geodesic ray. Letting {t,) be
any infinite sequence with t, -4 a- and setting q, = y(t,) gives the required
divergent sequence. D
8.3 DISCONNECTED SPACGTIMES, NONSPACELIKE LINES 289
With these preliminaries completed, we may now obtain a sufficient condi-
tion for strongly causal space-times to be causally disconnected. Minkowski
space-time shows that this condition is not a necessary one. Recall that a future directed, future inextendible, null geodesic y : [0, a) --+ M is said to be
a null geodesic my if d(y(O), y(t)) = 0 for all t with 0 _< t < a.
Proposition 8.18. Let (M,g) be strongly causal. If p E M is not the origin of any future [respectively, past] directed null geodesic ray, then (M, g)
is causally disconnected by the future [respectively, past] horismos E+(p) =
J+(p) - I+(p) [respectively, E- (p) = J- (p) - I- (P)] of p.
Proof. We first show that the assumption that p is not the origin of any
future directed null geodesic ray implies that E+(p) is compact. For suppose
that E+(p) is noncompact. Then there exists an infinite sequence {q,) C E+(p) which diverges to infinity by Lemma 8.17. Since q, E E+(p), we have
d(p, q,) = 0 for all n. As qn E Jf (p), there exists a future directed null
geodesic yn from p to q, by Corollary 4.14. Extend each yn beyond qn to a
future inextendible nonspacelike curve, still denoted by 7,. Let y be a future
inextendible nonspacelike limit curve of the sequence (7,) with y(0) = p,
guaranteed by Proposition 3.31. Using Proposition 3.34 and the fact that
the qn's diverge to infinity, it may be shown along the lines of the proof of
Theorem 8.10 that if q is any point on y with q # p, q 2 p, then L(y [p, q]) =
d(p, q). Thus y may be reparametrized to a null geodesic ray issuing from p,
in contradiction. Hence Ef (p) is compact.
We now show that E+(p) causally disconnects (M,g). Since E+(p) is com-
pact, the set {p) is future trapped in M. Thus by Corollary 8.16, there is a
future inextendible timelike curve y contained in D+(Ef (p)). Extend y to a
past as well as future inextendible timelike curve, still denoted by y. From
the definition of D+(Ef (p)), the curve y must meet E+(p) at some point r .
Since E+(p) is achronal and y is timelike, y meets E+(p) at no other point
than r. Now let {p,) and (9,) be two sequences on y both of which diverge
to infinity and which satisfy p, << r << q, for each n (cf. Proposition 3.13). To
show that {p,), {q,), and E+(p) causally disconnect (M,g), we must show
that for each n, every nonspacelike curve X : [O, 11 -+ .hri with X(0) = p, and
290 8 MAXIMAL GEODESICS, DISCONNECTED SPACE-TIMES
X(1) = q, meets E+(p). Given A, extend X to a past inextendible curve 5 by
traversing y up to p, and then traversing X from p, to q, (cf. Figure 8.2). As
qn E D f (E+(p)), the curve 5 must intersect Ef (p). Since y meets E+(p) only
a t T, it follows X intersects E f (p). Thus {p,), {q,), and K = E+(p) causally
disconnect (M, g) as required. 13
Combining Theorem 8.13 and Proposition 8.18, we obtain the following
result on the geodesic structure of strongly causal space-times with no null
geodesic rays. Examples of such space-times are the Einstein static universes
(Example 5.1 1).
Theorem 8.19. Let (M, g) be a strongly causal space-time such that no
point is the origin of any future directed null geodesic ray Then (M, g) contains
a timelike geodesic line.
Proof. By Proposition 8.18, (M,g) is causally disconnected. Thus (M,g)
contains a nonspacelike line by Theorem 8.13. By hypothesis, the line must
be timelike rather than null. 0
An equivalent result may be formulated using the hypothesis that no point
is the origin of any past directed null geodesic ray.
Using Propositions 8.14 and 8.18, we are now able to show that all two-
dimensional globally hyperbolic space-times are causally disconnected. We
first establish the following lemma.
Lemma 8.20. Let (M, g) be a two-dimensional globally hyperbolic space-
time. If E+(p) [respectively, E-(')] is not compact, then both of the future
[respectively, past ] directed and future [respectively, past] inextendible null
geodesics starting at p are maximal.
Proof. Assume E+(p) is noncompact for some p E M. Let cl : 10, a ) -+ M
and c:! : [0, b) + M be the two future directed, future inextendible, null
geodesics with cl(0) = cz(0) = p. Suppose that cl is not maximal. Setting
to = supit E [0, a) : d(p, cl (t)) = 0), we have 0 < t o < a since cl is not maximal
and (M,g) is strongly causal (cf. Section 9.2). Let q = cl(t0) and choose
t, --i t g with t, < a for each n. By the lower semicontinuity of Lorentzian
8.3 DISCONNECTED SPACE-TIMES, NONSPACELIKE LINES 291
FIGURE 8.2. In the proof of Proposition 8.18, the set E*(p) is
shown to causally disconnect (M,g) if p is not the origin of any
future directed null geodesic ray. The timelike curve y intersects
E+(p) in the single point T, and any nonspacelike curve X from p,
to q, must meet E+(p).
distance, w-e have d(p, q) = 0. Since cl(t,) E I+(p) and (M, g) is globally
hyperbolic, there is a maximal future directed timelike geodesic y, from p to
292 8 MAXIMAL GEODESICS, DISCONNECTED SPACETIMES
cl(tn) for each n. The sequence (7,) has a limit curve y which is a nonspacelike
geodesic, and y joins p to q by Corollary 3.32. Ikthermore, d(p, q) = 0 implies
that y is a null geodesic. Since dim M = 2, y is either a geodesic subsegment of
cl or a subsegment of cz. If y 5 cz, then cz passes through q at some parameter
value tb. In this case E+(p) = {cl 1 [O, to]) U {cz 1 [O, tb]) is compact and we
have a contradiction.
Assume y cl and let U(q) be a convex normal neighborhood about q.
Let U(q) be chosen so small that no nonspacelike curve which leaves U(q)
ever returns. The inextendible null geodesics of U(q) may be divided into
two disjoint families Fl and F2, each of which cover U(q) simply (cf. Section
3.4). Let Fl be the class which contains the null geodesic cl n U(q). Let c3
be the unique null geodesic of F2 which contains q. For some fixed large n
the timelike geodesic segment yn from p to cl(tn) must intersect c3 at some
point r. We must have q < r since if r < q, then p << r would yield p << q,
which is false. However, q < r, q = cl n cs, q < cl(tn), and r E cs imply
r $ cl(tn), [cf. Busemann and Beem (1966, p. 245)]. This contradicts the fact
that r << cl(tn) because r comes before cl(tn) on the timelike geodesic 7,.
This establishes the lemma. D
Theorem 8.21. All two-dimensional globally hyperbolic space-times are
causally disconnected.
Proof. Let (M, g) be a two-dimensional globally hyperbolic space-time. If
E+(po) is compact for some po E M, then each future directed null geodesic
starting at po has a cut point and thus fails to be globally maximal (cf. Section
9.2). Thus (M,g) is causally disconnected by Proposition 8.18. Now suppose
E+ (p) is noncompact for each p E M. Assume c : (a, b) -, M is a future
directed inextendible null geodesic. Let s , t with a < s 5 t < b be arbitrary.
Applying Lemma 8.20 at the point p = c(s), we find that c 1 [s, b) is maximal
and thus d(c(s), c(t)) = L(c] [s, t]) . This implies that c is maximal and hence
c is a null line. Using Proposition 8.14, it follows that (M,g) is causally
disconnected. O
8.3 DISCONNECTED SPACGTIMES, NONSPACELIKE LINES 293
We obtain the following corollary to Theorems 8.13 and 8.21.
Corollary 8.22. Let (M, g) be any globally hyperbolic space-time of di-
mension n = 2. Then (M, g) contains a nonspacelike line.
CHAPTER 9
THE LORENTZIAN CUT LOCUS
Let c : 1 0 , ~ ) -+ N be a geodesic in a complete Riemannian manifold
starting at p = c(0). Consider the set of all points q on c such that the portion
of c from p to q is the unique shortest curve in all of N joining p t o q. If this
set has a farthest limit point, this limit point is called the cut point of p along
the ray c. The cut locus C(p) is then defined to be the set of cut points along
all geodesic rays starting at p. Since nonhomothetic conformal changes do not
preserve pregeodesics, the cut locus of a point in a manifold is not a conformal
invariant.
The cut locus has played a key role in modern global Riemannian geometry,
notably in connection with the Sphere Theorem of Rauch (1951), Klingenberg
(1959, 1961), and Berger (1960). The notion of cut point, as opposed to the
related but different concept of conjugate point, was first defined by Poincari.
(1905). An observation of Poincark (1905) important in the later work of
Rauch, Klingenberg, and Berger was that for a complete Riemannian manifold,
if q is on the cut locus of p, then either q is conjugate to p, or else there
exist at least two geodesic segments of the same shortest length joining p to
q [cf. Whitehead (1935)l. Klingenberg (1959, p. 657) then showed that if q is
a closest cut point to p and q is not conjugate to p, there is a geodesic loop
at p containing q. Klingenberg used this result to obtain an upper bound for
the injectivity radius of a positively curved, complete Riemannian manifold
in terms of a lower bound for the sectional curvature and the length of the
shortest nontrivial smooth closed geodesic on N.
The importance of the cut locus in Ftiemannian geometry suggests inves-
tigating the analogous concepts and results for timelike and null geodesics in
space-times. The central role that conjugate points, which are closely related
296 9 T H E LORENTZIAN CUT LOCUS
to cut points, have played in singularity theory in general relativity (cf. Chap
ter 12) supports this idea. While there are many similarities between the
Riemannian cut locus and the locus of timelike cut points in a space-time,
there are also striking differences between the Riemannian and Lorentzian cut
loci. Most notably, null cut points are invariant under conformal changes.
Thus the null cut locus is an invariant of the causal structure of the space-
time (M,g). This may be used [cf. Beem and Ehrlich (1979a, Corollary 5.3)]
to show that if (M,g) is a Riedmann cosmological model, then there is a C2 neighborhood U(g) in the space C ( M , g) of Lorentzian metrics for M globally
conformal to g such that every null geodesic in (M,gl) is incomplete for all
metrics gl E U(g). Because there are intrinsic differences between null and
timelike cut points, we prefer to treat these cases separately. One such differ-
ence is that unlike null cut points, timelike cut points are not invariant under
global conformal changes of Lorentzian metric.
In Section 9.1 we consider the analogue of Riemannian cut points for time-
like geodesics. In the Lorentzian setting, timelike geodesics locally maxi-
mize the arc length between any two of their points. Thus the appropri-
ate question to ask in defining timelike cut points is whether the portion
of the given timelike geodesic segment from p to q is the longest nonspace-
like curve in all of M joining p to q. This may be conveniently formulated
using the Lorentzian distance function. Let y : [O,a) + M be a future di-
rected, future inextendible, timelike geodesic in an arbitrary space-time. Set
to = sup{t E 10, a) : d(y(O), y(t)) = L(y I [0, t])). If 0 < to < a, then y(t0) is
said to be the future timelike cut point of y(0) along y. The future timelike cut
point y(to) then has the following desired properties: (i) if t < to, then y I [O, t ]
is the only maximal timelike curve from y(0) to y(t) up to reparametrization;
(ii) y 1 [O,t] is maximal for any t 5 to; (iii) if t > to, there exists a future
directed nonspacelike curve a from y(0) to y(t) with L(a) > L(y 1 [O,t]); and
(iv) the future cut point y(to) comes at or before the first future conjugate
point of y(0) along y.
Since many of the theorems for Riemannian cut points are true only for
complete Riemannian manifolds, it is not surprising that the more "global"
9 THE LORENTZIAN CUT LOCUS 297
results given in the rest of Section 9.1 often require global hyperbolicity. What
is essential here is to know that chronologically related points at arbitrarily
large distances may be joined by maximal timelike geodesic segments. Even so,
the proofs are more technical for space-times than for Riemannian manifolds.
Instead of using the exponential map directly as in Riemannian geometry, it
is necessary to regard a sequence of maximal timelike geodesics as a sequence
of nonspacelike curves, extract a limit curve, take a subsequence converging
to the limit in the C0 topology (by Section 3.3), and finally, using the upper
semicontinuity of arc length, prove that the limit curve is maximal and hence a
geodesic. This technical argument, which is isolated in Lemma 9.6, yields the
following analogue of Poincark's Theorem for complete Riemannian manifolds.
If (M, g) is a globally hyperbolic space-time and q is the future cut point of p
along the timelike geodesic segment c from p to q, then either one or possibly
both of the following hold: (i) q is the first future conjugate point to p; or (ii)
there exist a t least two maximal geodesic segments from p to q.
In Section 9.2 we study null cut points. Even though null geodesics have zero
arc length, null cut points may still be defined using the Lorentzian distance
function. Let y : [O,a) -+ M be a future directed, future inextendible, null
geodesic with p = y(0). Set to = supit E [O,a) : d(p, y(t)) = 0). If 0 < to < a,
then $to) is called the future null cut point of y(0) along y. The null cut
point, if it exists, has the following properties: (i) y is maximizing up to and
including the null cut point; (ii) there is no timelike curve joining p to y(t) for
any t 5 to; (iii) if to < t < a, there is a timelike curve from y(0) to y(t); and
(iv) the future null cut point comes at or before the first future conjugate point
of y(0) along y. For globally hyperbolic space-times, it is true for null as well
as timelike cut points that the analogue of Poincark's Theorem for complete
Riemannian manifolds is valid. Thus if (M,g) is globally hyperbolic and q is
the future null cut point of p = y(0) along the null geodesic y, then either one
or possibly both of the following hold: (i) q is the first future conjugate point of
p along y; or (ii) there exist at least two maximal null geodesic segments from
p to q. We conclude Section 9.2 by using null cut points to prove singularity
theorems for null geodesics following Beem and Ehrlich (1979a, Section 5).
298 9 THE LORENTZIAN CUT LOCUS 9.1 T H E TIMELIKE CUT LOCUS 299
The nonspacelike cut locus, the union of the null and timelike cut loci of Definition 9.2. Let T-IM = {v E T M : v is future directed and g(v,v) =
a given point, is studied in Section 9.3. For complete Riemannian manifolds, -1). Given p E M , let T-lMlp denote the fiber of T-l M at p. Also given
if q is a closest cut point to p, then either q is conjugate to p or there is a v E T-1M, let c, denote the unique timelike geodesic with cV1(0) = v.
geodesic loop based a t p passing through q. The globally hyperbolic analogue It is immediate from the reverse triangle inequality that if y : [0, a) -+ M is
(Theorem 9.24) of this result has a slightly different flavor, however. If (M, g) a future directed nonspacelike geodesic and d(y(O), y(s)) = L ( y 1 [O, s]), then is a globally hyperbolic space-time and q E M is a closest (nonspacelike) cut d(y(O),y(t)) = L ( y 1 [O,t]) for all t with 0 I t 5 s. Also, the reverse triangle point to p, then q is either conjugate to p or else q is a null cut point to p. inequality implies that if d(y(O), y(s)) > L(y 1 [0, s]), then d(y(O), y(t)) > Thus there is no closest nonconjugate timelike cut point to p. We also show L(y 1 [O,t]) for all t with s < t < a. Hence the following definition makes
that for globally hyperbolic space-times, the nonspacelike and null cut loci are
closed (Proposition 9.29). It can be seen (Example 9.28) that the hypothesis Definition 9.3. Define the function s : T-lM -, R u {m) by s(v) = of global hyperbolicity is necessary here. supit 2 0 : d(r(v), c,(t)) = t).
In Section 9.4 we treat null and timelike cut points simultaneously, but
nonintrinsically, using a different tool than the unit timelike sphere bundle We may first note that if d(p,p) = m , then s(v) = 0 for all v E T-lM with
of Section 9.1. This approach, developed in collaboration with G. Gallow r(v) = p. Also S(V) > 0 for all v E T-1M if (M,g) is strongly causal. The
while non-intrinsic, enables the hypothesis of timelike geodesic completenes number s(v) may be interpreted as the "largest" parameter value t such that
to be deleted from Proposition 9.30. c, is a maximal geodesic between c,(O) and c,(t). Indeed from Lemma 9.1 we
know that the following result holds.
9.1 The Timelike Cut Locus Corollary 9.4. For 0 < t < s(v), the geodesic c, : [0, t ] -+ M is the unique
Recall that a future directed nonspacelike curve y from p to q is said t maximal timelike curve (up to repararnetrization) from c,(O) to c,(t).
be maximal if d ( p , q) = L(y). We saw above (Theorem 4.13) that a maxim The function s fails to be upper semicontinuous for arbitrary spacetimes as
future directed nonspacelike curve may be reparametrized to be a geode may easily be seen by deleting a point from Minkowski space. But for timelike
We also recall the following analogue of a classical result from Rieman geodesically complete space-times we have the following proposition.
geometry. The proof may be given along the lines of Kobayashi (1967, p. Proposition 9.5. Let v f T-1M with s(v) > 0. Suppose either that using in place of the minimal geodesic segment from pl to p2 in the Rieman (v) = +m, or ~ ( v ) is finite and c,(t) = exp(tv) extends to [0, s(v)]. Then s is proof the fact that if p << q and p and q are contained in a convex norm pper semicontinuous at v f T-1M. Especially, if (M, g) is timelike geodesi- neighborhood, then p and q may be joined by a maximal timelike geo ally complete, then s : T-1M -+ W U { ~ ) is everywhere upper semicontinuous. segment which lies in this neighborhood.
Proof. It suffices to show the following. Let v, -+ v in T-lM with (s(vn)) Lemma 9.1. Let c : [O, a] -+ M be a maximal timelike geodesic segm onverging in R U {a). Then s(v) 2 lim s(v,). If s(v) = m, there is nothing
Then for any s , t with 0 5 s < t < a , the curve c 1 [s, t ] is the unique maxi prove. Hence we assume that s(v) < lims(vn) = A and s(v) < co and geodesic segment (up to parametrization) from c(s) to c(t). rive a contradiction.
Before commencing our study of the timelike cut locus, we need to defi We may choose 6 > 0 such that s(v) + 6 < A is in the domain of c,
the unit future observer bundle T-IM [cf. Thorpe (1977a,b)]. d also assume that s(v,) 2 s(v) + 6 = b for all n. Let c, = tun. Since
300 9 THE LORENTZIAN CUT LOCUS 9.1 THE TIMELIKE CUT LOCUS 301
b < s(v,), we have d(n (v,), c,(b)) = b for all n. Since v, 4 v, we have by lower and q = c,(A + 6). Since v, 4 v and c, is defined for some parameter values semicontinuity of distance that d(r(v), ~ ( b ) ) 5 liminf d(r(v,), ~ ( b ) ) = b. beyond A + 6, the geodesics c, must be defined for some parameter values
Thus d(x(v), c,(b)) < b = L(c, 1 [0, b]), this last equality by definition past b, whenever n is larger than some N 2 No. Let q, = c,(b,) for n > N. of arc length. On the other hand, d(n(v), ~ ( b ) ) 2 L(q, 1 [0, b]) so that Now c, I [O, b,] cannot be maximal since b, > s(v,). Because M is globally d(r(v),q,(b)) = L(c, I [0, b]) = b. Hence s(v) 2 b = s(v) + 6, in contra- hyperbolic and c,(O) << c,(b,), we may find maximal unit speed timelike diction. 0 geodesic segments y, : [O,d(p,,q,)] M from p, to q,. Set w, = yn1(O).
Since c, ( [0, s(v)) is a maximal geodesic and thus has no conjugate points, it In order to prove the lower semicontinuity of s for globally hyperbolic space-
is impossible for v to be a limit direction of {w, : n 2 N). Thus the maximal times, it will first be useful to establish the following lemma.
geodesic c, joining p to q given by Lemma 9.6 applied to {w,) is different from Lemma 9.6. Let (M, g) be a globally hyperbolic space-time, and let {p,) c,. This then implies that s(v) < A + 6, which contradicts A + 6 < s(v).
and {q,) be two infinite sequences of points with p, p and q, --+ q where
p << q. Assume c, : [O,d(p,, q,)] 4 M is a unit speed maximal geodesic The following example of a strongly causal space-time which is not globally
segment from p, to q,, and set v, = ~ ' ( 0 ) E T-lM. Then the sequence {v,) hyperbolic shows that the hypothesis of global hyperbolicity is necessary for
has a timelike limit vector ur f T-l M. Moreover, c,,, : [0, d(p, q)] --+ M is a the lower semicontinuity of the function s in Proposition 9.7. Let W2 be given
maximal geodesic segment from p to q. the usual Minkowski metric ds2 = dx2 - dy2, and let (M,g) be the space-
time formed by equipping M = W2 - {(l,y) € R2 : 1 < y < 2) with the Proof. By Corollary 3.32, there is a nonspacelike future directed limit curve
induced metric (cf. Figure 9.1). Let p = (O,O), p, = (l/n,O), v = bldyl,, and c of c, from p to q. By Proposition 3.34 and Itemark 3.35, we have L(c) '2
v, = b/dylpn for all n 2 1. Then v, --+ v as n -+ m. Also let y(t) = (0,t) limsup L(%) = limd(~,, q,) = d(p,q) > 0. Thus as d(p, q) 2 L(c), it follows
and yn(t) = (p,, t) for all t 2 0. Conformally changing g on the compact set that L(c) = d(p, q) > 0. Hence Theorem 4.13 implies that the curve c may
C shown in Figure 9.1 which is blocked from I+(p) by the slit {(I, y) E It2 : be reparametrized to be a maximal timelike geodesic segment from p to q.
1 < y 5 21, we obtain a metric Zj for M with the following properties. First Finally, w = c'(0) / [-g(cl(0), c'(o))]'/~ is the required tangent vector.
y is still a maximal geodesic in (M,Zj) so that s(v) = +m. But for each n
We are now ready to prove the lower semicontinuity of s for globally hype there exists a timelike curve on passing through the set C and joining p, to
bolic space-times. a point q, = ( l l n , y,) on yn with y, 5 4 such that L(a,) > L(ynIpn, q,]).
Hence y,[p,, q,] fails to be maximal for all n so that s(v,) 5 4 for all n. Thus Proposition 9.7. If (M,g) is globally hyperbolic, then the function
s is not lower semicontinuous. Note that (M, g ) is strongly causal but (M, Zj) T- lM --+ W U {co) is lower semicontinuous.
is not globally hyperbolic since J + ( ( l , 0)) n J-((1,3)) is not compact. The
Proof. It suffices to prove that if v, -+ v in ThlM and s(v,) --+ A analogous construction may be applied to n-dimensional Minkowski space to W U {m}, then s(v) 5 A. If A = oo, there is nothing to prove. Thus supp roduce a strongly causal n-dimensional space-time for which the function s
A < m. Assuming s(v) > A, we will derive a contradiction. ils to be lower semicontinuous.
Choose 6 > 0 such that A + 6 < s(v). (Since A + 6 < s(v), the Globally hyperbolic examples also may be constructed for which the func- c,(A + 6) exists even if G, does not extend to s(v).) Define b, = s(v, tion s is not upper semicontinuous. However, this discontinuity may occur at and let No be such that b, < s(v) for all n 2 No. Put c, = cvn, pn = ~n a tangent vector v E T-lM in a globally hyperbolic space-time only if s(v) is
302 9 THE LORENTZIAN C U T LOCUS
finite and ~ ( t ) does not extend to t = s(v). Combining Propositions 9.5 and
9.7, we obtain the following result.
Theorem 9.8. Let ( M , g ) be globally hyperbolic, and suppose for v E
T- lM that either s (v) = +co, or s (v ) is finite and c,(t) = exp(tv) extends to
[0, s (v)] . Then s is continuous at v E T- lM. Especially, i f ( M , g ) is globally
hyperbolic and timelike geodesically complete, then s : T-1M 4 B U {CQ} is
everywhere continuous.
W e are now ready to define the timelike cut locus.
Definition 9.9. (Future and Past Timelike Cut Loci) The future timelike
cut locus in T,M is defined t o be r+(p) = {s(v)v : v E T-lMIp and 0 < s(v) < co). The future timelike cut locus C , f (p ) of p in M is defined to be
C t ( p ) = exp,(r+(p)). I f 0 < s(v) < co and c,(s(v)) exists, then the point
c,(s(v)) is called the future cut point o f p = c,(O) along c,. The past timelike
cut locus C;(p) and past cut points may be defined dually.
W e may then interpret s (v) as measuring the distance from p up to the
ture cut point along c,. Thus Theorem 9.8 implies that for globally hyperboli
timelike geodesically complete spacetimes, the distance from a fixed p E
t o its future cut point in the direction v E T-lMlp is a continuous function
v.
W e now give Lorentzian analogues of two well-known results relating cut a
conjugate points on complete Riemannian manifolds. The following proper
o f conjugate points is well known [Hawking and Ellis (1973, pp. 111-116
Lerner (1972, Theorem 4(6))].
Theorem 9.10. A timelike geodesic is not maximal beyond the first co
jugate point.
In the language o f Definition 9.9 this may be restated as follows.
Corollary 9.11. The future cut point o f p = c,(O) along c, comes no la
than the first future conjugate point o f p along c,.
Utilizing this fact, we may now prove the second basic result on cut an
conjugate points in globally hyperbolic space-times.
9.1 THE TIMELIKE CUT LOCUS
FIGURE 9.1. A strongly causal space-time ( M , 5) is shown with unit timelike tangent vectors v, which converge t o v, but with
s (v ) = +m and s(v,) 5 4 for alln 2 1. Hence s : T - I M -, R ~ { c o }
is not lower semicontinuous.
Theorem 9.12. Let ( M , g) be globally hyperbolic. I f q = c( t ) is the future
cut point o f p = c(0) along the timelike geodesic c from p to q, then either one
or possibly both o f the following hold:
(1 ) The point q is the first future conjugate point o f p along c.
304 9 THE LORENTZIAN CUT LOCUS
(2) There exist a t least two future directed maximal timelike geodesic seg-
ments from p to q.
Proof. Without loss of generality we may suppose that c = c, for some
v E T-lM and thus that t = d(p, q) = s(v). Let {t,) be a monotone decreasing
sequence of real numbers converging to t. Since c(t) E M, the points c(tn) exist
for n sufficiently large. By global hyperbolicity, we may join c(0) to c(tn) by
a maximal timelike geodesic c, = c,,, with v, E T-1 MIp. Since t, > t = s(v),
we have v # v, for all n. Let w E T-lM be the timelike limiting vector for
{v,) given by Lemma 9.6. If v # w, then c and c, are two maximal timelike
geodesic segments from p to q.
It remains to show that if v = w, then q is the first future conjugate point
of p along c. If v = w, then there is a subsequence {v,) of {v,) with v, -+ v.
If v were not a conjugate point, there would be a neighborhood U of v in
T-lMI, such that expp : U -+ M is injective. On the other hand, since c, and
c 1 [0, t,] join c(0) to c(t,) and v, -+ v, no such neighborhood U can exist.
Thus q is a future conjugate point of p along c. By Corollary 9.11, q must be
the first future conjugate point of p along c.
Theorem 9.12 has the immediate implication that for globally hyperbolic
spacetimes, q E C:(p) if and only if p E Cc(q).
The timelike cut locus of a timelike geodesically complete, globally hyper-
bolic space-time has the following structural property which refines Theorem
9.12. We know from this theorem that if q f C$tp) and q is not conjugate
to p, then there exist at least two maximal geodesic segments from p to q.
Accordingly, it makes sense to consider the set
Seg(p) = { q E C$(p) : there exist at least two future directed
maximal geodesic segments from p to q).
Since s : TP1M -+ W U (m) is continuous by Theorem 9.8 and maximal
geodesics joining any pair of causally related points exist in globally hyperbolic
space-times, it may be shown that Seg(p) is dense in C,f(p) for all p E M. The
proof may be given along the lines of Wolter's proof of Lemma 2 for complete
9.2 THE NULL CUT LOCUS 305
Riemannian manifolds [cf. Wolter (1979, p. 93)]. The dual result also holds
for the past timelike cut locus C c (p).
9.2 The Null Cut Locus
The definition of null cut point has been given in Beem and Ehrlich (1979a,
Section 5) where this concept was used to prove null geodesic incompleteness
for certain classes of space-times. Let y : [0, a) -+ (MI g) be a future directed
null geodesic with endpoint p = $0). Set to = supit E [0, a) : d(p, y(t)) = 0).
If 0 < to < a , we will say y(t0) is the future null cut point of p on y. Past null
cut points are defined dually. Let C$(p) [respectively, CG (p)] denote the future
[respectively, past] null cut locus of p consisting of all future [respectively,
past] null cut points of p. The definition of Cz(p) together with the lower
semicontinuity of distance yields d(p, q) = 0 for all q E C2(p). We then define
the future nonspacelike cut locus to be C+(p) = C$(p) U Cz(p). The past
nonspacelike cut locus is defined dually. For a subclass of globally hyperbolic
space-times, Budic and Sachs (1976) have given a different but equivalent
definition of null cut point using null generators for the boundary of I+(p).
The geometric significance of null cut points is similar to that of timelike
ut points. The geodesic y is maximizing from p up to and including the cut
oint ?(to). That is, L ( y 1 [O,t]) = d(p,y(t)) = 0 for all t 5 to. Thus there
no timelike curve joining p to y(t) for any t with t 5 to. In contrast, the
eodesic y is no longer maximizing beyond the cut point y(t0). In fact, each
oint y(t) for to < t < a may be joined to p by a timelike curve.
Utilizing Proposition 2.19 of Penrose (1972, p. 15) and the definition of
aximality, the following lemma is easily established.
Lemma 9.13. Let (M, g) be a causal space-time. If there are two future
irected null geodesic segments from p to q, then q comes on or after the null
cut point of p on each of the two segments.
The cylinder S1 x W with Lorentzian metric ds2 = dedt shows that the
assumption that (M,g) is causal is needed in Lemma 9.13.
We next prove the null analogue of Lemma 9.6.
306 9 THE LORENTZIAN CUT LOCUS
Lemma 9.14. Let (M, g) be globally hyperbolic, and let p, q be distinct
points in M with p 6 q and d(p,q) = 0. Assume that p, -+ p, q, - q and
p, < q,. Let c, be a maximal geodesic joining p, to q, with initial direction
v,. Then the set of direction {v,) has a limit direction w, and c, is a maximal
null geodesic from p to q.
Proof. Using Corollary 3.32 we obtain a future directed nonspacelike limit
curve X from p to q. Since d(p, q) = 0, the curve X must satisfy L(X) = 0. It
follows that X may be reparametrized to a maximal null geodesic.
We may now obtain the null analogue of Theorem 9.12.
Theorem 9.15. Let (M,g) be globally hyperbolic and let q = c(t) be the
future null cut point of p = c(0) dong the null geodesic c. Then either one or
possibly both of the following hold:
(1) The point q is the first future conjugate point of p along c.
(2) There exist a t least two future directed maximal null geodesic segments
from p to q.
Proof. Let v = c'(O), and let t, be a monotone decreasing sequence of real
numbers with t, -+ t. Since q E M, we know that c(t,) exists for all sufficiently
large n. Since (M, g) is globally hyperbolic, we may find maximal nonspacelike
geodesics c, with initial directions v, joining p to c(t,). By Lemma 9.14 the
set of directions {v,) has a limit direction w. If v # w, then the geodesic c,
is a second maximal null geodesic joining p to q as d(p, q) = 0. If v = w, then
q is conjugate to p along c. Since d(p,q) = 0, q must be the first conjugate
point of c (cf. Theorem 10.72). D
We now show how the null cut locus may be applied to prove theorems
on the stability of null geodesic incompleteness. The key ideas needed for
this application are first, that many physically interesting space-times may be
conformally embedded in a portion of the Einstein static universe (cf. Exa
ple 5.11) that is free of null cut points and second, that null cut points
irivariant under conformal changes of metric. A discussion of global con
ma1 embeddings of anti-de Sitter space-time and the Friedmann cosmologi
9.2 THE NULL CUT LOCUS 307
models in the Einstein static universe is given in Penrose (1968) [cf. Hawking
and Ellis (1973, pp. 132, 141)].
We now digress briefly to give an explicit computational discussion of the
well-known fact that null pregeodesics are invariant under global conformal
changes. An indirect proof of the conformal invariance of null cut points may
also be given using the Lorentzian distance function.
Recall that a smooth curve y : J -+ M is said to be a pregeodesic if y can
be reparametrized to a smooth curve c which satisfies the geodesic differential
equation VC~cr(t) = 0. Also recall the following definition.
Definition 9.16. (Global Conformal Dzffeomorphism) A diffeomorphism f : (MI, gl) --+ (M2, g2) of MI onto M2 is said to be a global conformal diffeo-
morphism if there exists a smooth function R : MI -+ R such that
The space-time (M1,gl) is said to be globally conformally diffeomorphic to an
open subset U of (M2,gz) if there exists a diffeomorphism f : M1 -+ U and a
smooth function R : Ml -r R such that
The purpose of using the factor e20 rather than just a positive-valued
smooth function in Definition 9.16 is to give the simplest possible formula
relating the connections V for (M1,gl) and 9 for (M2, 92). Explicitly, it may
be shown that if f*g2 = e20gl where j : (M1,gl) --+ (Mz,g2) is any smooth
function, then
: for any pair X, Y of vector fields on MI. Here "gradRn denotes the gradient
j vector field of C2 with respect to the metric g1 for MI. : Using formula (9.1), it is now possible to show that if y : J --+ MI is a null b geodesic in (MI, gl) , then a = f 0 y : J -+ M2 is a null pregeodesic in (M2,g2). h
308 9 T H E LORENTZIAN CUT LOCUS
The crux of the matter is that because yl(t) is null and V,q' = 0, formula
(9.1) simplifies to
?bjal(t) = 2y1(t)(R)a'(t)
for all t E J. Note, however, that if y were a timelike geodesic, then the factor
gl (y', y') f* (grad R) in formula (9.1) would prevent f o y from being a timelike
pregeodesic in (M2, g2).
Lemma 9.17. Let f : (MI, gl) -+ (M2, g2) be a global conformal diffeo-
morphism of MI onto M2. Then y : J -+ (MI, gl) is a null pregeodesic for
(MI, gl) iff f o y is a null pregeodesic for (M2,gz).
Proof. Since f -' : (M2, g2) + (MI, gl) is also a conformal diffeomorphism,
it is enough to show that if y : J + MI is a null geodesic, then a = f oy is a null
pregeodesic of (M2, 92). That is, we must show that a can be reparametrized
to be a null geodesic of (M2,g2). But it has already been shown that
for some smooth function f : J --t W. Just as in the classical theory of
projectively equivalent connections in Riemannian geometry, however, formula
(9.2) implies that a is a pregeodesic [cf. Spivak (1970, pp. G37 ff. )].
We are now ready to prove the conformal invariance of null cut points
under global conformal diffeomorphisms f : (M1,gl) -r (M2, g2). Notice that
if (MI, gl) is time oriented by the vector field XI, then (M2, g2) is time oriented
either by X2 = f*X1 or -X2. If M2 is time oriented by X2, then f maps future
directed curves to future directed curves and thus future (respectively, past)
null cut points to future (respectively, past) null cut points in Proposition
9.18. On the other hand, if M2 is time oriented by -X2, then f maps future
(respectively, past) null cut points to past (respectively, future) null cut points
since f maps future directed curves to past directed curves.
Proposition 9.18. Let f : (MI, gl) --t (M2, g2) be a global conformal
diffeomorphism of (Ml,gl) onto (M2,gz). Let y : [O,a) --t (MI, gl) be a
null geodesic of MI. If q = $to) is a null cut point of p = y(0) along the
9.2 THE NULL CUT LOCUS 309
geodesic y, then f (q) is a null cut point of f (p) along the null pregeodesic
f o r : I0,a) -+ (M2,92).
Proof. I t suffices to assume that q is the future null cut point of p along
y and that (M2,g2) is time oriented so that f o y is a future directed null
curve in M2. Let a : [0, b) + (M2,g2) be a reparametrization of f o y to a
future directed null geodesic guaranteed by Lemma 9.17 with f (p) = a(0).
Then f (q) = u(t1) for some tl E (0, b). Let d, denote the Lorentzian distance
function of (M,, g,) for i = 1,2.
We show first that d2(a(0), o(t)) = 0 for any t with 0 < t 5 t l . For suppose
d2(a(O),u(t)) # 0 for some t with 0 < t 5 t l . Then we may find a future
directed nonspacelike curve P in M2 from a(0) to a(t) with Lg,(P) > 0. Thus
f -' o p is a future directed nonspacelike curve in MI from p to f -l (a(t)) with
L,, (f -' o p) > 0. Since f (q) = cr(tl), we have f-l(u(t)) = y(t2) for some
t2 5 to. Hence dl(p, y(t2)) 2 L,, (f -' op) > 0, which contradicts the fact that
dl(p, y(t2)) = 0 since t2 5 to and y(t0) = q was the future null cut point to p
along y.
We show now that d2(u(O),a(t)) # 0 for any t > tl. This then makes
f (q) = u(t1) the future null cut point of f (p) along a as required. To this
end, fix t > tl. There is then a t2 > to so that f-'(a(t)) = y(t2). Since
y(t0) = q is the future null cut of p along y, we have dl (p, y(t2)) > 0. Hence
there is a future directed nonspacelike curve a from p to y(t2) with L,, (a) > 0.
Then f o a is a future directed nonspacelike curve from f (p) to u(t) Hence
d2(f(p),a(t))2Lg,(f oa)>Oasrequired. !J
With Proposition 9.18 in hand, we may now apply the null cut locus to study
null geodesic incompleteness. Recall that a geodesic is said to be incomplete if
it cannot be extended to all values of an affine parameter (cf. Definition 6.2).
Let R and Ric denote the curvature tensor and Ricci curvature tensor of
(M,g), respectively. Recall that an inextendible null geodesic y will satisfy
the generic condition if for some parameter value t, there exists a nonzero
tangent vector v E T,(t)M with g(v, yl(t)) = 0 such that R(v, yl(t))y'(t) is
nonzero and is not proportional to yl(t) (cf. Proposition 2.11, Section 2.5). In
particular, if Ric(v, v) > 0 for all null vectors v E TM, then every inextendible
310 9 T H E LORENTZIAN CUT LOCUS 9.3 T H E NONSPACELIKE CUT LOCUS 311
null geodesic of (M,g) satisfies the generic condition. In Section 2.5, it was (M, g) is globally conformally diffeomorphic to some portion of the subset M' noted that dim M 2 3 is necessary for the null generic condition to be satisfied. of the Einstein static universe, then d l null geodesics of (M, g) are incomplete. Thus we will assume that dim M 2 3 in the following proposition.
Since Minkowski space-time is free of null cut points, the above result re- Proposition 9.19. Let (M,g) be a space-time of dimension n > 3 such mains valid if we replace M' with Minkowski space-time.
that all inextendible null geodesics satisfy the generic condition and such that Friedmann space-times are used in cosmology as models of the universe. In Ric(v, v) 2 0 for all null vectors. If (M, g) is globally conformally diffeomorphic these spaces it is assumed that the universe is filled with a perfect fluid having to an open subset of a space-time (MI, g') which has no null cut points, then zero pressure. We will also assume that the cosmological constant A is zero for all null geodesics of (M, g) are incomplete. these models. These space-times are then Robertson-Walker spaces (cf. Sec-
Proof. Proposition 4.4.5 of Hawking and Ellis (1973, p. 101) shows that if tion 5.4) with p = A = 0. These space-times may be conformally embedded in
the Ricci curvature is nonnegative on all null vectors, then each complete null the subset M' of the Einstein static universe defined above [cf. Hawking and
geodesic which satisfies the generic condition has conjugate points (cf. Propo- Ellis (1973, pp. 139-141)). Furthermore, Ric(g)(v, v) > 0 for all nonspacelike
sition 12.17). Since maximal geodesics do not contain conjugate points, we vectors in a F'riedmann cosmological model (M, g). By Proposition 7.3, there
need only show that all null geodesics of (M, g) are maximal to establish their is a C2 neighborhood U(g) of g in C(M,g) such that Ric(gl)(v, v) > 0 for all
incompleteness. Assume that y is a future directed null geodesic from p to q gl E U(g) and all nonspacelike vectors v in (M, gl). Since R ~ C ( ~ I ) ( V , v) > 0
which is not maximal. Then there is a timelike curve from p to q. Since con- implies that the generic condition is satisfied by all null geodesics in (M, gl),
formal diffeomorphisms take null geodesics to null pregeodesics and timelike Corollary 9.20 yields the following result.
curves to timelike curves, the image of y must be a nonmaximal null geodesic Corollary 9.21. Let (M, g) be a Friedmann cosmological model. There in (MI, g'). This implies that (M', g') has a null cut point and hence yields a is a c2 neighborhood U(g) of g in C(M, g) such that every null geodesic in contradiction. (M, gl) is incomplete for all gl E U(g).
Recall that the four-dimensional Einstein static universe (Example 5.11) is
the cylinder x2+y2+~2+w2 = 1 embedded in R5 with the metric induced from 9.3 The Nonspacelike C u t Locus
the Minkowski metric ds2 = -dt2 +dx2 +dy2 + dz2 +dw2. The Einstein static Recall the following definitions. universe is thus B x S3 with a Lorentzian product metric. The geodesics and
null cut points are easy to determine in this space-time. The null cut locus of Definition 9.22. (Future and Past Nonspacelike Cut Loci) The future
the point (t, x, y, t, w) merely consists of the two points ( t i n , -x, -y, -2, -w) nonspacelike cut locus C+(P) of p is defined as C+(p) = C:(P) U C$(P). The
past nonspacelike cut locus is C-(p) = CL(p) U Cg(p), and the nonspacelike Consequently, the subset cut locus is C(p) = C-(p) U C+(p).
M ' = ( ( t , s , y , z , w ) : O < t < n , X ~ + ~ ~ + Z ~ + W ~ = ~ )
We mentioned in Chapter 8 that a complete noncompact Riemannian man- has the property that C N ( ~ ) n M' = 0 for all p € M'. Proposition 9.19 th ifold admits a geodesic ray issuing from each point. Thus at each point there has the following consequence. a direction such that the geodesic issuing from that point in that direction
Corollary 9.20. Let (M, g) be a space-time such that all null geod as no cut point. For strongly causal space-times, the Lorentzian analogue of
satisfy the generic condition and such that Ric(v, v ) >_ 0 for alI null vectors is property follows similarly from the existence of past and future directed
312 9 THE LORENTZIAN CUT LOCUS
nonspacelike geodesic rays issuing from each point. This may be restated as
the first half of the following proposition.
Proposition 9.23. (1) Let (M,g) be strongly causal. Then given any
point p E M, there is a future and a past directed nonspacelike tangent vector
in T,M such that the geodesics issuing from p in these directions have no cut
points.
(2) Let (M,g) be globally hyperbolic. Given any point p f M , p has no
farthest nonspacelike cut point.
Proof. As we have already remarked, part (1) follows immediately from
Theorem 8.10. For the proof of part (2), assume q E M is a farthest cut
point of p. Then q is a cut point along a maximal geodesic segment y from
p to q. Choose a sequence of points {q,) such that q << q, for each n and
q, --+ q. Since (M,g) is globally hyperbolic, there exist maximal timelike
geodesic segments c, : [O, d(p, q,)] + M from p to q, for each n. Extend each
c, to a future inextendible geodesic. Since q is a farthest cut point of p and
d(p, q,) 2 d(p, q) +d(q, q,) > d(p, q), for each n the geodesic ray c, contains no
cut point of p. The sequence {G) has a limit curve c which is a future &rected
and future inextendible nonspacelike curve starting at p by Proposition 3.31.
By passing to a subsequence if necessary, we may assume that {c,) converges
to c in the topology on curves (cf. Proposition 3.34). Using the global
hyperbolicity of (M,g) and the fact that q, -+ q, we find that q E c. If
r E c and r, E c, with r, -+ r , then d(p,r) = limd(p,r,). Using the upper
semicontinuity of arc length for strongly causal space-times, we find that the
length of c from p to r is at least as great as limsupd(p, r,) = limd(p, r,) =
d(p, r). Thus c is a maximal geodesic ray. Since q is a cut point of p on 7,
the geodesics c and y are &stinct maximal nonspacelike geodesics containing
p and q. Either Lemma 9.1 or Lemma 9.13 now yields a contradiction to the
maximality of c beyond q. 0
We now turn to the analogue of Klingenberg's observation that, for Rie
mannian manifolds, if q is a closest cut point to p and q is nonconjugate to p,
then there is a geodesic loop a t p passing through q. For Lorentzian manifolds,
however, a different result is true. If {q,) 5 Cz(p) converges to q E C$(p),
9.3 THE NONSPACELIKE CUT LOCUS 3 13
then d(p, q,) -4 0 in a globally hyperbolic space-time since the Lorentzian
distance function is continuous. Thus it is not unreasonable to expect that,
for globally hyperbolic space-times, there is no closest tirnelike cut point q to
p that is nonconjugate to p.
Theorem 9.24. Let (M,g) be a globally hyperbolic space-time, and as-
sume that p E M has a closest future (or past) nonspacelike cut point q. Then
q is either conjugate to p, or else q is a null cut point of p.
Proof. Let q be a future cut point of p which is a closest cut point of p
with respect to the Lorentzian distance d. Assume q is neither conjugate
to p nor a null cut point of p. Then p << q, and by Theorem 9.12 there
exist a t least two future directed maximal timelike geodesics cl and cz from
p to q. Let y : [O,a) -+ M be a past directed timelike curve starting at
q. By choosing a > 0 sufficiently small, we may assume the image of y lies
in the chronological future of p. Then p << y(t) << q for 0 < t < a implies
d(p, q) > d(p, y(t))+d(y(t), q) > d(p, y(t)) using the reverse triangle inequality.
Since q is a closest cut point, the point y(t) comes before a cut point of p on
any timelike geodesic from p to y(t). Thus any timelike geodesic from p to
y(t) is maximal. Since q is not conjugate to p along cl, there is a timelike
geodesic from p to y(t) near cl for all sufficiently small t. Similarly, there
exists a timelike geodesic from p to y(t) near cz for all small t. The existence
of two maximal timelike geodesics from p to y(t) implies y(t) is a cut point of
p and yields a contradiction since d(p, y(t)) < d(p, q). O
For Riemannian manifolds, the set of unit tangent vectors in any given tan-
gent space is compact. It is then an immediate consequence of the Riemannian
analogues of Propositions 9.5 and 9.7 above that the cut locus of any point in
a complete Riemannian manifold is a closed subset of M [cf. Gromoll, Klin-
genberg, and Meyer (1975, pp. 170-171), Kobayashi (1967, pp. 100-101)]. For
Lorentzian manifolds, the timelike cut locus is not a closed subset of M in
general as the Einstein static universe (Example 5.11) shows. However, for
globally hyperbolic space-times it may be shown that the nonspacelike cut
locus and the null cut locus of any point are closed subsets of M [cf. Beem and
Ehrlich ( 1 9 7 9 ~ ~ Proposition 6.5)]. It may be seen by Example 9.28 given below
314 9 THE LORENTZIAN CUT LOCUS 9.3 THE NONSPACELIKE CUT LOCUS 315
that the assumption of global hyperbolicity is necessary for the nonspacelike
cut locus to be a closed subset of M. An "intrinsic" proof for Lorentzian man-
ifolds is more complicated than the proof for Riemannian manifolds as a result
of the lack of unit null tangent vectors.
We now turn to the proof of the closure of the nonspacelike and null cut loci
for globally hyperbolic space-times. We first establish a technical lemma. Here
the assumption that expp(v) = q is essential to the validity of the conclusion
of this lemma.
Lemma 9.25. Let (M, g) be a strongly causal space-time. Assume p < q, and q, + q # p. If expp(vn) = q,, exp,(v) = q, and the directions of the
nonspacelike vectors v, converge to the direction of v, then u, + v.
Proof. Choose numbers a, > 0 such that the vectors w, = a,v, converge
to v. Then the sequence of points r, = expp(wn) must be defined for large n
and must converge to q. Since (M, g) is causal, there is only one value t , of t
such that expp(tnwn) = q,, namely t, = a;'. Using r, -+ q, q, -+ q, and the
strong causality of (M, g) near q, we find that t, -+ 1. Since v, = t,w,, the
sequence {u,) converges to v.
Fix a point p E M. A tangent vector v E T,M is a conjugate point to p
in T,M if (exp,), is singular at v. The conjugate points to p in T,M must
form a closed subset of the domain of exp, because (exp,), is nonsingular on
an open subset of T,M. A point q f M is conjugate to p along a geodesic c if
there is some conjugate point v E T,M such that exp, v = q and c is (up to
reparametrization) the geodesic expp(tv). If q f M , p < q, and q is conjugate
to p along a nonspacelike geodesic, then we will say q is a future nonspacelike
conjugate point of p. Past nonspacelike conjugate points are defined dually.
If (M,g) is a causal space-time, then p cannot be in its own future or past
nonspacelike conjugate locus because a nonspacelike geodesic passes through
p a t most once and (expp), is nonsingular at the origin of T,M.
Remark 9.26. It is possible even in globally hyperbolic space-times for a
point to be conjugate to itself along spacelike geodesics, however. This happens
in any Einstein static universe of dimension n 2 3.
Lemma 9.27. Let (M, g) be a globally hyperbolic space-time. Then the
future (respectively, past) nonspacelike conjugate locus in M is a closed subset
of M.
Proof. Assume that {q,) is a sequence of future nonspacelike conjugate
points of p with q, + q. Then p # q, p < q, and p < q, for each n. Let
v, E T,M be a nonspacelike vector such that (exp,), is singular at v, and
exp,(v,) = q,. Then c,(t) = exp,(tv,) is a future directed, future inextendible
geodesic starting a t p and containing q,. The geodesics c, must have a limit
curve y by Proposition 3.31. Since (M, g) is globally hyperbolic, y must contain
q. Using the strong causality of (M,g) and the fact that y is a limit curve
of nonspacelike geodesics, we find that y is itself a nonspacelike geodesic. Let
v be the unique (nonspacelike) vector tangent to y at y(0) = p such that
expp(v) = q. Since y is a limit curve of {c,), there must be some subsequence
{m) of {n) such that the directions of the vectors urn converge to the direction
of v. Lemma 9.25 then implies that v, -+ v. Since the vectors urn belong to
the total conjugate locus of p in T,M and this conjugate locus is a closed
subset of the domain of exp,, the vector v is a conjugate point of p in T,M.
Because v is a future directed nonspacelike vector, the point q = expp(v) is a
future nonspacelike conjugate point of p in M. This establishes the closure in
M of the future nonspacelike conjugate locus. The dual proof shows that the
past nonspacelike conjugate locus is also closed. Ci
At this juncture we stress that Lemma 9.27 reflects a specifically non-
Riemannian phenomenon. Contrary to well-established folk lore, the (first)
conjugate locus, even for a compact Riemannian manifold, need not be closed
in general [cf. Magerin (1993)l. In the space-time case we are rescued by the
non-imprisonment property that a nonspacelike geodesic must escape from any
compact subset of a globally hyperbolic space-time in finite affine parameter.
On the other hand, examples may be constructed of stably causal space-times
which do not have closed nonspacelike conjugate loci by deleting appropriate
subsets from the four-dimensional Einstein static universe.
Example 9.28. Let M = S1 x IR = ((9,t) : 0 5 0 5 2 ~ , t E R) with
the flat metric ds2 = do2 - dt2. This is the two-dimensional Einstein static
316 9 THE LORENTZIAN C U T LOCUS
universe which is globally hyperbolic. There are no conjugate points in this
space-time. If p = (0, O), the future null cut locus C$(p) consists of the single
point (T, n). The future timelike cut locus Cz(p) consists of {(n, t) : t > T} and is not closed. On the other hand, C+(p) = C$(p) U C$(p) is closed.
If we delete the two points (f n/4,277) from M, we obtain a strongly causal
space-time (M', glM,) which is not globally hyperbolic. In (MI, glM,) the
future nonspacelike cut locus C+(p) is not closed.
Proposition 9.29. Let (M,g) be a globally hyperbolic space-time. For
any p E M, the sets C$(p), Ci(p) , C+(p), and C-(p) are all closed subsets
of M. In particular, the null cut locus and the nonspacelike cut locus ofp are
each closed.
Proof. Since the four cases are similar, we will show only that C$(p) is
closed in M . To this end, let q, E C$(p) and q, -+ q. Since (M,g) is
globally hyperbolic, q # p and q E Jt(p). From the definition of C$(p) we
see that d(p, q,) = 0. Using the continuity of Lorentzian distance for globally
hyperbolic space-times (Lemma 4.5), it follows that d(p,q) = limd(p, q,) = 0.
Let 7 be any nonspacelike geodesic from p to q. Then d(p, q) = 0 implies that
y is a maximal null geodesic and that the cut point to p along y cannot come
before q.
There are two cases to consider. Either infinitely many points q, are future
nonspacelike conjugate to p or else no q, is future nonspacelike conjugate to p
for large n. In the first case we apply Lemma 9.27 and find that if q is future
nonspacelike conjugate to p, then if this conjugacy is along y, q is the cut point
along y because the cut point along y comes before or a t the &st conjugate
point along y. If q is a future nonspacelike conjugate point to p along some
other nonspacelike geodesic y', then y' must be null, and Lemma 9.13 shows
that q is the cut point along both y and y'.
Assume now that q, is not future nonspacelike conjugate to p for all suffi-
ciently large n. Theorem 9.15 implies that for large n there exist a t least two
maximal null geodesics from p to q,. Thus for large n there are two future
directed null vectors v, and w, with expp(vn) = exp,(w,) = q,. The future
directed nonspacelike directions a t p form a compact set of directions. Thus
9.3 THE NONSPACELIKE CUT LOCUS 317
the directions determined by {v,) and {w,) have limit directions v and w
respectively. If v and w determine different directions, we apply Lemma 9.14
to obtain two maximal null geodesics from p to q. In this case q is a cut point
of p. On the other hand, v and w may determine the same direction. If this is
so, we first note that there are constants a > 0 and b > 0 such that av = bw
and expp(av) = expp(bw) = q. Applying Lemma 9.25, it follows that for some
subsequence (m) of (n) we have urn -+ av and w, + av. However v, # w,
and expp(vrn) = expp(wrn) then yield that (exp,), must be singular at av.
Thus q is conjugate to p along y(t) = exp,(tav), and since d(p, q) = 0 we find
that q is the cut point of p along y(t). !I
Surprisingly, the above result holds without any assumptions about the
timelike or null geodesic completeness of (M,g). In particular, the null cut
locus and the nonspacelike cut locus in globally hyperbolic space-times are
closed by Proposition 9.29, even though the function s : T T I M -+ R U {co)
may fail to be upper semicontinuous.
It is well known [cf. Kobayashi (1967, pp. 100-101)] that using the cut locus,
it may, be shown that a compact Riemannian manifold is the disjoint union
of an open cell and a closed subset (the cut locus of a fixed p E M) which
is a continuous image (under expp) of an (n - 1)-sphere. Thus the cut loci
inherit many of the topological properties of the compact manifold itself. For
Lorentzian manifolds, cut points may be defined (using the Lorentzian distance
at least) only for nonspacelike geodesics. Hence a corresponding result for
space-times must describe the topology not of all of M itself, but only of J+(p)
for an arbitrary p E M. To obtain this decomposition with the methodology of
this section, we need to assume that (M, g) is timelike geodesically complete
as well as globally hyperbolic, so that the function s : T - I M --t R U {co)
defined in Definition 9.3 will be continuous and not just lower semicontinuous
(cf. Propositions 9.5 and 9.7).
Recall also that the future horismos E+(p) of any point p E M is given by
E+(p) = J+(p) - I+(p) and that C+(p) denotes the future nonspacelike cut
locus of p.
318 9 THE LORENTZIAN CUT LOCUS 9.4 THE NONSPACELIKE CUT LOCUS REVISITED 319
Proposition 9.30. Let (M, g) be globally hyperbolic and timelike geodesi- cut loci for globally hyperbolic spacetimes could not draw on the continuity
cdly complete. For each p E M the set J+(p) - [C+(P) U E+(P)] is an open properties of the s-function established in Proposition 9.7. Rather, we relied
cell. on a more indirect proof method which called upon the characterization of
PTOO~. Let B = J+(p)- [C+(p)uE+(p)]. Then q E B if and only if there is a nonspacelike cut points given in Theorems 9.12 and 9.15. In this section wc
maximal future directed timelike geodesic which starts at p and extends beyond will give a more elementary method to derive the closure of the nonspacelike
q. Thus B = I+(p) - C+(p) which shows that B is open. Now T-'MI, = {v E cut loci for globally hyperbolic spacetimes which treats both null and timelike
TpM : v is future directed and g(v,v) = -1) is homeomorphic to lRn-l. Let cut points simultaneously but is non-intrinsic. The treatment given in this
H : T-'MIP + IRn-' be a homeomorphism, and define 3 : Itn-' -+ R U {m) section has been worked out in discussions with G. Galloway.
by 3 = s o H-'. There is an induced homeomorphism of B with {(x,t) E We have noted in Section 9.3 just prior to Example 9.28 that Magerin (1993)
wn-1 x R : 0 < t < ~ ( x ) ) defined by q + (H(v), t), where v is the vector has given examples for Riemannian spaces that show that, contrary to folk lore,
in T-IM such that exp,(sv) is the unique (up to reparametrization) maximal the first conjugate locus of a point in a compact Riemannian manifold need not
geodesic from p to q, and expp(tv) = q with v f T-IM. Let f : [0, m] + [o, 11 be closed. Thus for completeness, we present an elementary proof that the cut
be a homeomorphism with f (0) = 0 and f (m) = 1. Then the map (x, t ) -4 locus of any point in a complete Riemannian manifold is closed, nonetheless.
(x, f ( t ) / f ( ~ ( 2 ) ) ) shows B is homeomorphic to Etn-' x (0, I), which establishes Theorem 9.31. Let (N, go) be a complete Riemannian manifold, and let
the proposition. p in N be arbitrary. Then the cut locus C(p) of p in N is a closed subset of
Since I+(p) J+(p ) , Proposition 9.30 yields some indirect topological in-
formation about I+(p). The Einstein static universe shows that I+(P) need not Proof. We have the basic fact that the distance to the cut point in unit be an open cell even if (M, g) is globally hyperbolic. However, (I+(p), glI+(p)) direction v, which we will denote as is customary by s : SN -, W u {rn) is globally hyperbolic whenever (M, g) is globally hyperbolic. Thus I+ (p) may corresponding to Definition 9.3 for unit timelike vectors, is continuous pro- be expressed topologically as a product I+(p) = s x W, where S is an (n - 1)- vided (N, go) is complete. Now let {q,) G C(p) be given with qn -+ q in M. dimensional manifold (cf. Theorem 3.17). Represent q, = expp(s(vn)vn) with llvnll = 1. Since the set of unit vectors in
9.4 T h e Nonspacelike C u t Locus Revisited TpM is compact, we may assume by passing to a subsequence if necessary that
un -, v f TpM. By continuity of the s-function, lims(vn)vn = s(v)v. Since
In Sections 9.2 and 9.3 we treated the null and timelike cut loci separate1 (N,go) is geodesically complete, exp,(s(v)v) is defined. Hence
because the intrinsic unit observer bundle T-1M was used as the prim technical tool for investigating the timelike cut locus. This approach s q = lim qn = lim exp,(s(vn)vn) = exp,(s(v)v).
the drawback, however, that if {q,) is a sequence of future timelike cut poi to p and q, = exp(s(vn)v,) converges to a null cut point q to p, then Therefore, q is a cut point to p.
(9.3) lim s(vn) = 0. Now Galloway suggested a non-intrinsic way to deal with the nonspacelike
cut locus in such a manner as to bring an analogue of (9.4) to bear. The As a result of the technical differences in handling the null and timelike wo facts that (i) future inextendible nonspacelike geodesics are not trapped
loci, the proof in Proposition 9.29 of the closure of the nonspacelike and n m compact sets in strongly causal spacetimes, and (ii) a modified s-function
I 320 9 THE LORENTZIAN C U T LOCUS
defined for all future causal directions to be constructed below is lower semi-
continuous for globally hyperbolic space-times suffice to ensure that the basic
ideas o f the above proof are valid in the space-time setting.
Fix throughout the rest o f this section an auxiliary complete Riemannian
metric h for the space-time ( M , g). Let
U M = {v E T M : h(v, v ) = 1 and v is future directed ).
Then we define the modified s-function on U M , which will be denoted by s l ,
as would be expected.
Definition 9.32. Define the function sl : U M + W U { m ) by
where c,(t) = exp(tv) as before.
Evidently, for timelike v in UM
and further it is not difficult to see that the semicontinuity proofs given for s
and T-lM apply equally well to sl and U M with minor modifications. Also,
i f ( M , g) is strongly causal, then s l (v ) > 0 for any v in U M .
Proposition 9.33.
(1) Let v E UM with s l (v ) > 0. Suppose either that s l ( v ) = +m, or
s l (v ) is finite and cv(t) extends to [O,sl(v)]. Then sl is upper semi-
continuous at v in U M . Especially, i f (M,g) is future nonspacelike
geodesically complete, then sl : U M -+ JR U { m ) is everywhere upper
semicontinuous.
(2) I f ( M , g) is globally hyperbolic, then sl : U M --t R U { m ) is lower
semicontinuous.
Before giving the elementary proof o f the closure o f the nonspacelike cut
locus for globally hyperbolic space-times, it is helpful t o note the following
technical result. I t is not required a priori in this result that either of a or b is
finite. Similar issues are addressed in a more general context and on a more
systematic basis in Lemma 11.20 and the proof o f Theorem 11.25.
9.4 THE NONSPACELIKE CUT LOCUS REVISITED 321
L e m m a 9.34. Let ( M , g ) be a globally hyperbolic space-time. Suppose
that c, : [0, b,] -+ ( M , g) is a sequence o f future nonspacelike maximal geodesic
segments with v, = cnl(0) in U M and c,(O) = p. Put q, = cn(bn), and
suppose that q, --t q in M , b, -+ b > 0 in W U { m ) , and v, -+ v in U M . Let
c : 10, a ) + ( M , g) denote the maximally extended nonspacelike limit geodesic
given by c(t) = exp,(tv). Then
(1) b < a, so that b is finite and positive,
(2 ) c(b) = q, and
(3 ) s1(v) 2 b.
Proof. ( 1 ) Suppose a 5 b. Fix any q' with q << q'. Then for any s with
0 < s < a, we have c,(s) defined for all n sufficiently large, and c,(s) E I-(9')
for all such large n. Since v, -+ v and cn(s) is defined for all sufficiently large n,
we have c(s) = limc,(s), whence c(s) E J-(9'). Thus the future inextendible,
nonspacelike geodesic c 1 10, a) is contained in the compact set J+(p) n J-(q') ,
which is impossible since ( M , g) is strongly causal. Thus b < a, and b = limb,
must be finite as well.
(2) Since b < a, we find that c(b) is defined and q = lim q, = lim c,(b,) = c(b)
by uniform convergence.
(3) Since the geodesic segments c, 1 10, b,] are assumed to be maximal and b =
limb,, the limit segment c I [0, b] must also be maximal, whence s l (v ) 2 b. O
T h e o r e m 9.35. Let (M,g) be globally hyperbolic. Then the future non-
spacelike cut locus and the future null cut locus o f any point p in M are closed
subsets o f M .
Proof. The proof may be given in a uniform fashion either for (q,) a se-
quence o f null cut points to p or a sequence o f nonspacelike cut points to p, so
we will not specify. Suppose qn -+ q in M . Choose v, E U M with
and put b, = sl(v,). Because {w E UM : w E T p M ) is compact, by taking
successive subsequences we may suppose that v, --t v in U M and that b, -+ b
in W U { m ) . Put c( t ) = expp(tv). By Proposition 9.33-(2), we have b 2 s l ( v ) ,
322 9 THE LORENTZIAN CUT LOCUS
so that b > 0. By Lemma 9.34, b is finite? c(b) = q, and s l ( v ) 2 b. Hence
s l (v) = b, and thus q is a cut point of the required type.
G. Galloway has also remarked that with non-intrinsic methods, the hy-
pothesis of "timelike geodesic completeness" may be removed from Proposition
9.30.
Proposit ion 9.38 (Galloway). Let (M, g) be globally hyperbohc. Then
for each p in M the set B = Ji(p) - [C+(p) U E+(p)] is an open cell.
Proof. -4s before, B = I + ( p ) - C+(p). Fix c > O sufficiently small that if
we set V = ( v E T M , h ( v , v ) = c, v future timelike), then C - expp(V) is
contained in B. Also, fix a diffeomorphism f : U + Rn-l. Define a projection
map r, : B i U by letting ?(m) denote the point of intersection of U with
the unique future timelike gcodcsic in M from p to m. Now choose a second
auxiliary complete Riemannian metric hl just for the open subset B of M.
considered as a manifoid in its own right. For each q in U, let "14 denote the
g-geodesic cexp;~(q) reparametrized as an hl-unit speed curve in B with
efq(0) = q and parametrized with the same orientation as the corresponding
g-geodesic. By Lemma 3.65. y, : W -+ 3 for each q in U. Now define t : B -+ R
by the requirement that
(t(m)) = m
Then we may define a homeomorphism F : B -+ R x Rn-' by F(m) =
. f ?
CHAPTER 10
MORSE INDEX THEORY ON EORENTZIAN MANIFOLDS
Given a nonspacelike geodesic y - [O. a ) -+ M In a causal space-tlme, Xe
have seen in Chapter 9 that if ?(to) is the future cut point to 7 ( 0 ) along 2 , then
for any t < t o the geodesic segment y I [O, t ] is the longest nonspacclike curve
from ~ ( 0 ) to y(t) In all of M . We could ask a much less stringent quest~on:
among all nonspacelike curves u from $0) to "((to) sufficiently "closc" to 7,
is L(y ) 2 L(u)? If so, y(to) comes at or before the first future conjugate
point of y(0) along y. The crucial difference here is between "in all of M"
for cut points and "close to y" for conjugate points. The importance of t h ~ s
distinction is illustrated by the fact that while no two-dimensional space-time
has any null conjugate points, all null geodesics in the two-dimensional Einstein
static universe have future and past null cut points.
Since only the behavior of "nearby" curves is considered in studying con-
jugate poirits, it is natural to apply similar techniques from the calculus of
variations to geodesics in arbitrary Rlernannian manifolds and to nnnspace-
like geodesics in arbitrary Lorentzian manifolds. To ~ndicate the flavor of the
Lorentzian index theory. we sketch the Ivlorse Index theory of an arbitrary
(not, necessarily complete) Riemannian manifold (iV, go). Let c : [a, b j -+ N be a fixed geodesic segment and consider a one-parameter family of curves a,
starting s t c(a) and ending at c(b). More precisely, let a : 'a, bl x ( - c , E ) --, AT
be a continuous, piecewise smooth map with C Y ( ~ . 0) = ~ ( t ) for all t E [a, b ] ,
and &(a, s ) = c(a), a(b, s) = c(b) for all s f (-t, t) Thus each neighbor~ng
clirve a, . t 4 cr,(t) = a(t, s ) is piecewlse smooth. The vanatzon vector field
(or s-parameter denvatzve) V of this deformation is given by
324 10 MORSE INDEX THEORY ON LORENTZIAN MANIFOLDS
Since c is a smooth geodesic, one calculates that
and the second variation works out to be
d2 -(b(ffs))1 ds2 S=O
b
bo(V1, V') - go(R(V, c1)c', V) - c'(go(V, cl))l I t d t =a
b + 90(VvV,c1) I,.
This second variation formula naturally suggests defining an zndez form
on the infinite-dimensional vector space V$ (c) of plecewise smooth vector fields
X along e orthogonal to e with X(a) = X(b) = 0 It IS then shown that
necessary and sufficient cond~tlon for e to be free of conjugate points is tha
I ( X , X ) > 0 for all nontr~vial X E V&(c).
T h ~ s suggests that for a geodeslc segment c : [a, b] -+ N wlth conjugat
points to t = a, the index Ind(c) of c with respect to I : Vg'(c) x V$(c) -+
should be defined as the supremum of dimensions of all vector subspaces o
V&(c) on which I is negative definite. Even though e ( c ) is an infinit
dimensional vector space, the Morse Index Theorem for arbitrary Riemannia
manifolds asserts that.
(1) Ind(c) is finlte; and
(2) Ind(ef equals the geodesic index of c, l.e., the number of conjugat
points along c counted wlth multlphattes.
More precisely, if we let Jt(c) denote the vector space of smooth vector fie1 c - 0 w i Y along c sat~sfying the Jacob1 differential equation Y" + R(Y, c') ' -
boundary conditions Y ( a ) = Y(t) = 0, then (2) is equivalent to the formula
(3) Ind(c) = dlm Jt (c).
10 MORSE INDEX THEORY ON LORENTZIAN RlANIFOLDS 325
Also, c has only finitely many points in (a, bj conjugate to c(a).
With the Morse Index Theorem in hand, the homotopy type of the loop
space for complete Riemannian manifolds may now be calculated geornetri-
cally [cf. Milnor (1963, p. 95)]. One obtains the result that if (N,go) is a
complete Riemannian manifold and p, q E N are any pair of points which are
not conjugate along any geodesic, then the loop space fl(p,,) of all contiriuous
paths from p to q equipped with the compact-open topology has the homotopy
type of a countable CW-complex which contains a cell of dimension X for each
geodesic from p to q of index A.
The purpose of Sections 10.1 and 10.3 is to prove the analogues of (1)
through (3) for nonspacelike geodesics in arbitrary space-times Let c : [a, b] -,
&I [respectively, P : [a.b] -+ M ] denote an arbitrary timelike [respectively,
null] geodeslc segment in ( M , g ) . Let V , (c ) [respectively, V$(P)] denote the
infinite-dimensional vector space of piecewise smooth vector fields Y along c
[respectively, P] perpendicular to c [respectively, P] with Y ( n ) = Y(b) = 0.
The timelike index form I : q ( c ) x Vo-(c) -+ R may be defined by
in analogy with the Riemannian index form. It may also be shown that c . [a. b] - A4 has no conjugate points in (a. b) if and only if I . *(c) x V$(c) i
R is negative definite.
Similarly, an index form I : V&(P) x @(P) + R may be defined by
r(X, Y) = - J ,~[~ (x ' , Y ' ) - g(R(X . P1)P1, Y)] dt. But since P is a null geo-
desic, g(P1, @') = 0. Consequently, vector fields of thc form V(t) = f (t)P1(t)
with f : [a. b] -+ R piecewise smooth and f ( a ) = f (b) = O are always
in the null space of I : V;(f?) x e(P) -- R yet never give rise to null
conjugate points. One way around this difficulty is to consider the quo-
tient bundle Xo(P) of V$(P) formed by identifying YI and Y2 in V$(/3) if
Yl - Y2 = fP/ for some piecewise smooth function f . [a. b ] --+ W. The in-
dex form I : Vo-(@) x Vo-(6) -+ R may also be projected to a quotient index
form 7 : Xo(Pj x Xo($) 4 R. It may then be shown that the null geo-
desic segment ,B : [a, b ] -+ M has no conjugate points in [a, b ] if and only if
326 10 MORSE INDEX THEORY ON LORENTZIAN MANIFOLDS
- I : Xo(P) x Xo(P) -+ ?It is negatlve definite [cf. Hawking and Ellis (i973. Sec-
tion 4.5), Bolts (1977, Chapters 2 and 4)j. In the case of a congruence of null
geodesics, the procedure of forming the quotient bundle has also been utilized
in general relativity since the 1960's. and the term "screen space" has been
ernployed [cf. Robinson and Trautman (1983) for a recent reference].
Let Jt (c) [respectively, Jt (P)] denote the space of Jacob1 fields along c [ r e
spectively, ,Dl with Y ( a ) = Y ( b ) = 0. Define the index Ind(c) of c [respecti.rrely,
Ind(B) of ,!?I to be the supremum of dinlensions of all vector subspaces of V$(c)
[respectively, Xo(B)] on which I [respectively, -I] is positive definite. We es-
tablish the Morse Index Theorem
Ind(c) = dim Jt (c) t E ( a , b )
and
for the timelike geodesic c : [a, b ] -+ M and the null geodesic ,5' . [a, b ] -+ n/f in
Sections 10.1 and 10.3, respectively. The reason for ronsidering the timelike
and null cases separately is that the quotient index form f : Xo(j?) x Xo(P) -+ R must be used to obtain the Null, but not the Timelike. Morse Index Theorem.
In Section 10.2 we study the theory of timelike loop spaces for globally
hyperbolic space-cimes, summarizing some results of Uhlenbeck (1975). Since
completeness is necessary to develop the Riemannian loop space theory. it is
not surprising that global hyperbolicity is needed for the Lorentzian theory.
Yet a significant difference occurs between the Lorentzian and Riemannian
loop spaces. Let (.&?,go) be a complete Riemannian manifold with positive
Ricci curvature bounded away from zero. Thus N is compact by Myers' The-
orem. It is shown in Milnor (1963, Theorem 19.6) that if p and q in such
a Riemannian manifold are nonconjugate along any geodesic: then the loop
space O(p,p) has the homotopy type of a CW-complex having only finitely
many cells in each dimension. But the loop space may still be an ~nfinzte
CW-complex as is seen by the example of Sn with the usual complete metric
of constan: sectional curvature. On the other hand. if (M,g) is an arbztrary
10.1 THE TIMELIKE MORSE INDEX THEORY 327
globally hyperbolic (hence noncompart) space-time, and p, q M are any two
points with p << q such that p and q are not conjugate along any nonspacelike
geodesic, then the timelike loop space C(,,,) has only f int felg many cells
This striking difference between the Lorentzian and Riemannian loop spaces
is a result of the following basic difference between Lorentzia~ and Riemannian
manifolds. If y is any nonspacelike curve from p to q ir an arbitrary space-
time and d ( p , q ) 1s finite. then y has bounded Lorentzian arc length since
L(y) 5 d ( p , q). In contrast. any curve a in the Riemannian path space saticifies
Lo(a) 2 do(p, q), hence has arc length bounded from below but not above
10.1 The Timelike Morse Index Theory
In this section we give a detailed proof of the Morse Index Theorem lor
timelike geodesics in an arbitrary space-time. Several different approacl~eb to
Morse index theory for timelike geodesics under various causality conditions
have been published by Uhlenbeck (1975), Woodhouse (1976), Everson and
Talbot (1976). and Beem and Ehrlich (1979~). Were we give a treatment which
parallels the proof of the Morse Index Theorem for an arbitrary bcmannian
manifold in Gromoll, Kllngenberg, and hfeyer (1975, Sections 4.5 and 4.6).
A sim~lar treatment of the results in this section through Theorem 10 22 has
been given in Bolts (1977).
Let (M, g) be an arbztrary space-time of dimens~on n 2 2 throughout this
section. We will denote by ( , ) the Lorentzian metric g IR this section. Also.
all timelike geodesics c : [a, b] 4 M will be assumed to have unit speed, 1.e..
(cl(t), c1(t)) = -1 for all t E [a, b ] .
Definition 10.1. (Paecewzse Smooth Vector Fzeld) A pzece.iiilse smooth
vector field Y along (the timelike geodesLC; c [a, b ] - Jf is a coniin~ioiis
map Y : [a.b] -* I M . with Y(t) E Tc(t+' for all t E [a. h ] , such that there
exists some finite partition a = to < t l < < t k m l < t k = h of [a, b j so that
Y/[,, ,,+,\ : [t,,t,+l] 4 TM is smooth for each z with 0 5 2 i, k - 1. Let VL(c)
denote the R-vector space of all piecewise smooth vector fields Y along c with
{Y(t). c l j t ) ) = 0 for all 0 < t 5 b. Also let fib'(c) = {Y t VL:c) Y(a\ =
Y(b) = 0 ) and Ar(c(t)) = ( u E T,(L)&f. (v ,c i ( t ) ) = 0).
328 10 MORSE INDEX THEORY ON LORENTZIAN MANIFOLDS
G~ven Y E V L ( c ) , there is a finite set of points I. (a , b) such that Y 1s
differentiable on [a, b] - lo. Define Y' : [a, b] + T M by Y 1 ( t ) = V,tY(t) for
t E [a, b] - I. and extend to points t , E lo by secting
Y 1 ( t i ) = lim Y 1 ( t ) t-t;
Thus Y' is extended to [a, bj by left continuity but is not necessarily continuous
at points of Io.
Remark 10.2. Since cl(t) is timelike, N(c( t ) ) is a vector space of dimension
n-1 consisting of spacelike tangent vectors, and thus (v E N(c(t)) : (v: w) 5 1)
is compact.
Definition 10.3. (Conjugate Point) Let c : [a, b] + M be a timelike
geodesic. Then t l , tz E {a, b ] with tl # t2 are conjugate with respect to c
if there is a nontrivial smooth vector field J along c with J ( t l ) = J ( tn ) =
0 satisfying the Jacobi equation J" + R(J,cl)c' = 0. Were J' denotes the
covariant derivative operator on vector fields along c induced by the Levi-
Civita connection of ( , ) on M. Also tl E (a , b) is said to be a conjugate
poznt of the geodesic segment c : [a, b] --+ M if a and tl are conjugate along
c. The geodesic segment c is said to have no conjugate points if no tl E (a: b ]
is conjugate to t = a along c. A smooth vector field J along c satisfying the
differential equation J1' + R(J, c1)c' = 0 1s called a Jacob% fieEd along c
We now define the Lorentzian index form I : V L ( c ) x V L ( c ) 4 R.
Definition 10.4. (Lorentzian Index Form) The indez form I : V'fc) x
V L ( c ) --t R is the symmetric bilinear form given by
for any X : Y E V L ( c )
If X E V L ( c ) is smooth, we also have
10.1 THE TIMELIKE LIORSE INDEX THEORY 329
Thus if Y E VoL(c) and X is smooth, one obtains the formula
5
(10.3) (X" + R ( X , cl)c', Yj d t
linking the index form to Jacobi fields.
Remark 10.5. Given a piecewise smooth vector field X E V L ( c ) . we may
choose a partition a = t o < t l < < tk-I < t k = b such that X / It,, is smooth for each i = 0, l . 2 , . . . , k - 1. Let
A,, ( X I ) = X1(a+),
A,, ( X I ) = -X1(b-), and
A , ( X I ) = lirn X 1 ( t ) - lim X 1 ( t ) t-t: t-t;
for i = 1,2, . . . , k - 1. It may then be seen by applying (10.2) on each subin-
terval (t,, that
This is the form of the second variation formula given in Hawking and Ellis
(1973. p. 108) and Bblts (1977. pp. 86-87) [cf. Cheeger and Ebin (1975, p. 21)]
In order to give geometric applications using the index form, it is useful to
make the following standard definition.
Definition 10.6. (Variatzon) Let c : [a, b] -+ M be a smooth curve. A
vanatzon (or homotopy) of c is a smooth mapping a [a, b ] x ( - E . E ) -+ M .
for some F > 0, with cr(t 0 ) = c ( t ) for all t E :a, b j The variation cu is said to
be a proper vanatton if cr(a.9) = c(o) and cr(b, s) = c(h) for all s E (-E, E ) A
continuous mapping a : [a, b] x ( - E . e ) -+ M is said to be a pteceui7se f m 00th.
va7.~atzon, of c if there exists a finite partition a = to < t i < . . . < tk -1 < t b = h
of [a. b ] such that cr I [t,, t,+l] x ( - E , E ) is a smooth variation of c / [t,, t,-l] for
eachz=0,1 ,2 ,..., k - 1 .
If we set a,(t) = cu(t.s) in Definition 10 6. then for a smooth variation a,
each curve a, : [a, b ] -+ M is a smooth curve and thus :he mapping s - cu,
330 10 MORSE INDEX THEORY ON LORENTZIAN MANIFOLDS
is a deformation of the curve c through the "neighboring curves' a,. If a
is a piecewise smooth variation, the neighboring curves a, will be 2iecewise
smooth. If a is a proper variation, all of the neighborlng curves a, begin at
c(a) and end at c(b). It is customary in defining variations of timelike curves
to ratr ict consideration to variations having the additional property that all
neighboring curves or, : [a, b] --, M are timelike. However, if we use Definition
10.6, this restriction is unnecessary by the following lemma.
Lemma 10.7. Let a : [a, b ] x ( - 6 , ~ ) -, A4 be a piecewise smooth variation
of the tirnelike geodesic segment c : [a, b ] -+ Al. Then there exists a constant
O > 0 depending on a such that the neighboring curves a, are timelike for all
s with Is] 5 6
Proof We first suppose that a is a smooth variation. Choose any el with
0 < €1 < E. Then a is differentiable on the compact set [a,b] x [-el,el] by
Definition 10.6. Hence by definition of differentiability, a extends to a smooth
mapping of a larger open set containing [a, b] x [-el, el]. Since c is a timelike
geodesic, the vectors cl(a+) and cl(b-) are timelike. It follows from this and
the extension of N to an open set containing [a, h ] x [-el, el] that there exists
a constant El > 0 such that the tangent vectors a,'(ai) and cuS1(b-) to the
xeighboring curves a, are timelike for all s with (s j < bl.
Suppose now that no 6 > 0 can be found such that all the curves a, are
timelike for Is] < 6. Then we could find a sequence s, --+ 0 such that the
curves a,,, failed to be timelike. Thus there would be t, E [a,b] so that
g (orin (t,), a:, (t,)) 2 0 for each n. Since [a, b ] x [-el. el] is compact, the
sequence ((t,, 5,)) has a poirit of accumulation ( t , s). Since s, - 0 , this
point must be of the for111 (t,O). and also the existence of 61 above shous
that t # a, b. But then as g (a:,, (t,), a,'n (t,)) 2 0 for each n, it follows that
g(cJ(t) , cl(t)) >_ 0, in contradic~ion to the fact that c was a timelike gecdesic
segment. Thus we have seen that ~f a : [a. b ] x (-E,E) 4 M is a smooth
variation of the timelike geodesic c : [a, b] 4 M , then there is a constant 5 > 0
10.1 THE TIMELIKE MORSE INDEX THEORY 331
such that a,'(af) and a,'jb-) are timelike vectors for ail 1s 5 6 and the curves
a , are timelike for all 1s 5 6.
Now let (Y : [a, b j x (-E.E) -+ M be a piecewise smooth variation of the
timeilke geodesic c : [a, b ] -+ lif. There is by Defin~tion 10 6 a finite partition
a = to < t l < ... < tk = b such that a 1 [t:, t,+( x ( - E , F ) 4 hf is a smooth
variation of c / :t,, t,+l]. By the above paragraph. we may find a constant 6, > 0
such that for all s with Is/ 5 E,, the tangent vectors as1(t:) and a,'(t;+,) are
timelike and a, I [t,, t,,~] 1s a timellke curve. Taking b = min(6~. b l , . . . , 6k-1)
shen yields the required 6.
Remark 10.8. There is no result corresponding to Lemma 10.7 for varia-
tions of null geodesics (cf. Definition 10.58 ff.).
The index form may now be related to variat~ons of timelike geodesic seg-
ments c : [a, b] -, ht as follows. Given Y E V ~ ( C ) , define the canonzcal proper
variation a : [a, b ] x (-6, E) -+ M of c by setting
(10.5) 4 t . s) = exp,(z)(sy(t)).
I t should first be noted that slnce c([a,G]) is a compact subset of M , and
the differential of the exponential map, exp,-. is nonsingular at the origin of
T,M for all p E M, it is possible given c([a, b] ) to find an E > 0 such that
exp,(,)(sY(t)) is defined for all s with Is/ 5 c and for each t E [a, b]. Secondly,
fro111 Definition 10.1 it follows that a( t , s) defined as in (10.5) 1s a piecewise
sn~ooth variation of c. Nence given Y t VoL(c), we know from Lemma 10.7
that there exists some constant 5 > 0 such that all the neighborlng curves
a, : t -+ a(t , s) are timelike for all s with -6 < s < 6. Given an arbitrary smooth variation a . [a, bj x ( -e , E ) -, M of c : [a, b ] -+
M . the van'atzon vector field V of a is defined 50 be the vector field V(t) along
c given by the formula
More precisely, lettlng a l a s be the coordinate vector fieid on [a, b ] x (-e, e)
corresponding to the s parameter, the variation vector field is given 'ny
332 10 MORSE INDEX THEORY ON LORENTZIAN MAXIFOLDS
For a piecewise smooth variation a : [a. b ] x (-E, E ) --+ M , one obtains a
continuous piecewise smooth varlatlon vector field as follows. Let a = to < t2 < < < tk = b be a partit~on of [a, b ] such that cu / It,. t,+l] x (-E,E)
IS smooth for z = 0,1,. . . , k - 1. Given t E [a, b], choose an index z such that
t, < t < t,+l, and set
~ ( t ) = [ a I it,, t,+ll x (-E, ~11 ,
The canonical variation (10.5) has the property that each curve s -+ a ( t , s) 1s
just the geodesic s -i exp,(,)(sY(t)) which has initial direct~on Y(t) at s = 0.
Hence the variation vector field of the canonical variation (10.5) is just the
glven vector field Y E V$ (c).
If we put L(s) = L(a,) = L(t -+ a ( t , s)), then L'(0) = y \ 8 = o = 0 since
c IS a smooth tlmelike geodesic, and
(10.8) L"(0)=2L(s) =I(Y,Y). d Z Lo
Thus if I(Y, Y) > 0 for some Y E Vg'(c), then the canonical proper variation
a ( t , s) defined by (10.5) using Y will produce timelike neighboring curves a,
joining c(a) to c(b) with L(a,) > L(c) for s sufficiently small. Thus if the
timdike geodesic c : ;a, b] -i M is maximal [i.e., L(c) = d(p,q)], then I . V$(c) x V&(c) --+ W must be negative semidefinite. Before proving the Morse
Index Theorem for tirnelike geodesics, we must establish the following more
precise relationship between corljugate points, Jacobi fields, and the index
form. First, the null space of the index form on V$ (c) consists of the smooth
Jacobi fields in VO-(c), and second, c has no conjugate points on [a, b] if and
only if the index form is negative definite on V&(c).
We first derive an elementary but important consequence of the Jwobi
differential equation.
Lemma 10.9. Let c : [a, b ] --+ M be a timelike geodesic segment an
Y be any Jacob] field along c. Then (Y(t), cl(t)) 1s an affine function of
(Y ( t ) , cl(t)) = a t + /3 for some constants a, p E R.
Proof. First (Y, c')' = (Y', c'} + (Y, c") = (Y', c'), as c" = V,lcr =
along the geodesic c. Differentiating again, we obta~n (Y, c')" = (Y", d )
10.1 THE TIMELIKE MORSE INDEX THEORY 333
(Y', c"} = (Y", c') = -(R(Y, cl)c', c') = 0 by the skew symmetry of the
Riemann-Ghristoffei tensor Thus (Y: c') is an affine function.
Corollary 10.10. If Y is any Jacob1 field along the timelike geodesic
c : [a,b; --+ IM and Y ( t ~ j = Y(tz) = 0 for distinct t ~ . t n E [a,bj, then
Y E VL(c).
Corollary 10.11. If Y E @ ( c ) is a Jacobi field; then Yf E VL(c).
Using the canonical variation, we are now ready to derive the following
geometric consequence of the existence of a conjugate point to E (a, b) to t = a
along c.
Proposition 10.12. Suppose that the timel~ke geodes~c c : [a. b ] -+ M
contains a conjugate point to E (a, b) to t = a along c Then there exists a
piecewise smooth proper variation a, ofc such that L(a,) > L(c) for all s # 0.
Thus c : [a, b] --+ M is not maximal.
Proof. In view of (10.5), (10.8), and Lemma 10.7, it is enough to construct
a piecewise smooth vector field Y E V$(c) with I(Y, Y) > 0 and let a be
the canonical variation associated with Y. To this end, let Yl be a nontrivial
Jacobi field along c with YI (a) = Yl (to) = 0. By Corollary 10.10, Yl E VL(c).
Hence as Yl(a) = &(to) = 0, we have YI' E VL(c) by Corollary 10.11. Since
Yl(to) = 0 and Yl is a nontrivial Jacobi field. it follows that YI1(to) is a
(nonzero) spacelike tangent vector.
Let I( , ): denote the restriction of the index form to cj [a, s ] , that IS.
Then since Yl is a Jacobi field, we have from (10.2) of Definition 10.4 that
r any Z E Vi(c).
We are now ready to construct a piecewise smooth vector iield Y 6 V$(c)
ith IfY, Y) > 0. Let $J : [a, b] --, IR be a smooth function with @(a) = @(b) =
0 and +(to) = 1. Also let Z1 be the uniqae smooth parallel vector Seld along
334 10 MORSE INDEX THEORY ON LORENTZIAN MANIFOLDS
c with &(to) = -fil(to). Then Z = ~ Z 1 E V ~ ( C ) . Define a oneparamete1
family Y, E @(c) by
YI(~) + €Z(t) i f a s t t o , K(t) =
t:Z(t) if to < t < b.
Then using (10.9) we obtain
I ( K , K ) = I(Y,, Y,)2 + I(Y,, yE)fo
= I(Y1 + €2, Yl + E Z ) ~ + I(&, EZ):~
= I(Y1, fi)? + 2tI(Yl, Z)? + C~I(Z,Z)? + t21(2, z) i0
= - (Yll, YI)I: - 26 ( ~ 1 ' . z)(: + E~I(Z, 2).
Since Yl(a) = Yl(to) = 0, this simplifies to
I ( X . Y,) = -2€(Y11(to), Z(t0)) + €21(2. 2)
= 2E 11 Yjl(fJo) / I 2 E'I(Z, 2).
As Yll(to) is a (nonzero) spacelike tangent vector and I (Z , Z) is finite, it follows
that there is some t: > 0 such that I ( K , K ) > 0. Put the required vector field
Y = YE for this value of E . @
We now turn to an important characterization of Jacobi fields in terms of
the index form. The same characterization holds for Riemannian spaces with
an identical proof. It is important to note that the index form characterizes
smooth Jacob1 fields among all pzecewzse smooth vector fields in VoL(c) and
not just among all smooth vector fields in Vg'(c).
Proposition 18.13. Let c : [a, b ] -+ ( M , 9) be a timclike geodesic segment.
Then for Y E Vo-(c), the following me equivalent:
(1) Y is a (srnootl-i) Jacobi field along c.
(2) I (Y, Z) = 0 for all Z E V$ (c) .
Proof. First (1) + (2) is immediate, since for smooth vector fields Y and
arbitrary Z the index form may be written as
I (Y.Z)=-(Y1,z) / :+ (Y1l+R(Y,e')e',Z)dt. i"
10.1 THE TIMELIKE MORSE INDEX THEORY 335
If Y is a Jacobi field. I(Y, Z) = - (Y', 2) 1:. Thus l(Y, Z) vanishes for all
z E VoL(c).
To show (2) =+ (1). we first note that since c is a tirnelike geodesic segment
and Y E v ( c ) , we have (Y', c') = 0 and (Y" + R(Y, c')cl, cl) = 0 at all points
where Y is differentiable. Taklng left-hand limits. we have (Y1(t,), cl(t,)) = 0
also at the finitely many polnts of d~scontinuity a = to < tl < , . < tk = b of
Y. By contmuity, the right-hand limit lim,+,;(Y1(t),c'(t)) = 0 as well. Hence
the vectors Ot*(Y1) defined in Remark 10.5 also satisfy (At,(Y').cl(t,)) = 0
By (10.4) of Remark 10.5, the index form may be calculated as
k b
(10.10) I(Y. Z) = X ( O t , (Y'), Z(t,)) + (Y" + R(Y, cl)c', 2 ) dt ,=o
Let p : [a, b] -+ [0, 11 be a smooth function with d( to) = +(tl) = . . . = Q(tk) =
0 and #(t) > 0 elsewhere. Then the vector field Z1 = d(Y" + R(Y, c')cl) is In
V$(c) and Zl(t,) = 0 for all 2. Since it 1s assumed that I(Y. 2) = 0 for all
Z E V&(c), we obtain from (10.10) that
As 21 is a spacelike vector field, smooth except at the t,'s, and ~ ( f ) > 0 if
t 4 ita) . we obtain Y1'(t) + R(Y(t). cl(t))c'(t) = 0 if t d it,). Thus Y is a
piecewise Jacobi field and formula (10.10) reduces to
Recalling from above that (At,(Y').cl(t.)) = 0 for each 2, a vector field Z2 E
V$(c) may be constructed with Z2(t,) = At, (Y') for 1. = 1,2. . . , k - 1. Then
we have k-1
0 = I(Y, 22) = /lAl%(Y1) l 2 2=1
Since all of the tangent ~ectors in this sum are spacelike. it follows that
At,(Y1) = 0 for z = 1,2, . . . , k - 1. This :hen implles that Y1 has no breaks
at the t,'s. Since for any t E [a, bj there IS a .lniqu3 Jacobi field along c
336 10 MORSE INDEX THEORY Oh' LORENTZIAN MANIFOLDS
with Y(t) = v and Y 1 ( t ) = tour? it follows that the Jacobi fields Y / [ti, ti+,] fit
together to form a smooth Jacobi field.
In view of Propositions 10.12 and 10.13, it should come as no surprise that
the negative definiteness of the Lorentzian index form should be relzted to the
absence of conjugate points just as the positive definiteness of the Riemannian
index form is guaranteed by the nonexistence of conjugate points [Gromoll,
Klingenberg, and Meyer (1975, p. 145)j. The negative semidefiniteness of the
index form in the absence of conjugate points has been given in Hawking
and Ellis (1973, Lemma 4.5.8). It has been noted in Bolts (1977. Satz 4.4.5)
and Beem and Ehrlich (1979c, p. 376) that the negative definiteness of the
index form in the absence of conjugate points follows "algebraically" from the
semidefiniteness just a s in the proof of positive definiteness for the Eemannian
index form. In order to give a proof of the negative semidefiniteness of the
Lorentzian index form in the absence of conjugate points, we need to obtain
the Lorentzian analogues of several important results in Riemannian geometry
[cf. Gromoll, Klingenberg, and Meyer (1975, pp. 132, 136-137, 140)].
For the purpose of constructing Jacobi fields, it is useful to introduce some
notation for the identification of the tangent space T,(TpM) with TpM its
by "parallel translation in T,M."
Definition 10.14. (Tangent Space T,(T,M)) Given any p E M an
v E TpM, the tangent space T,(T,M) to the tangent space T'M at v is give
by
T,(T,M) = (4, : R -+ T,M)
where
&(t) = v i r t u .
Then T,(TpM) may intuitively be identified with T,M by identifying th
image of 9, in TpM with the vector w. More formally, let 1 be given th
usual manifold coordinate chart x ( r ) = r for all r E W, i.e., s = id. The
B/dz is a vector field on R. Since 4, : R -. TpM and &(0) = v, we hav
+,, : TOR -+ T, (T,hf). We may then make the following definition.
10.1 THE TIMELIKE MORSE INDEX THEORY 337
Definition 10.15. (Canonical Isomorphzsm) The canonzcal isomo7;iihssm
r, : TpM -+ T, (T,M) is given by
where, as in Definition 10.14,
In particular, let v = Q,, the zero vector in the tangent space T,M. Then
4, : R -+ TpM is the curve &(t) = tu: in Top(TpiZif). and we find that
ro,(w) = 4,. Thtis To,(T,M) is often canonically identified with T,M itself
by identifying the vector w E TpM and the map 4,. If p E M and v E T p M ,
then since exp, : T'M -t M . the definition of the differentlal gives
In particular, for v = Op we have
exp,, : To, (TpM) --t T,IM
since expp(O,) = p. If E = ~ o p (v) E Top (TpM) and we define Q : R - T, bf hy
q(t) = tu, then exp,,(b) = exp,.(q5, d/dzlo), where a/& is the usual bask
bector field for TR defined above. Thus we obtain
hus exp, orv = I ~ T , M . This fact 1s commonly stated as "the differential .%'
the exponential map at the origin of TpJVi 1s the identity."
The following proposition shows how the differential of the exponeritial map
ay be used to construct Jacobi fields.
338 10 MORSE INDEX THEORY ON LORENTZIAN MANIFOLDS
Proposit ion 10.16. Let c : [ O , b] -, M be a geodesic with c(0) = p. Let
w E T,M be arbitrary. Then the unique Jacobi field J aiong c with J(O) = 0
and J1(0) = w is given by
J(t) = ex?,- ( t ~ i ~ ~ ( ~ ) ~ ) .
Prooj. Set v = cl(0). We may find an E > 0 so that the smooth variati
a : [a, b ] x (-E, E ) + M of c : [a, b] -+ M given by a(t, s) = exp,(t(v + sw
is defined. Since this is a variation of c whose s-parameter curves t + a(t, s
are geodesics, it follows that the variation vector field of this deformation i
a Jacobi field. Since a, D/dsl(,,,) = expp* (rt(v+sw) (tw)), the variation vecto
field is just J ( t ) = ex?,. (rt,tw) = exp,. (t ~ ~ , w ) . Since a(0, s) = c(O) for a1
s, we have J(0) = 0, and a calculation also gives J1(0) = w Icf. Gromoll
Klingenberg, and Meyer (1975, p. 132)].
As in the Riemannian theory, it is now possible to prove the Gauss Eemm
using Proposition 10.16. We first need to put a natural inner product ( ( , ) on T,(T'M) using the given Lorentzian metric ( , ) on TpM and the canonic
isomorphism.
Definition 10.17. (Inner Product on T,(T,M)) The inner product
(( , )) : Tv(Xp~v) x Tv(TpAd) -+ R
associated with the Lorentzian metric ( , ) for M is given by
((a, b ) ) = (7,'(a),.r,-l(b))
for any a, b E T,(TpM).
Theorem 10.18 (Gauss Lemma). Let v E T'M be a t ange~~t vect
in the domain of definition of the exponential mapping and let a = T,(%)
T,(T,M), Then for any b E T,(TpM), we have
(10.12 ) ((a, b:: = (exp,. a , exp,. b),
Thus the exponential map is a "addial isometry."
10.1 THE TIMELIKE MORSE INDEX THEOW 339
Proof. If d(t) = tz, then a = dl(l) . If c is the geodesic c(t) = exp,(tv) =
exppc+(t), we then have expp.a = cl(l) . Also set w = rV-l(b) E T,M. Let
Y be the unique Jacobi field along c with Y(0) = 0 and Y1(0) = w. By Proposition 10.16, we know that Y ( t ) = exppw ( t ~ ~ , w ) . 1x1 particular, Y ( l j =
expp, (rvw) = expp. (b).
From Definition 10.17, we have ((a,b)) = (.r;'(a),r;'(b)) = ( v , ~ ) . Hence
the Gauss Lemma IS proved if we show that (z., w) = (cl(l), Y(1)). But by
Lemma 10.9, the function f (t) = (cl(t),Y(t)) = cyt + /3 for some constants
a , P E R. Since Y(0) = 0, we have O = O and j ( t ) = t f l ( 0 ) = t(cf(0), Y1(0)) =
t(v, w). In particular, ( c l ( l ) , Y(1)) = f (1) = (L, w) % required. D
The Gauss Lemma has the following geometric consequences. The proofs of
these corollaries, which are given along the lines of Gromoll. Klingenberg, and
Meyer (1975, pp. 137-138) in Bolts (1977, pp 75-77), will be omitted. The
use of the Gauss Lemma here replaces the use of a synchronous coord~nate
system in Penrose (1972, p. 53).
Corollary 10.19. Let U be a convex normal neighborhood in M, and let
c : [O, lj -, U be a future directed timel~ke geodesjc segment from p = c(O) to
q = c(1) in U. Then if P : [0, 11 - U is any future directed timelike piecewise
srnooth curve from p to q, we have L(P) 5 L(c), and L(O) < L(c) unless is
ju s t a reparametrization of c.
The basic idea of the proof is that since lJ is convex, B and c may be lifted
to rays : [0,1] + TpM and F: [O, 11 -, T,M. with F(t) = tcl(0) Then the
Gauss Lemma may be appl~ed to compare P I = expp* OF and d = exp,. o Z and hence the lengths of P and the geodesic c.
An alternative formulation of Corollary 10.19 is also given in Bolts (1977.
pp. 75-77) as follows.
Corollary 10.20. Let v E T,M be a timelike tangent vector In the domain
of definition of exp,. Let : [0,1] + T,M be the curve +(t) = tv. Let + : ;0, lj + TpM be a piecewise smooth curve with d(0) = q(0) and +(1) = d(1)
such that exp, 3w : j0,11 --$ M is a future directed nonspaceljke curve Then
340 10 MORSE INDEX THEORY ON LORENTZIAN MANIFOLDS
L(exp o+) 5 L(exp 04); and moreover,
provided that there is a to E (O,1] such that the component b of +'(to) per-
pendicular to ~+(~,)($(to)/ll$(t~)/l) satisfies expp.b # 0.
We are now ready to show that if the timelike geodesic segment c has
no conjugate points (recall Definition 10.3), then the length of c is a local
maximum.
Proposit ion 10.21. Let c : [a, b] 4 M be a future directed timelike geo-
desic segment with no conjugate points to t = a and let a : [a, b] x (-E, 6) - M
be any proper piecewise smooth variation of c. Then there exists a constant
6 > 0 such that the neighboring curves a, : [a, b] -+ M given by cr,(t) = a(t , s)
satisfy L(cr,) 5 L(c) for ail s with Is/ < 6. Also L(a,) < L(c) ~f 0 < 1st < 6
unless the curve a, is a reparmetrizat~on of c.
Prooj. Reparametrize cr to a variation n : [O.P] x ( - c , ~ ) M . By
Lemma 10.7, there is an €1 > 0 such that all the neigllboring curves a,
of a 1 [0,/3] x (-tl,el) are tirnelike. We may then restrict our attention to
I [O,P] x ( - ~ 1 , ~ 1 ) .
Set p = c(0) and let 4 : [O, $ 1 4 TpM be the ray +(t) = tcl(0). Since
c has no conjugate points, expp has maximal rank at p(t) E TpM for each
t E [O, p ] . Thus by the inverse function theorem, there is a neighborhood of
+(t) in TpM which is mapped by exp, diffeomorphically onto a neighborhood
of c(t) in M. Since +([O, PI) is compact in TpM, we can find a finite partition
0 = to < t l < - - . < t k = P and a neighborhood U, > 6([t3, t,+l]) in T,M for
each J = 0,1,. . . , k - 1 such that h, = exp I : U, 4 M is a diffeomorphism p u3
of U3 onto its image. By continuity, we may then find a constant 6, > 0
such that a([t,, t,+l] x (-6,, 6,)) exp,(U,) for each j = 0,1, . . . : li - 1. Set
6 = min(61,62,. . . ,6k).
Then we may define a pierewise smooth map Q : [0,/3] x (-6.6) -+ T,M
with exgp o@ = N and @(t. 0) = d(t) for all t E [O. 41 as follows. Given (t, s) E
IO,@] x [-6,6]. choose j with t E [t,,t,+l], and set @(t, s) = (h,)-"a(t, s)).
10.1 THE TIMELIKE MORSE INDEX THEORY 341
Corollary 10.20 then implies that L(a,) = L(expp oQ) 5 L(exp, od) = L(c)
for each s with Is1 5 6, and equality holds only if a, is a reparametrization of
c. C
With Proposition 10.21 in hand, we may now show that the negative defi-
niteness of the index form on V$(c) is equivalent to the assumption that c has
no points conjugate to t = a along c in [a, b] .
Theorem 10.22. For a given future directed tirnelike geodesic segment
c : la, b ] M , the following are equiva!ent:
(1) The geodesic segment c has no conjugate points to t = a in (a, b ] .
(2) The index form I : V/(c) x V$(c) -, R is negative definite.
Proof. ( 1 ) 3 (2) Suppose Y # 0 in V$(c) and I(Y. Y) > 0. Let cr(t. s) =
exp,(,)(sY(t)) be the canonical variation associated to Y . Then we have
L1(0) = 0, L1'(0) = I ( Y . Y ) > 0, so that for all s # 0 sufficiently small,
L(cY,) > L(c). Bu& this then contradicts Proposition 10.21. Hence if Y # 0,
then I (Y. Y) 5 0, so that the index form 1s negative semidefinlte.
It remains to show that if Y E V&(c) and I(Y, Y) = 0. then Y = 0. To this
end, let Z E V$(c) be arbitrary. By Remark 10.2, we have Y - t Z E V$(c)
for all t E R. Hence I(Y - tZ, Y - t Z ) 5 0 for all t E W by the negative
semidefiniteness of the index form just established. Since I(Y - tZ, Y - tZ) =
-2tI(Y, Z) + t21(Z, Z), it follows that I(Y, Z ) = 0. As Z was arbitrary. this
then implies by Proposition 10.13-(2) that Y is a Jacobi field. Since c has no
conjugate points, Y = 0.
(2) + (1) Suppose Y is a Jacobi field with Y (a ) = Y(tl) = 0, a < tl 5 b. By
Corollary 10.10, Y E VL(c). Extend V to a nontrivial vector field Z E V$(c)
by setting
Then I (Z , 2) = I(Z. 2): + I(Z, Z)& = - (Z', z)/: + 0 = 0.
A consequence of Theorem 10.22 that is crucial to the proof of the Timelike
Morse Index Theorem is the following maximality pro?erty of Jacobi fields with
respect to the index form for timelike geodesic segments without conjugate
342 10 MORSE INDEX THEORY ON LORENTZIAN MANIFOLDS
points. This result is dual to the minimality of Jacobi fieids with respect to the
index form for geodesics without conjugate points in Riemannian manifolds.
Theorem 10.23 (Maximality of Jacobi Fields). Let c : [a, b] + M
be a timelike geodesic segment with no conjugate points to t = a, and let
J E VL(c) be any Jacobi field. Then for any vector field Y E VL(c) with
Y # J , and
(10.13) Y(u) = J(a) and Y(b) = J(b),
we have
Proof. The vector field W = J - Y E V&(c) by (10.13) and W # 0 as
Y # J by hypothesis. By Theorem 10.22. we thus have I(W, W) < 0. Now
calculating i(W, W) we obtain
I(W, W) = I(Y, Y) - 2I(J, Y) + I (J , J )
= I(Y,Y) + 2 (J', Y)/! - (J', J)J;.
Since Y(a) = J (a ) and Y(b) = J(b), we have (J',Y)I: = (J'. J)/:. Thus
I(W,W) = i (Y ,Y ) + 2 { J 1 , ~ ) l b , - (J', J)/:
= I(Y,Y) + (J', J)I: = I(Y, Y) - I ( J , J ) .
As i(W, W) < 0, this establishes (10.14).
Now that Theorem 10.23 is obtained, a Morse Index Theorem may be estab-
lished. First we must define the ~ndex of any timelike geodeslc c : [a, b] --+ M.
The definition given makm sense because Vo-(c) is a vector space.
Definition 10.24. (Indez and Extended Indez) The index Ind(cf and
the extended zndez Indo(c) of the timelike geodesic c : [a, b ] -+ M are defined
by
Ind(c) = lub(dim A : A is a vector subspace of ~ & ( c )
and Il A x A is positive definite),
10.1 THE TIMELIKE MORSE INDEX THEORY
Indo(c) = iuWdii11A . A is a vector subspace of v$(c)
and I / .A x A is posltlve sem~definite).
Also let Jt(c) denote the W-vector space of sclooth Jacobi fields Y along c with
Y(a) = Y( t ) = 0. a < t 5 b.
Mre now relate Ind(c) to Indo(c) and establish their finiteness in Proposition
10.25. The maximality of Jacobi fields with respect to the index form for
timelike geodesics without conjugate points plays a key role in the proof of
this proposition. The basic idcas involved in the proof of Proposition 10.25
and Theorem 10.27 are due to Marston Morse (1934)
Proposition 10.25. Let c : [a, b ] -+ M be a future directed timellke geo-
desic segment. Then Ind(c) and Indo(c) are finite and
(10.15) Indo(c) = Ind(c) + dim Jb(c).
Proof. The proof follows the usual method of dpproxirnating V$(c) by
finite-dimensional ~ec to r spaces of piecewire srriooth Jacob! fields. To this
end, dioose a finite partition a = to < t l < . . . < t k = b so that c 1 [t,, t,+l]
has no conjugate poirlts to t = t , for each z with 0 5 z 5 k - 1. Let J ( t , ) denote
the subspace of Vj'(c) consisting of all Y E V&(c) suck1 that Y / [t,,t,+l] is a
Jacobi field for each z with 0 5 z < k - 1. Since c / [t,, t,+l] has no conjugate
points for each 2, it follows that dim J{t,) = (n - l ) (k - 1).
Mre now define the approximation 4 : ~ & ( c ) 4 J{tz) of V,'-(c) by J { t , ) as
follows. For X E V&(c) let (pX) I [t,,t,+l] be the unique Jacobi field along
c / [tl,tB+l] with (q5X)(tE) = X(tb) and (4X)(t,+l) = X(t,,l) for each z with
0 ( i 5 k - 1. Thus X is approximated by a piecewise smooth Jacob] field
OX such that (dX)(tt) = X(t t ) at each t,, 0 < 5 k This approximat~on is
useful in t h ~ s context a s 9 is iqdex nondecr~asing KIore explicitly, Q 1 . J { t , )
is just the identity map, so that I ( X , X) = I(4X. q X ) ~f X E J{t,). On the
other hand, if X $ J j t , ) then the inequality
344 10 MORSE INDEX THEORY ON LORENTZIAX MANIFOLDS
may be obtained by applying Theorem 10.23 to each subinterval It,, t,+l] and
summing.
We estabiish the following sublemma which shows the finiteness of Inda(c)
and Ind(c) and also allows us to replace Vo-(c) by J{ti) in calculating these
indexes.
Srrblemrna 10.26. Let Indb(c) and Indl(c) denote tbe extended index and
index, respectively, of I1 J( t i ) x J(ti). Then
Indo(c) = Indb(c) and Ind(c) = Indl(c).
Hence Indo(c) and Ind(c) are finite.
Proof. First it is easily seen by the uniqueness of the Jacobi field J along
c ] [t,,t,+l] with J(t,) = v and J(t,+l) = w that ihe map 4 . VOL(c) -+ J{t,)
is R-linear. That is, if XI , X2 E V$(C) and a,P E R, then @(ax1 + PX2) =
aQ(Xl) + P+(Xa). Thus + maps a vector subspace of Vd(c) to a vector
subspace of J{t,).
In order to establish the sublemma, we first show that if A is any subspace
of q ( c ) on which I I A x A is positive semidefinite, then & I A : -4 + J(t,) is
injective. Thus suppose X E A and &(X) = 0. If X E J(t,), then + ( X ) = X,
so that X = 0. If X 4 J(t,), then I(+(X),p(X)) > I (X ,X) by (10.16).
Thus $X = 0 implies that I (X, X) < 0 which contradicts the assumption that
I ] A x A is positive semidefinite Thus if +X = 0: then X = 0.
Now that we know that c$ is injective on subspaces of V$(C) on which I is
positive semidefinite, we may prove the sublemma. For if A is a subspace of
V/(c) on which 11 A x A is positive semidefinite, inequality (10.16) implies
that the index form of J ( t , ) is positive semidefinite on the subspace d(A) of
J{t,). Since q5 is injective on such a subspace from above, dim A = dim $(A) .
Hence Indb(c) 2 Indo(c). However, since J( t , ) is a vector subspace of q ( c ) ,
we have Ind&(c) 5 Indo(c). Thus Indo(c) = Indb(c). The same argument
shows Ind(c) = Ind(cl). 3
To conclude the proof of Proposition 10.25, we must establish the equal-
ity Indo (c) = Ind(c) + dim Jb(c). To this end, we choose a second partition
10.1 THE TIMELIKE MORSE INDEX THEORY 345
a = so < s l < . . . < s,, = b so that {sl,sa, .. .,sm-3} n (t1,t2,. . . . t r - l ) = Q and c / [s,, s,+lj has no conjugate points for each i with 0 5 z < m - 1. Lct
J{s ,) denote the vector subspace of V ~ ( C ) consisting of all vector fields Y such
that Y 1 [s:, sz4 11 is a Jacobi fieid for each z with 0 < i < m - 1. Since the two
partitions (3,) and {t,) are distinct except for a = so = to and b = s, = tk,
it fallows that
J(tJ n JGE) = ~ ~ ( 4
where .Jb(c) denotes the vector space of all (smooth) Jacobi fields J along c
with J (a) = J ( b ) = 0. By Corollary 10.10, we have Jb(c) V$(c). Also, if
X E J(s,) but X $4 Jb(c), we have by (10.16) that
Applying the proof of Sublemma 10.26 to the partition {s,) of [a! b ] , we may
choose a vector subspace BA of J{s,) with Il Bh x BA positive semidefinite
and with Indo(c) = dim B;. Since dim BA = Indo(c) < m, it follows that Jb(c)
is a vector subspace of BA. By (10.17) and the proof of Sublemma 10.26, the
map
bjBA : BA - J{t,}
is injective. Thus if we set Bo = &(B&), we have d!m Bo = dimBA = Indo(c).
Since Bo is a finite-dimensional vector space and Jb(c) is a vector subspace,
we may find a vector subspace B of Bo such that Bo = B 9 Jb(c), where 9
denotes the direct sum of vector spaces
We claim now that TI B x B is positive definite By construction. we
know that Il Bh x B; is positive semidefinite. Also if 0 # Z E B and we
represent Z = $(X) with X E Bb, then X 4 .ib(c). (For if X E Jb(c) . we
have b(X) = X so that Z = 4(X) = X E J ~ ( c ) also, which is impossible since
B r Jb(c) = {O) by construction.) Hence I (Z . Z ) = I($X, 4X) > I (X ,X) ,
the last inequality by (10.17). Thus as I Bh x B& is positive semidefinite,
we obtain I(Z, 2) > I (X ,X) 2 0 so that I ( Z , Z ) > 0 . This shows that
I1 B x B is positive definite. With the notation of SubIernma 10.26, we then
have Indl(c) 2 dim B.
346 10 MORSE IXDEX THEORY ON LORENTZIAN MANIFOLDS
From the direct sum decomposition Bo = B @ Jb(c). we obtain the equality
Indo(c) = dim Bo = dim B + dim Jb(c),
and we also know that Indl(c) 2 dim B. Thus the proof of Proposition 10.25
will be complete if we show that Indl(c) 5 dirn B.
To establish this inequality, suppose B' J(t,) is a vector subspace with
I j B' x 3' positive definite and dim B' = Indl(c). Suppose that dim B' > dim B. Then I is positive semidefinite on B' @ &(c) SO that dim(B1@ Jb(c)) 5 Indo(c). On the other hand, since dim B1 > dim B we obtain
This contradiction shows that dimB1 5 dimB, whence Indl(c) 5 dim B as
required. 13
We are now in a position to prove a Morse Index Theorem for tirnelike geo-
desic segments. The proof we give here 1s modeled on the proof for Riemannian
spaces given in Gromoll. Klingenberg, and bfeyer (1975, pp. 150-152).
Theorem 10.27 (Timelike Morse Index Theorem). Let c : [a, b ] 4
A4 be a timelike geodesic segment, and for each t E [a, b] let Jt(c) denote the
R-vector space of smooth Jacobi fields Y along c with Y (a) = Y (t) = 0. Then
c has only finitely many conjugate points, and the index Ind(e) and extended
index Indo(c) of the index form I : V&(c) x V ~ ( C ) --+ W are given by the
formulas
(10.18) Ind(c) = dim Jt(c) tE(a ,b )
and
(10.19) Indo(c) = dim Jt(c), t ~ ( a , b j
respectively.
Proof. We first show th2t xCtE(a,bj dim Jt(c) is a finite sum. We know that
dim Jt(c) 2 1 if arid only if c(t) is a conjugate point of t = a along c. We may
also define embeddings
i : Jt(c) - I#(.)
10.1 THE TIMELIKE MORSE INDEX THEOXY
for each t E [a, b ] by
for a 5 s 5 t , i(Y) (s) =
0 for t 5 s 5 b.
Evidently dirn Jt(c) = dirni(Jt(c)) for any t 6 (a. b ]
Recall that Indo(c) is finite by Proposition 10.25 Thus to show that c has
only finitely many conjugate points, it suffices to prove that if I t l , t2, . . tk) is
any set of pairwise distinct conjugate points t o t = a along c, then k 5 Indo(c).
To this end, set Ag = z ( J t , ( c ) ) for each j = 1,2, . . . , k and A = A1 B . . . @ Ak.
Then A is a vector subspace of V&(c), and ilecon~posing Z E A as Z =
c:,~ X3ZJ with A3 E W, Zg 6 A3, we obtain I (Z, 2) = C,,l A,AII(Z,, 21).
But if t3 _< t l , we obtain I(Z2, Zl) = I(Zj. 21): + I (Z3. Zl)Zi = -(&I, ZJ)l; + I(0, &);, = 0 using (10.21, Zl 6 At, and Z,(a) = ZJ(tl) = 0. Hence i ( Z . 2 ) = 0
from the symmetry of the index form. Thus I I A x A is positive semidefinite.
Hence
as required. Therefore c : [a, b ] -+ A4 has only finitely many conjugate points
in (a. b] , which we denote by tl < tz < ... < t , Except for t E ( t l , ta , , t,), we have dim Jt(c) = 0, and thus CtE(a,b dim .7t(r) i.; a finite sum
Since Jndo(c) = Ind(c) + dim Jb(c) from Proposition 10 25, !t is enough to
establish the equality
Indo(c) = dim Jt(c)
* to prove Theorem 10.27. Let Z denote the integers with the discrete topology
and define f , fo : (a , b] -+ Z by f (t) = ind(c 1 [a, t ] ) arid fo(t) = Indo(c / [a, t ] ) .
We now show that (10.20) holds if we establish the left ~oritinuity of f arid
the right continuity of So. liere we mean that iirntnrt f ( t n ) = f ( t ) a11d
Emtnlt fo(tn) = fo(t). Using (10.15) it follows that f (tj- fo(t) = - aim Jt(c) =
0 if t $! ( t i , t z , . . . , t,). Assuming we have shown f is left continuous and f U
is right cont~nuous, we also have f (t3+l) = fO(t3) f ~ r each 3 = 1,2 , . . . ,T - 1.
348 10 MORSE INDEX THEORY ON LORENTZIAN MANIFOLDS
Thus
By Theorem 10.22 the index form is negative definite if c has no conjugate
points, so that f ( t ) = 0 for all t < t l . Hence f(t l) = 0 since f is left
continuous. Thus CtE(a,bl dim Jc(c) = fo(tr). Since fo is constant on [t,, b],
we have fo(t,) = jo(b). Hence
which establishes (20.20).
We have thus reduced the proof of the Morse Index Theorem to showing
that f is left continuous and fo is right continuous. First note that f and fo
are nondxreasing (i.e., f ( t ) 1 f (8) if t 2 s). For suppose we fix t l , t2 E (a, b]
with tl 5 tz. Let cl = c 1 [a, t l ] and 6.2 = cl [a,tz]. We then have an R-linear
embedding i : v$(c~) + ~ d ( c 2 ) given by
This map has the property that I(Y, Y) = I(i(Y), i(Y)), where the indexes are
calculated with respect to cl and c2, respectively. Thus if A C @(cl) is a vec-
tor subspace on which the index form of cl is positive (semi) definite, then i ( A )
is a vector subspace of V&(cz) on which the index form of c2 is positive (serni)
definite and dim A = dim i (A) . Thus f (tl) = Ind(c I [a, tl]) < Ind(c I [O, tz]) =
f jt2) and similarly fo(tr ) 5 Soft2). Thus fo and f are nondecreasing.
To obtain the continuity properties of f and fo, we fix an arbitrary E
(a , b] and study the continuity off and fo at ;using the same approximation
techniques as in the proof of Sublemma 10.26. Since c([a, b ] ) is a compact
subset of M , there is a constant 5 > 0 such that for any sl,sz E la, b] with
Is1 - szl < 6 and sl 5 8 2 , the geodesic segment c 1 [sl, s2] has no conjugate
10.1 THE TIMELIKE MORSE INDEX THEORY 349
- points. Choose a partition a = to < tl < . .- < t k - l < tk = t (unrelated
to the above enumeration of conjugate points) such that It, - tgtll < S for
each z = 0,1,2,. . . , k - 1. Let J C [a, b ] be an open interval containing - t with It - tk-11 < E for ail t E J . FOT each t E J , let J"(c,) denote the
finite-dimensional R-vector subspace of V&(c/ [a, t]) consisting of all vector
fields Y E 'ybi(cj[a,t]) such that Yl[t,,t,+i] for each j = 0.1.. . . k - 2
and Y I [tk-1, t] are Jacobi fields. By Sublemma 10.26, f ( t ) is the index of I
restricted to J"(ct) x y(ct) and fo(t) is the extended index of 1 restricted to
T(ct) x T(ct).
Now let E = N(c(t1)) x N(c(t2)) x . . - x N ( ~ ( t k - ~ ) ) The set E is ciosed
since each N(c(t,)) = (v E T,(t,)M : (v,cl(t,)) = 0) is a closed set of space-
like tangent vectors. We may define a Euclidean product metric (( , )) : E x E -+ R by ( ( v , ~ ) ) = ~f iT , ' (v , , w,) for v = (vi, 212, - . , , uk-1). w =
(ul, w2 ,... ,wk-l) E E. Then by Remark i0.2, S = {v E E : lltrll = 1) is
compact.
If Y E T(ct), then Y ( a ) = Y(t) = 0 by definition. Also since c j [t,,t,+l]
has no conjugate points, for any v E N(c(t,)) and w f N ( c ( ~ , + ~ ) ) there is a
unique Jacobi field Y along c with Y(t,) = v and Y(t,+l) = ZL. Since (Y,cl)l,
is an aEne function of t and (v. cf(t,)) = (w, ~ ' ( t , + ~ ) ) = 0. it follows that
(Y, c') 1, = 0 for all t. Hence the map
4t : ?(ct) + E
defined for t E J by
is an isomorphism. For each t t J we also have a quadratic form Qt : E x E -4
R given by QQt (u, v) = I(#;' (u), #tf (v)). Sublemma 10.26 implies that jo(t)
the extended index of the quadratic form &t on E x E and f ( t ) is the index
o f Q t o n E x E f o r e a c h t ~ J .
Each Qt may then be used to define a map Q . E x E x J -+ W given
by Q(u. v, t) = Qt(u, v). We want to show that Q is continuous in order to
prove that f and fo have the d ~ i r e d continuity properties. To this end, let
350 10 MORSE INDEX THEORY ON LORENTZIAN MANIFOLDS
B = (Y / [a,tkPl] : Y E Z(c,)). Then B is isomorphic to E via the mapping
4 : B -+ E given by
Thus
where and YVat are the Jacobi fields along c given by
and
YV,t = $';'(v) I [ t k - 1 , t l .
Since the map (u, v) -+ I(p-'(u). $-'(v)) from E x E 4 W is a bilinear form,
the map ( u , ~ , t ) + I($-1(u),6-1(v)) is certainly continuous. By Proposition
10.16, the map (u,v,t) 4 (Xu,t,~:,,)/:k-l is continuous. This establishes the
continuity of Q : E x E x 3 -+ W. We are finally ready to show that fo is right continuous and f is left con-
tlnuous at the arbitrary t € (a, b]. Since Sublemma 10.26 implies that f
is finite-valued, we may choose a subspace A of E with dimA = f(t) and
Q ( u , u , t ) > 0 for all u E A, u f 0 Since Q : E x E x .J -+ R is contirnions
and S1 = S n A = (u E A : l/ull = 1) is compact, there is a neighborhood 30
of Fin J such that Q(zl.u, t ) > 0 for all t E Jo and u E S1. Hence Qt 1 A x A
is positive definite for dl t E Jo. Thus f(t) 2 f ( r ) for all t f Jo. Since f is
nondecreasing, it follows that f ( t ) = f (z) for all t E Jo with t 5 z Hence f 1s
left continuous.
It remains to show the right continuity of fo at Let (s,) be a sequence
In J with s, > t for all n and s, -+ Since Jo is nondecreasing and integer-
valued, we may suppose that fo(s,) = k for all n. By the monotonicity of
fo, we then have f o ( t ) 5 k. It thus rernains to &how that f o f t ) > k . To
accomplish this, choose for each n a k-dimensional subspace A, of E such
10.1 THE TIMELIKE MO=E INDEX THEORY 351
that Q,,, / A, x A, is positive semidefinite. Let (a' (n), a2(n), . . . . ak(n)) be an
orthonormal basis of A, for each n. Thus al(n), uz(n), . . . . ak(n) are contained
in the compact subset S of E. By the compactness of S, we may assume
a,(n) -+ a, 6 S for each j By continuity of the innel product, it follows
that the vectors (51, ar , . . . ,a,) form an orthonormal subset of S. Let A =
span{al, az. . . . , ak), which is thus a k-dimensional subspace of E. Given u E k k A. we may write u = A3a3. Let u(n) = C,=l Xla,(n). Obviously
~ ( n ) --I u as n -+ oo. Thus using the continuity of Q : E x E x J - R. me
obtain
QZ(u,u) = lim Q(u(n),u(n).s,) 2 0 11-00
since Q,- 1 A, x A, is positive semidefinite for each n. Hence Qt I A x A is
positive semidefinite. Thus fo( t ) 2 dim A = k as required. O
We cnnclude this section with an application of the Timelike Morse Index
Theorem to the structure of the cut locus of future one-connected, globally
hyperbolic space-times [cf. Beem and Ehrlich (1979c, Section 8)j.
Definition 10.28. (Future One-Connected Space-Tzme) A space-time
( M , g ) is said to be juizlkre one-connected if for all p,q E M, any two future
directed timelike curves from p to q are homotoplc through smooth future
directed timelike curves with fixed endpoints p and q.
This concept, a Lorentzian analogue for simple connectivity, has been stud-
ied in Avez (1963), Smith (1960a), and Flaherty (1975a, p. 395). The van-
ishing of the Lorentzian fundamental group implies that ( M , g ) is future orie-
connected. However, the simple connectivity of M as a topological space does
not imply that ( M , gj is future one-connected, as the following example of Ge-
roch shows. Consider B3 with coordinates (x, y, t ) and the Lorentzian metric
ds2 = -dt2 + dz2 + dy2. Let T = ((z, 0,O) E W3 : x 7: 0) and set M = B3 - T
with the induced Lorentzian metric from (W3, ds2) Then M is simply con-
nected. On the other hand, let p = (2,0, -1) and q = (2.0,1). Then p and q
may be joined by future directed timelike curves yl and lying on opposite
sides of the positive z axis (cf. Figure 10.1) But -/I and 72 are not homotopic
through fcture directed timelike curves starting at p and ending at, q since such
352 10 MORSE INDEX THEORY ON LORENTZISN MANIFOLDS
FIGURE 10.1. A space-time M = W3 - T which is simply con-
nected but not future one-connected is shown. The future directed
timelike curves yl and 7 2 from p to q are not homotopic through
timelike curves with endpoints p and q.
a homotopy would have to go around the point (0. 0 , 0), which would introduce
spacelike curves.
Note that if (M, g) is future one-connected. then the path space of smooth
timelike curves from p to q is connected. Thus Lemma 4.11-(2) of Cheeger
and Ebin (1975, p. 85) and the standard path space Morse theory [cf. Everson
and Talbot (1976), Uhlenbeck (1975), Woodhouse (1976)] imply the followmg
proposition.
10.1 THE TIMELIKE MORSE INDEX THEORY 353
proposit ion 10.29. Let (M,g) be future one-connected and globally hy-
perbolic. F i x p 6 M and suppose that every future directed timelike geodesic
segment starting at p has index zero or index greater than or equal to two.
Given q E I+@) such that q is not conjugate to p along any future directed
timelike geodesic from p to q, there is exact1.y one future directed timelike ge-
odesic from p to q of index zero, namely, the unlque maximal geodeslc from p
to q.
We are now ready to prove the Lorentzian analogue for globally hyperbolic,
future one-connected space-times of a theorem of Cheeger and Ebin (1975.
Theorem 5.11) on the cut locus of a complete Riemannian manifold which
generalized a theorem of Crittenden (1962) for simply connected Lie groups
w ~ t h bi-invariant Riemannian metrics. The global hyperbolicity is used in The-
orem 10.30 to guarantee the existence of maximal geodesic segments joining
chronologically related points. Here the order of a conjugate point t = to to
p along the t~rnelike geodesic y with $0) = p is defined as dim J,, (y) lrecali
Definition 10.241.
Theorem 10.30. Let (M, g ) be future one-connected and globally hyper-
bolic. Suppose that for p € A4, the first future conjugate point ofp along every
timelike geodesic y with y(0) = p is of order tu-o or greater. Then the future
timelike cut locus of p and the locus of first future timelike co~~jugate po~n t s of
p coincide. Thus all future timelike geodesics from p maximize up to the first
future conjugate point.
Proof. All geodesics will be unit speed and future timelike during the course
of this proof. It suffices to show that ~f y : [O, t ] -+ M wlth y(0) = p has index
zero and y(t) is not conjugate to p along y 1 10, t ', then y 1s maximal. Since
the set of singula points of expp is closed, it follows from Theorem 10.27 that
there exist E I , E ~ > 0 such that if <(u,y1(O)) < €1 and c < € 2 , then the future
timelikc geodesic c- : [O, t + €1 --t M nit11 al(0) = u is of index zero. Here Q
denotes angle measure using an auxiliary Ttiernailrlidn metric.
By Sard's Theorem, we may find a sequence of points { p , ) E, I t ( p ) with
p, -4 y ( t ) such that every timelike geodesic segment from p to pz has concon-
jugate endpoints. Thw, by hypothesis, every such segment has index zero or
354 10 MORSE INDEX THEORY ON LORENTZIAK MANIFOLDS
Index two or greater. By Proposition 10.29, there is a unique timelike maximal
geodesic segment 7% from p to p, of index zero.
Since (exp,), is nonsingular at t?'(O), for ? suEcie~ltly large there are geo-
desic segments 7% from p to p, with 7% -+ y. Since 6(Tt1(0), yf(D)) -+ 0, these
segments have index z ~ r o for large i. It follows that for i sufficiently large, - y, = y, and hence 7, is maximal. Thus 7 is maximal as a limit of maximal
geodesics. t
10.2 The Timelike Path Space of a Globally Hyperbolic
Space-time
In this section we discuss the Morse theory of the path space of future
directed timelike curves joining two chronologically related points in a globally
hyperbolic space-tirne following Uhlenbeck (1975) [cf. also Woodhouse (1976)j.
Both of these treatmerits are modeled on Milnor's exposition (1963, pp. 8&92)
of the Morse theory for the path space of a complete Riemannian manifold in
which the full path space is approximated by piecewise smooth geodesics. A
different approach has been given to the Morse theory of nonspaeelike curves
in globally hyperbolic space-time by Everson and Talbot (1976, 1978). They
use a result of Clarke (1970) that any four-dimensional globally hyperbolic
space-time may be isometrically embedded in a high-dimensional hlinkowski
space-time to give a Wilbert manifold structure to a subclass of timelike curves
in M .
We now turn to Uhlenbeck's treatment of the Morse theory of the path
space of piecewise smooth timelike curves joining points p << q of any globally
hyperbolic space-time (M, g) of dimension n 2 2
Definition 10.31. (The Tzrnelzke Path Space C!,,,)) Given p, q E. ( M , g)
with p << q, let C(,,,) denote the space of future directed piecewise smooth
timelike curves from p to q, where two curves which differ by a parametrization
are identified.
While C:p,q) is not a manifold modeled on a Banach space, it does possess
tangent spaces consisting of piecewise smooth vector fields along the given
piecewise smooth curve assumed to be parametrized by arc length. Functionah
10.3 HYPERBOLIC SPACE-TIME: TIMELIKE PATE SPACE 355
F . C(p,,) -t R may then be considered from the pomt of vlew of the calculus
of variations. Thus a cntzcal poznt of F is a polnt at ujhlch all first variations
vanish, and a cntzcal value is the image under F of a critical polnt. The
functional F on C(,%,) is said to be a homotopzc A4orse jknctzon i f for any
b > a which is not a critical value of F, the topological space F-I ( -m, b) 1s
homotopically equivalent to the space F-"-w. a) with celis adjoined, where
one cell of dimension equal to the index of the cor~esponding critical polnt 1s
adjoined for each critical point of F with critical talue in (a , b).
We will show in this section that the Lorentzian arc length functional is a
homotopic Morse function for C(,,,) provided that p and q are nonconjugate
along any nonspacelike geodesic. This result is analogous to Morse's result
[cf. Milnor (1963, Theorems 16.3 and 17.3)) for complete Riemannian mani-
folds. Namely, if p and q are any two dlstinct points not conjugate along any
geodesic, then the space R(,,,) of piecewise smooth curves from p to q has the
homotopy type of a countable CW-complex with a cell of dimension X for each
geodesic from p to q of index A.
Given that p and q must be nonconjugate along any geodesic for L to be
a Morse function, it is of interest to know that such pairs of points exist. As
for Riemannian spaces, conjugate points in an arbitrary Lorentzian manifold
may be viewed as singularities of the differentia! of the exponential mapping.
Hence Sard's Theorem [cf. Hirsch (1976, p 69)] may be applied. Here a subset
X of a manifold is said to have measure zevo if for every chart (li. 4): the set
O(U n X ) E R" has Lebesgue measure zero in Wn, n = dim A 4 Also a subset
of a manifold is said to be residual if it contains the intersection of countably
many dense open sets. A residual subset of a complete metric space is dense
by the Baire Category Theorem [cf. Kelley (1955, p. 200):. Sard's Theorem
then implies the following result,
Theorern 10.32. Let ( M , g ) be a globally hyperbolic space-time, and let
p E M be arbitrary. Then the set of points of M conjilgate to p along some
geodesic has measure zero. Thus for a residual set of q E M , p and q are
nonconjugate along ail geodesics between them.
356 10 MORSE INDEX THEORY ON LORENTZIAN MANIFOLDS
Recall the following properties of the Lorentzian arc length functional L :
C(p,q) -" R. First, from the cdculus of variations, the critical points of L on
C(p,,) are exactly the future dlrected timelie geodesic segments from p to q
with parameter proportional to arc length. Second, the Timelike Morse Index
Theorem (Theorem 10.27) implies that the index of a critical point of L is
just its index rn a geodesic, i.e , the number of conjugate points t o p along the
geodesic, counting multiplicitim, from p up to but not including q.
In order to show that L is a homotopic Morse function, it is necessary to a p
proximate C(p,p) by a subset such that there exists a retraction of C(p,q)
onto -W(,,,) which increases the length functional L. This step corresponds to
the finite-dimensional approximation of the loop space in Riemannian Morse
theory. The corresponding Lorentzian approximation makes crucial use of the
global hyperbolicity of (n/i,g), as will be apparent from Lemma 10.34 below.
First it is useful to make the following definition.
Definition 10.33. (Timelike Chain) Let p. q E ( M , g ) with p << q. A
finite collection (xl,x2,. . . , x3) of points of A4 is said to be a timelike chazn
fvom p to q if p << x1 <( xz << . . . <C xJ << q.
The next lemma follows from the existence of convex normal neighborhoods
(cf. Section 3.1), the compactness of J f (p) n J-(q) in a globally hyperbolic
space-time, and Theorem 6.1.
Lemma 10.34. Let (M, g) be a globally hyperbolic space-time with a fixed
smooth globally hyperbolic time funct~on f : M -+ R. Let p, q E M with p < q
be given. Then there exists ( t l , tz , . , . , tk) with f ( p ) < tl < tz... < tk < f ( q )
satistying the following properties:
( 1 ) If z E f - -'(f(P), tl] m d p < x, then there is a unique maximal future
directed nonspacelike geodesic segment from p to x;
(2) For each i with 1 5 i 5 k - 1, if x E f-'(tz) and y E f-'(t,,t,+l]
with p < x ,< y < q. then there is a unique maximal future directed
nonspaceiike geodesic segment from z to y:
(3) If y E. f-l[tk. f ( q ) ) and y G q, then there is a unique maximal future
dlrected nonspacelike geodesic segment from g to q;
10.2 HYPERBOLIC SPACGTIME: TIMELIKE PATH SPACE 357
(4) In paxticular. if {zl, x2,. . . , s k ) is any timefike chain from p to q with
x, E f-l(t,) for each i = 1.2, . . . , k , then there is a unique maximal
future directed tlrnelike geodesic segment from p to sl , zk to q, and s,
to x , ,~ for each E with 1 5 i 5 k - 1.
Since p , q E M with p < g are given, we may now fix i t l , t 2 , . . . . t k ) sat-
isfying the conditions of Lemma 10.34. Denote by S, the Cauchy surface
S, = f -I(&) for each i = 1,2, , . . , k. We may now define a space McP,4) of
broken timelike geodesics to approximate C(p,q) from the point of view of the
arc length functional.
Definition 10.35. (Tzmeizke Guwe Space M(,,,)) Let M(PPq be the
space of all continuous curves y : [O, 11 -+ M such that y(0) = p, y (1 ) = q. and
y ( i / ( k t 1)) E St for each i = 1.2,. . . . k and such that the restricted curve
7 1 [ i / ( k + 1) . (z -I- 1)/(k + I ) ] is a future direczed timellke geodeslc for each
i = 0 , 1 , . . . . k.
Since each y / [ i / ( k + I ) , (i + 1 ) / ( k + l ) ] must be the unique maximal seg-
ment joining its endpoints by Lemma 10.34, it follows that
(10.21) J W ( ~ , ~ ) = { ( Z ~ , T Z , . . . , zk) : x, 6 S, for 1 < i 5 k ,
p <C 2 1 , x, < s,+l for each 1 s i 5 k - 1, arid xp, << y ) .
Since chronological future and past sets arc open. M(p,g) viewed as in (10 21)
is an open submanifold of S1 x Sz x - - x &. Let 7 i ~ : M(p,q) + S2 be the
projection map given by 7 i , ( ~ ~ , x 2 , . . . . x k ) = z1 for each z = 1,2. . . , k . S~nce
I+(p) n I - ( q ) h a compact closure by the global hyperbolicity of M , and
~ , ( h f ( ~ , ~ ) ) c I+(p) n I - (q ) , ~t follows that T~(A~(,,,)) has compact closure In
S, for each i = 1,2.. . . . k. A length-nondecreasing retraction of C(p,q) onto hf(p,,) may now be estab-
lished along the lines of Riemannian Morse theory [cf M~lnor (1963, p Ql)]
Proposit ion 10.36. There exists a retraction Qx, 0 < X 5 1, of C(,,,)
onto hf(p,q) which is Lorentzian arc length nondecreasing.
Proof. Let y E C(,,,; be arbitrary W e may suppose that y is parametrized
so that f ( y ( t ) ) = t where f : M -+ B is the globally hyperbolic time function
358 10 MORSE INDEX THEORY ON LORENTZIAN MANIFOLDS
fixed in Lemma 10.34. Tnus y : [f (P), f (q)] - M For each X E [O. I], define
Qx(y) : [f (P), f (411 -" M as follol-s Set B = (1 - X ) f (PI + X f (q) , and put
to = f (p) and tk+: = f (4). If P satisfies t , < 0 5 tl+l for i with O < 'L j k, let
Q x ( ~ ) ( t ) be the unique broken timelike geodesic joining the point p to r ( t l ) ,
y(tl) to y(tz). . . . , y(t,-1) to y(t,), and y(t,) to yfP) successiveiy for t 5 p. and let Q:,(r)(t) = y(t) for t 2 p (cf. Figure 10 2). I t is ~mmediate from the
definltlon that Qo(y) = y and that Ql(y) E M(,,,). Since Q~(y) [y ( t , ) ,~ (P) ] is
maximal from y(t,) to r (P) by Lemma 10.34, we have
where d denotes the Loreiltzian distance function of (M, g) and we have used
Notational Convention 8.4. Similarly, since Qx(y) is the unique niaximal ge-
odesic segment from y(t3) to y(t,+l) for each j with 0 5 j < i - 1, we have
L($x(y)[y(t,),y(t,+~)]) 1 L(y I [t,,t,+,]) for each j with 0 L j I i - 1. Sum-
ming, we obtain L(Q:,(y)[p, y(P)]) 1 L(y I [f ( P ) , PI). Since Qx(y)(t) = ;dt) for
t 2 p, we thus have L(Qx(y)) 2 L(y). Moreover, it is clear from the above
argument that L(QA(y)) = L(y) for all X if and only if y is a broken time-
like geodesic from p to q with breaks possible only at the s , ' ~ . In particular.
Q:, ( IV~,,,) = Id for all X E [O,1]. Finally, the continuity of the map X --, Qx
is clear from the differentiable dependence of geodesics on their endpoints in
convex neighborhoods.
As in Uhlenbeck (1975, p. 791, we denote by L, = L 1 A4(,,,) the restriction
of the Lorentzian arc length functional to the subset M(p,q) of C(p,q). Just as
in the Riemannian case [cf. Milnor (1963. Theorem 16.2)], it can be seen that
L, : Id(,,,) 4 R is a faithful model of L : C(,,,) 4 W in the following sense
Proposition 10.37. lf(-k?. g) isglobally hyperbolic, then the critical points
of L, = L 1 Af[,,,) are smooth timelike geodesic segments from p to q. These
rritical points are nondegenernte iffp is not conjugate to q dong any timelike
geodesic from p to q. Moreover, the index of each critical point is the same in
C(,,,) and M(,,,). namely, the index by conjugate points.
Proof. Identify M(p,q) with k-chains (zl, xz, . . . xk) from p to q with z, .E S,
for each i, as in (10.21). Set so = p and xk+l = q. Let y, : [0, I] -r M
10.2 HYPERBOLIC SPACE-TIME: TIMELIKE PATH SPACE 359
FIGURE 10.2. In the proof of Proposition 10 36. the curve Qx(y)
is used to approximate the given curve 7 .
denote the unique maximal timelike geodesic segment from r, to rz+l for
z = 0,1. . . , k. Then
k r l 1
L. ( r l , TZ, . . . , I*) = C J j - ( -hf j t ) . yti,'(i)) d t . 2 = 0 0
Suppose we have a smooth deformation (xl ( t ) . s2(t), . . , sk ( t j ) of the given
chain {xl , 22, . . . . xk}. Then since each t -r x, ( t ) is a curve in SZ, the vari-
360 10 MORSE INDEX THEORY ON LORENTZIAN MANIFOLDS
ation vector field I/ of the deformation must lie in T,,(S,) at x,. Also as
we are deforming the glven chain, which represents a piecewlse smooth time-
like geodesic, through plecewise smooth timelike geodesics with discontinuities
only at the St's, the space of deformations of (xl, 22, , xk) may be den-
tified with all vector fields Y along is1. 12,. . , xi,) so that Y (z,) E Tx3 (S,)
for z = 1,2, . , k . Y ( p ) = Y ( q ) = 0, and Y 17, is a smooth Jacobi field
along y, for each z = 0,1, , k. By choice of the S,'s as in Lemma 10.34,
no y, . [0, l] -- Af has any conjugate points. Tllus given any v, E T,* (St )
and zu, E T,,,, (S,+lJ, there is a unique Jacobi field J along [O. 1, -+ iM
with J(0) = ?it and J(1) = w,. Thus the space of 1-jets of deformations of
the given chain isl, x2, , sk) may slmply be identified with the Cartesian
product T,, (SI) x Tx,(Sz) x x T,, (Sk).
Let u : [a, b ] --+ M be a smooth future d~rected timelike curve parametrized
so that J-(al(s). al(s)) = A is constant for all s E ( a , b) and so that ol(a+)
and ol(b-) are timelike tangent vectors Let n : [a, h ] x (-c.c) + M be a
smooth variation of a through nonspacelike curves with variation vector field
V . If A * : [a. b ] + itl denotes the curve a,(t) = o( t , s) , then the first variation
formula for ~ ' ( 0 ) = (d/ds)L(~~.)],=o is given by
6 Thus if o is a timelike geodesic. ~ ' ( 0 ) = (-l/A)(V,
\'Vc now apply (10.22) to calculate the first variation of a proper deformation
a of an element {s l ,zz , . ,zk) E M(P,,,). As before, let y, : [O. 11 -+ M be
tlmelike geodesics from x, to r,+l for 2 = 0,1, k Then {xl, x2 , xk) IS
represented by the piecewise smooth timelike geodesic y = -0 * yl * ., . * y k
Let V be the variation vector field of a along y and set y, = V(rr,) E T,, (S,)
As we mentioned above the piecewise smooth Jacobi field V along y may then
be identified with (yl,y2, . l /k) f TT, (5'1) x ... x T l k (Si,). Restricting V to
each y, and applying formula (10.22), we obtain the first variation formula
10 2 HVPERSOLIC SPACE-TIME TIMELIKE PATH SPACE 361
where
for L, = L 1 -44(,,,). Rom (10.23), it is a standard argument to see that
6L,(yl,y2,. . . . yk) = 0 for all (yl, yz.. . . . yk) E 'Iz, (5'1) x . . . x Tzls(Sk) if and
only if the tangent vectors
7i1(1) and 7:+1(0) 1/-(7%1(1)-711(1)) J- kL'+1(0)? pf%'-l (0))
in T,> (M) have the same projection into the subspace T,, (S,) of T,, (M) for
each i = 1: 2,. . . , k. This then implies that y = yo x y, * . . . * yk can be
reparametrized to be a smooth timelike geodesic from p to g.
Thus we have seen that the critical points of L, : M(,,,) -+ R and L :
C(,,,) -+ R coincide and are exactly the smooth timelike geodesics from p to q.
By our proof of the Timelike hlorse Index Theorem (Theorem 10.27) by ap-
proximating V&(c) by spaces of piecewise smooth Jacobi fields (cf. Sublemma
10.26), it follows that the indices of L, : -M(,,,) -t R and L : C( ,,,, i R
coincide. The Timelike Morse Index Theorem (Theorem 10.27) then implies
that a critical point c is degenerate if and only i f p and q are conjugate along
c (cf. also Proposition 10.13).
With Proposition 10.37 in hand, the followiilg result rnay now be estab-
lished.
Proposition 20.38. Let ( M g) be globally hyperbolic. If p and g are
nonconjugate along any nonspacelike geodesic. then L : C(p,,) -+ lR and L, . M(,,,) i R have only a finite number of critical points. ln partielljar, there
are only finitely many future directed timelike geodesies from p to q.
ProoJ. By Proposition 10.37, it is enough to show that L, : ~%f(,,~) + R
has only finitely many critical points. Suppose L, has infinitely many critical
points. Then there would be an infinite sequence of smooth timelike geodesics
from p to q. Since each ? T ~ ( M ( ~ , ~ , ) has compact closure in S,, the h ' s
have a subsequence which must then converge to a timelike or null geodesic c
362 10 MORSE INDEX THEORY ON LORENTZIAN MANIFOLDS
from p to q. Since c is a limit of an infinite sequence of geodesics from p to q,
it follows that p is conjugate to q along c, in contradiction. 0
Another interesting property of L, : M(,,,) -+ R is given by the following
result.
Proposition 10.39. L, : &I(,,,) - B assumes its maximum on every
component of hf(,,,).
Proof. While M(,,,) is an open submanifold of SI x Sz x . . x Sk, its closure - iM(,,,) is compact in S1 x S2 x . . . x Sk. Also
where xo = p and xk+l = q for any timelike chain {xl, x2,. . . , xn) from p to q.
Thus as the Lorentzian distance function is finite valued for globally hyperbolic
space-times, it follows that L, is bounded on each connected component U
of Ad(,,,). Let (7,) be a sequence in the connected component U of M(p,q)
with L,(y,) -+ sup(L,(y) : y E U) as n -+ a. By compactness of Z(,,,) in S1 x Sz x - . . x Sk, the sequence {y,) has a limit curve y, in SS, x
$2 x . x Sk. Since {y,) was a maximizing sequence for L, ] U, it follows that
L,(y,) 2 L,(n) for any a E g, and L,(y,) > 0. If y, is represented by the
element (sl, 22,. . . , xk) E S1 x . . . x Sk, we have p < zl < 3.2 < . . . ,< r k 6 q,
and if a is the unique maximal geodesic from 2% to x,+l guaranteed by Lemma
10.34, then each 7% is either null or timelike. If some y, is null. it follows from
the first variation (10.23) that y, may be deformed to a curve y E jl with
L, (y) > L, (7,). This then contradicts the maximality of L, / at y,. Thus,
in fact, the Iimit element r,, E U . U
Again, consider LW(~,,) .ca a subset of S1 x . . . x Sk. Let P, : T,(M) --+ T, (St)
denote the orthogonal projection map for any x E S, and each i = 1,2.. . . , k .
The set M(p,q) regarded as an open subset of S1 x - x 31, may thus be given
a Riemannian metric induced from the Lorentzian metric restricted to the
spacelike Cauchy hypersurfaces S1, Sz, . . . , Sb It then follows from formula
(10.23) that at the point of &I(,>,) represented by the broken timelike geodesic
10.2 HYPERBOLIC SPACE-TIIvIE: TIMELIKE PATH SPACE 363
y = 70 * 71 * . . - * */kt with each y, parametrized on [O. 11 arid A,, B, s above.
the gradient of L. is given by the formula
Using this formula. Uhlenbeck (1975, pp. 80-81) then establishes the following
lemma.
Lemma 10.40. Let /3 : (a, b) -+ M(,,,) be a maximal integral curve for
grad L,. Then b = oo and limt,, P( t ) lies in the cri~ical set of L,.
With the behavior of grad L, established, ~t now follows [Uhlenbeck (1975,
p. 81)] that if p and 4 are nonconjugate along any timelike geodesic. then
L, : M(,,,) -+ R is a Morse function for A4(,,,). The main result now follows
from this fact and Propositions 10.36 and 10.38.
Theorem 10.41. Let (M, g) beglobally hyperbolic, and let p, q he any pair
of points in (34, g) with p << q and such that p and q are not conjugate along
any nonspaceiike geodesic from p to q. Then there are only finitel-v many future
directed timelike geodesics in (M, g) from p to q, and the arc length functional
L : C(,,,) -+ W is a homotoplc Morse function. Thus ~f b > a are any two
noncritical values of L, then L-I( -m, b) is homotopically equivalent to the
space L-'(-03, a ) with a cell attached for each smooth timellke geodesic y
from p GO q with a < L(-f) < b. where the dimens~on of the attached cell is
the (geodesic) index of y Thus C(,,,) hds the hornotopy type of a finite CW-
complex with a cell of d~mension X for each srnooth future directed timellke
geodesic 2 from p to q of index A.
Note that in Theorem 10.41 the topology of Ci,,,) is not related to the
given rnanifold topology. But Uhlenbeck (1975, Theorem 3) has shown for a
class of globally hyperbolic spacetimes sat~sfying a metric growth condition
[cf. Uhlenbeck (1975. p. 7 2 ) ] that the hamotopy of the loop space of M itself
may be calculated geometrically as follows. Let (M, g) be a globally hyperbol:~
space-time satisbing Chlenbeck s metric growth condition. Then there is a
364 10 MORSE INDEX THEORY ON LORENTZIAN MANIFOLDS
class of smooth timelike curves y : [0. 03) + M with the following property
For any such timelike curve y, there is a residual set of points p E M such
that the loop space of Al is homotopic to a cell compiex with a cell for each
null geodmc from p to y, where the dimension of the cell corresponds to the
conjugate polnt index of the geodes~c.
It will be clear from the proof of Proposition 10.42 below that the finiteness
of the homotopy type of the timelike loop space C(,,,) in Theorem 10.41 follows
from the assumption that p is not conjugate to q dong any null geodesic
together with the fact that L(7) 5 d ( p , q ) < co for a l ly E R(,,,). For complete
Riemannian manifolds (N.go) on the other hand, it is known from work of
Serre (1951) that if iV is not acyclic [i.e., H,(N;Z) + 0 for some i > 01. then
the loop spaces f l ( , ,q) are infinite CW-complexes for all p, q E N . Thus if p
and q are nonconjugate along any geodesic, there are infinitely many geodesics
c, : [O, 11 4 N from p to q. Since p and q are nonconjugate along any geodesic
and La(?) > do(p, q) > 0 for all y E a(,,,), it may be seen that Lo(&) -+ oo
asn-+w.
For completeness, we now give a different proof from Proposition 10.38 of
the existence of only finitely many critical points for L : C(,,,) + P. Instead of
using the finite-dimensional approximation of Cb,,) by W,,,). we work directly
with C(,,,) using the existence of nonspwelike limit curves as in Section 3.3.
Proposition 10.42. Let ( M , g) be globally hyperbolic, and suppose that
p, q E M with p << q are chosen ssuch that p and q are not conjugate dong
any future directed nonspacelike geodesic. Then there are only finitely many
cimelike geodesics from p to q.
Proof. Suppose that there are infinitely many future directed timelike ge-
odesic segments c, : [0, l] A4 in C(p,g). Using Corollary 3.32 and the
arguiients of Section 3.3 we obtain a nonspacelike geodesic c : [O, 11 + A4 with
c(0) = p and c(1) = q which is a limit curve of the sequence (h) and such
that a subsequence of {c,) converges to c in the CO topology on curves. Since
a subsequence of the pairwise distinct tangent vectors ( ~ ' ( 0 ) ) converges to
cl(0), it follows that q is conjugate t o p along c, in contradiction. El
10.3 T H E NULL MORSE INDEX THEORY 365
Suppose that it is only assumed that p and q are not conjugate along any
timelike geodesic in the hypotheses of Prcpositioc 10.42. If there are infinitely
many tirxelike geodesics {c,) in C(p,q), we then have L(c,) -+ 0 as n - m,
and the limit curve c in the proof of Proposition 10.42 is a, null geodesic such
that q is conjugate t o p along c. In particular, c contains a future null cut point
t o p (cf Section 9 2) Thus, Theorem 10.41 or the proof of Proposition 10.42
also yields the following result which applies. in particular, to the Friedmann
cosmological models with p = A = 0.
Corollary 10.43. Suppose that (M,g) is globally hy-perboljc and that
p,q E M are chosen such that p << q, p is not conjugate to q along an^;
timelike geodesic, and the future null cut locus of p in M is empty, Then there
are oniy finitely many timelike geodesics from p to q.
Prooj. If there were infinitely many timelike geodesics from p to q, there
would be a null geodesic c from p to q such that q is conjugate to p along c.
But then p contains a future null cut point along c, in contradiction. R
If d imM = 2, then (M,g ) contains no null conjugate points (cf. Lemma
10.45). Thus we also have the following result.
Corollary 10.44. Suppose that (M, g) is any two-dimensional globally
hyperbolic space-time and that p. q E M are chosen such that p < q and q is
not conjugate to p along any ~imelike geodesic. Then there are only finitely
many timelike geodesics from p to q.
10.3 The Nu11 Morse Index Theory
Thi9 section is devoted to the proof of a Morse Iedex Theorem for null geo-
desic segments @ : [a, h ] + (M,gj in an arbitrary space-time. The appropr~ate
index form we will use. however, 1s not the standard index form defined on
piecewise smooth vector fields orthogonal to 0' but rather its proj~ction to the
quot~ent bundle formed by identifving vector fields differing by a miiltipie of
p'. The tdea to use the quotient bundle as the domain of definition for the null
index form is zmplicitly contained in the discussion of variatmn of arc length
for null geodesics in Hawking and E!hs (1973, Sectlon 4.5) and is further de-
366 10 MORSE INDEX THEORY ON LORENTZIAN MANIFOLDS
veloped in Bolts (1977) [cf. Robinson and Trautman (1983)]. The first part
of this section develops the basic theory of the index form along the lines of
Chapters 2 and 4 of Bolts (1977) In the second part of this section, we give
a detailed proof of the Morse Index Theorem for null geodesics sketched In
Beem and Ehrlich (1979d).
Instead of working with the energy functional. Uhlenbeck (1975, Theo-
rem 4.5) has constructed a Morse theory for nonspacelike curves in globally
hyperbolic space-times as follows. Choosing a globally hyperbolic splitting
M = S x (a, b) as in Theorem 3.17. Uhlenbeck projected nonspacelike curves
y ( t ) = ( c l ( t ) , cZ(t)) onto the second factor and showed that the functional
J ( 7 ) = S[c2'(t)l2dt y~elded an index theory.
It should be mentioned at the outset that the null index theory, unlike the
timelike index theory, is interesting only if dim M 2 3 for the following reason
Lemma 10.45. No null geodesic ,# in any two-dimensional Lorentzian man-
ifold has any null conjugate poin~s.
Proof. Let p : (a, b) + ( M , g ) be an arbitrary null geodesic. Suppose that
J is a Jacobi field along /3 with J(tl) = J ( t 2 ) = O for some t I # t2 in ( a , b).
Just as in the proof of Lemma 20.9, we have ( J , ,@)I' = - ( R ( J : ,B')P1, 9') = 0
so that ( J ( t ) , P1 ( t ) ) = 0 for all t E (a, b). Since spacelike and null tangent
vectors are never orthogonal when dimM = 2, the space of vector fields Y
along j3 perpendicular to @' is spanned by P' itself. Hence J(tf = f ( t )P1( t )
for some smooth function f : (a, b) + R. The Jacobi equation then becomes
0 = J" + R(J. @)P' = f"P1 + fR(P' , P')P = f1I/?' by the skew symmetry of
the curvature tensor in the first two slots. Hence f"( t ) = 0 for all t E (a, b)
Since J ( t i ) = J ( t z ) = 0, it follo~vs that f = 0. Thus J = 0 as required.
In view of Lemma 10.45. we will let ( M , g ) be an arbitrary space-time of
&rnension n 2 3 throughout this section and let 0 . (a, b ] -+ M be a fixed
null geodesic segment in -44. If we let V1(P) denote the W-vector space of
all piecewise smooth vector fields Y along ,!3 with (Y(t),P1(t)) = 0 for all
t E [a, b ] , then pl( t ) and t,B1(t) are both Jacobi fields in V1(P). k t h e r ,
10.3 THE NULL MORSE INDEX THEORY 367
suppose we consider an index form I : Vj(P) x Vt(f i ) -+ R defined by
I ( X 7 Y ) = - / [(x', Y' ) - ( R ( X , 01)/3' .Y)] dt
analogous to the index Lorm (10.1) for timelike geodesics In Section 10.1. Then
A = {f ( t )P i ( t ) 1 f . [a, b ] -+ R is a smooth function with f ( a ) = J ( b ) = 0) 1s
an infinite-dimensional vector space such that I (Y . Y) = 0 for all Y E A. Thus
the extended index of /3 defined using the index form I : Yo'-(@) x V:($) -+ R
is always infinite. Also while I ( f P', Y) = 0 for any Y E VZ(0) and f 9' E A.
the vector field fP1 is not a Jacobi field unless f" = 0. Thus the relationships
we derived for the index form of a timelike geodesic in Section 10.1 linking
Jacobi fields, conjugate points, and the definiteness of tlie index form fail to
hold for I : voA(a) x V,-(B) - R
The crux of the difficulty is that P' and tP1 are both Jacobi fielda in ~ ' ( 0 ) . Thus by ignoring vector fields in A, it is possible to define an ~ndex form for
null geodesics nlcely related not only to the second variation formula for the
energy functional but also to conjugate points and Jacobi fields. This may
be accomplished by working with the quotlent bundic VL(/3)/[/3'] rather than
V1(P) itself Using this quotient space, an index form 7 may be defined so
that ,9 has no conjugate points if and only if? is negative definite [cf. Hawking
and Ellis (1973, Proposition 4.5.11). Bolts (1977, Satz 4.5.5)j. Also, a Morse
Index Theorem for null geodesic segments in arbitrary space-timcs may be
obtained for the iridex form 7 [cf. Beern and Ehrlich (1979d)j.
Since we are interested in studying conjugate points along null geodesics, it
is important to note that Lemma 10 9 and Corollar~es 10.10 and 10.11 carry
over to the null geodesic ease with exactly the same proofs
Lemma 10.48. Let /3 : [a, b ] -+ (M,g ) be a null geodesic segment. and
let Y be any Jacobi field along P. Then (Y ( t ) , y(t)) is an afine function of f
Thus i f Y ( t l ) = Y(t2) = 0 for distinct t l , t z E [a. b j . then ( Y ( t ) , f l ( t ) ) = 0 for
all t E [a, b ] .
Accordingly, we may restrict our attention to the following spaces of vector
fields.
368 15 MORSE INDEX THEORY ON LORENTZIAK MANIFOLDS
Definition 10.47. Let V L ( P ) denote the R-vector space of all piecewise
sinooth vector fields Y along P with ( Y ( t ) , f l ( t ) ) = 0 for all t E [a, b] . Let
V$ (0) = ( Y E v'@) : Y (a) = Y {b) = 0). Also set N(P(t) ) = (v E Tp(tjM :
(v. pl(t)) = 01, and let
= I.) N(P(t) ) . a s t s b
For any Y E V L ( @ ) , the vector field Y' E V L ( P ) may be defined using
left-hand limits just as in Section 10.1. Since /3 is a smooth null geodesic.
P1( t ) E N ( p ( t ) ) for all t E [a. b ] . Thus, following Bolts (1977, pp. 39-44). we
make the follo\ving algebraic construction. Since N(P( t ) ) is a vector space and
is a vector subspace of N(P( t ) ) for each t E [a, b] , we may define the quotient
vector space
and the quotient bundle
Elements of G(P(t) ) are cosets of the form v + [P1(t)], where 1; E N ( B ( t ) ) and
the vector subspace [P1(t)] is the zero element of G(P(t)) for each t E [a, b].
Also 7: + [PJ ( t ) ] = w + [@'(t)] in G(P( t ) ) if and only if v = w + XP1(t) for some
X E R. We may define a natural projection map 2.r : :V(P(t)) 4 G(B(t) ) by
(10.27) K ('u) = TJ + [P1(t)].
The projection map 7i on each fiber induces a projection map 2.r : N(P) -+ G(P)
given by r ( Y ) = Y + [P'], i.e., r ( Y ) ( t ) = Y ( t ) + [P1(t)] E G(P(t)) for each
t E [a, b ] .
This quotient bundle construction may be given a (non-unique) geomet-
ric realization as follows. Let n E Tp(o)M be a null tangent vector with
15.3 THE NULL MORSE INDEX THEORY 369
(n, fll(0)) = -1. Parallel translate n along P to obtain a null parallel field
q along with ( ~ ( t ) , P1( t ) ) = -1 for all t E [a, b ] . Choose spacelike tangent
vectors el. en,. . . , en-2 6 Tp(o)M such that (n, e,) = (01(0), e3 j = 0 for ali
j = 1.2, . . . . n - 2 and (e,, e2) = 6,, for all J = 1,2, . . . , n - 2. E x t e ~ d by par-
allel translation to spacelike parallel vector fields El, E2,. . . , En-2 E vJ-(P), and set
n-2
(10.28) CA,E,(~) :A, ER for 1 < j s n - 2 3=1
Then V ( P ( t ) ) is a vector subspace of N(P( t ) ) consisting of spacelike tangent
vectors, and we have a direct sum decomposition
for each t E [a: b ] . Set V ( P ) = lJa<t5b V(D(t) ) , and let %(a) = {Y E V(O) : - Y ( u ) = Y ( b ) = 0). If P'(t) and ~ ( t ) are given, V ( P ) 1s independent of the
particular choice of {el, en, . . . , en-2) in Tp(o)lVI. However. if n E T p p ) M
and hence 17 are changed: a different direct sum decomposition (10.29) may
arise since the given Lore~ltzian metric g restricted to .?-(,B(t)) 1s degenerate.
Nonetheless, we may regard V(P( t ) ) as being a geometric realization of the
quotient bundle G(@) via the map Z --+ Z + (@'I from V(P) :nto G(P). It 1s
easily checked that this map is an isonlorphisin since (10.29) is a d~rect sum
decomposition.
We may dcfine the inverse isomorphism
for each t E la, b ] as follo~vs. Given v E G($(t ) ) , choo~i. any rr E N ( P ( t ) ) with
~ ( x ) = TJ. Decomposing 3: uniquely as x = X/j"(t) + zo with vo E V ( P ( t ) ) by
(10.29), set 6(v) = vo. If any other X I E N(P(t) ) with 7i(zl) = .r, had been
chosen. we would still have X I = pP(t) + vo w ~ t h the same ' ~ ' 0 E V ( B ( t ) ) for
some /I E R. Thus 6 is well defined.
As a first step toward defining the index form on the quotient bundle
G(P), we show how the Lorentzian metric, covariant derivat:ve, and curvature
370 10 MORSE INDEX THEORY ON LORENTZIAN MANIFOLDS
tensor of (M, g) may be projected to G(P). First, given any u , w f G(P(t)).
choose x, y f N(P(t)) with n ( x ) = v and n(y) = w. Define B(v. w ) by
Suppose we had chosen X I , yl E N(P(t)) with ~ ( 2 1 ) = v and ~ ( y l ) = w.
Then x = z1 + XPf ( t ) and y = yl + p/3'(t) for some X,p E R, arid thus
g(x, Y) = g(al,yi)+Xg(yl. ,3'(t)) f ~ g ( x ~ , P ' ( t ) ) + ~ X g ( P ' ( t ) , P ' ( t ) ) = y(z l ,y l ) . Hence g is well defined. I t is also easily checked that B(v, w) = g(B(u), B(w)) for
all v, zli E G(P(t)). Hence the metric on G(P(t)) may be identified with the
given Lorentzian metric g on V(O(t)). Since g 1 V ( P ( t ) ) x V ( P ( t ) ) is positive
definite for each t 6 [a, b ] . the induced metric 3 : G(P(t)) x G(P(t)) -t R thus
has the following important property.
Remark 10.48. Fhr each t E [a, b ] , the metric jj : G(P(t)) x G(P(t) ) -+ R
is positive definite.
We now extend the covariant derivative operator acting on vector fields
along P to a covariant derivative operator for sections of G(F). We first intro-
duce the following notation.
Definition 10.49. Let X(P) denote the piecewise smooth sections of the
quotient bundle G(P), and let
Given V E X(@, choose X E V'(6) with n ( X ) = V. Set
If XI E V L ( P ) also satisfies: x ( X 1 ) = V , then XI = X + jPf for some f :
[a, b ] + R, and we obtain X I 1 = XI + f1p since P is a geodesic. Thus
x ( X l f ( t ) ) = x ( X 1 ( t ) ) for all t E [a, b] . Hence the covariant, derivative for X ( P )
given by (10.32) is well defined. It may also be checked that this covariant
differentiation is compatible with the metric g for G(P) and satisfies thc usual
properties of a covariant derivative.
10.3 THE NULL MORSE INDEX THEORY 371
Given V E X ( P ) , choose X E V L ( 0 ) with T ( X ) = V . We then have
X ( t ) = f(t)O1(t) + B(V)(t) with 6 as in (10.30). According to (10.32), we may
thus calculate V1( t ) using B(V) as V f ( t ) = 7(VP(B(V)) ( t ) ) . Now B(V) satisfies
(P1(t), O(V)(t)) = ( ~ ( t ) , B(V)(t)) = O for all t E [a, 61. Since 0 is a geodesic and
7 j is parallel along 3, we obtain on diserentiating that (O1(t), VptB(V)( t ) ) =
( ~ ( t ) , V p B ( V ) ( t ) ) = 0 for all t E [a, b ] . Thus 'G?$'O(V) E V(P) . Hence we
obtain
(10.33) 6 ( V f ( t ) ) = (B(V))'(t) for all t E [a, b ]
where the differentiation on the left-hand side is in G(P) and on the right-
hand side is in V(/3). Thus if we identify G(0) and V(P) using @, covariant
differentiation in G(P) and in V(0) are the same.
To project Jacobi fields and the Jacobi differential equation to G(/3). it 1s
necessary to define a curvature endomorphism
for each t E [a, b ] . This may be done as follows. Given v E G(P(t)), choose
any 3: E N(P( t ) ) with ~ ( x ) = v, and set
This definition is easily seen to be independent of the choice of x 2: f (B( t ) )
with ~ ( x ) = v since R(P1, fi l)Pf = 0. if 71 = T ( X ) and u: = n(y) with x, y E
N(P( t ) ) , it follows from (10.31) and (10.34) that
Finally horn the symmetry properties of g(R(x, g)z, w) we obtain
for all u, w E G(P(t))
With these preliminary considerat!ons completed, we are now ready to de-
fine Jacobi classes in G(P) [cf. Boltr (1977, pp 43-44)]
372 10 MORSE INDEX THEORY ON LORENTZIAN MANIFOLDS
Definition 10.50. (Jacobi Class) A smooth section V E X(P) is said to
be a Jacobi class in G(P) if V satisfies the Jacobi differential equation
where the covariant differentiation is given by (10.32) and the curvature endo-
morphism a by (10.34).
Given a Jacobi class W E X(P) with W ( a ) # [,@(a)] and W(b) # [ P ( b ) ] ,
we will see in the next series of lemmas that there IS a two-parameter family
Jx , of Jacobi fields of the form JA,, = J i- A/? + ptP1 in V i ( P ) , A. /I E R,
with r ( J x ,) = W . Nonetheless, it should be emphasized that given a Jacobi
class W E X(P), there may be no Jacobi field J in any geometric realization
V ( G ) for G ( P ) with z ( J ) = W . On the other hand, there will always exist
a Jacobi field J E VL(P) with r ( J ) = W . But it may be necessary for J
to have a component in ! /? ' I . The reason for this is made precise in Lemma
10.52. In the next series of lemrnas, we will use the geometric realization
VL(p ) = [pl] @ V ( p ) defined in (10.29).
Lemma 10.51. Let W be a Jacobi class of vector fields in G(@). Then
there is a smooth Jacobi field Y E VL(0 ) with K(Y) = W . Conversely, i f Y is
a Jacobi field in VL (P) , then r (Y) is a Jacobi class in G(B).
Proof. The second part of the lemma is clear, for if Y" + R(Y,Pt)j3' = 0,
then [P'] = r ( 0 ) = 7i(Y1' + R(Y, PI)@) = ( a ( Y ) ) l l + z(=(Y), PI)@.
I t remains to establish the first part of the lemma. Given the Jacobi class
W. let Yl be a smooth vector field in the geometric realization V ( P ) with
.ir(Y1) = W . Slnce W 1 ' + ~ ( W , PI)@' = [,@I in X(@), there is a smooth function
f : [a, b ] 4 R such that Yl" + R(Y1,pt)p' = fP1. Let h : [a,b] -. R be a
smooth function with h" = f and set Y = Yl - hp'. Then r (Y) = W , and
f 0' = Yl " + R(&, /'3")P1 = (Y + hfi')" + R ( Y + hot , P1)P'
= Y" + h"P1 + R(V, Pt)P' = Y" + fP'+ R(Y,P1)/3'.
Therefore 0 = Y ' I -+ R(V, ,B')/?' as required. O
10.3 THE NULL MORSE INDEX THEORY 373
A more precise relationship between Jacobi fields in the geometric realiza-
tion V(0) of G ( P ) and Jacobi classes in X(0) is given by the following lemma.
Lemma 10.52. Let W he a Jacobi class in X(P). Then there is a Jacohi
field J E V ( P ) with r ( J ) = W iff the geometric realization 6(W) of W in V(0) satisfies the condition R(O(W),/3')p'/t E V ( P ( t ) ) for all t E [a, b ] .
Proof. If J E V(P) and n ( J ) = W , we have @(W) = J. But as J is a Jacobi
field, we then obtain
since J E V(P).
Now suppose that R ( @ ( W ) , /?')dl E V ( @ ) and J = 0 ( W ) . Then R ( J , /3')P1 =
R(O(W),,B')P1. Since W " + f i(W, P1)j3' = [PI] in G(0), we know that J must
satisfy a differential equation of the form J" + R(J,P1)P1 = fP1 in V L ( P ) .
But if R(0(V),j3')fi1 = R(J,P1)@' E V(P). the vector field J" -R(J,P)/3 ' is in
V ( P ) Hence by the deconlposition (10.29), J" + R(J,P1),D' = 0. 0
For the purpose of studying conjugate points, it is helpful to prove the
following refinement of Lemma 10.51 [cf. Bolts (1977, pp. 43-44 )].
Lemma 10.53. Let W E X(P) be a Jacobi class with W ( a ) = [P1(a)] and
W ( b ) = [/3'(b)]. Then there is a unique Jacobi field Z E vL(P) with ~ ( 2 ) = W
and Z(a) = Z(b) = 0.
Proof. From Lemma 10.51, we know that there exists a Jacobi field Y E
VL(/3) with a ( Y ) = W . However, we do not know that Y ( a ) = Y ( b ) = 0. But
for any constants A, p E R, the vector field Y + XB' + /1t/3' is also a Jacobi field
in V-(a) with 7i(Y + X P t + pt?) = W . Since x ( Y ) = W , W ( a ) = [/.?'(a)],
and W ( b ) = [,B'(b)], we know that Y ( a ) = c1,O1(a) and Y ( b ) = c2,B1(b) for
some constants cl,c2 E W. Choosing X = (cza - c ~ b ) ( b - a)-' and p =
b-I [(clb - c2a) ( b - a)-' - c2], it follows easily that Z = Y + X,Bt - pt9' satlefies
Z ( a ) = Z(b) = 0.
For uniqueness, suppose that Zl is a second Jacobi field in VL(@) with
r ( Z 1 ) = W and Zl ( a ) = 21 (b) = 0. Then X = Z1 - Z is a Jacob) field of the
form X = hp' with X ( a ) = X ( b ) = 0. Since 0 = X" + R(X,f l ) ,B1 = h"O1 i
374 25 MORSE INDEX THEORY ON LORENTZIAN MANIFOLDS
hR(P1, P1)F = h1'pP', it follows that h is an xffine function. As h ( a ) = h(b) = 0,
we must have iz = 0. Therefore Z 1 = Z as required. C3
With Lemma 10.53 in hand, it is now possible to make the following defi-
nition. Recall also that if J is any Jacobi field along ,f3 with J ( t l ) = J ( t 2 ) = 0
for t l # t2, then J E Vi(p) (cf. Lemma 10.46).
Definition 10.54. (Conjugate Point) Let ,B : [a, b] -+ (M.g ) be a null
geodesic. For tl # t2 in [a, b ] , tl and t2 are said to be conjugate dong /3
if there exists a Jacobi class W # [P'] in X(0) with W ( t l ) = [B1( t l ) ] and
W(t2) = [pl( t2)] . Also t l E (a, b ] is said to be a conjugate point of ,B if t = a
and tl are conjugate along 8. Let
Jt(P) = (Jacobi fields Y along P : Y ( a ) = Y(t) = 0)
and
- J@) = (Jacobi classes W E X(P) : W ( a ) = [P1(a)] and
W(t) = IB1(t)j).
The11 J t (P) C vi(@) and tl and t 2 are conjugate along f l in the sense of
Definition 10.54 if and only if there exists a nontrivial Jacobi field J along /3
with J( t1 ) = J( t2 ) = 0. Also we obtain the following important result from
the uniqueness in Lemma 10.53.
Corollary 10.55. The natural projection map T : J t@) - J,(,L?) is an
isomorphism for each t E (a, b ] . Thus Tt(P) is finite-dimensional, and also
dim Jt(@ = dim &(p) for all t E (a, b].
We are now ready to study the index form of the null geodesic /3 : la, b ] --+
(M,g) . For the geometric interpretation of the index form, ~t will be useful
to introduce a functionai analogous to the arc length functional for timelike
geodesics, namely the energy functional. The reason for using energy rather
than arc length is simply that the derivative of the function f (2) = f i does
not exist at x = 0, but J--g(,l3'(t), ,@(t)) = 5 for all t E [a, b] if /3 is a null
geodesic.
15.3 THE NULL MORSE INDEX THEORY 3
Definition 10.56. (Energy Functzon) Let y : [ c , d ] -t (M,g) be
smooth nonspacelike curve. The smooth mapping E- : [c, d ] -+ R given by
is called the eneqy finct~on of 7, and the number E ( y ) = E,(d) is called ti
energy of y.
The energy of a piecewise smooth nonspacelike curve y is calculated 1
summing the energies over the intervals on which -/ is smooth just as in formu
(4.1) for the arc length functional. If we let Ly( t ) = L(y jc , t ] ) , then tl
Cauchy-Schwarz inequality yields
Also. equality holds in (10.38) if and only if Ily'(t) 1 1 1s constant. Thus equals
holds for null or timelike geodesics.
Recall that V-(8) denotes the space of piecewise smooth vector fields
along ,B that are orthogonal to P', and @(@) = (Y E ~ ' ( 9 ) : Y ( a )
0 and Y ( b ) = 0).
Definition 10.57. (Index F o n ) The index f o n I : v-'(@) x V L ( P ) -
W of the null geodesic P : [a, b ] M is given by
b
(10.39) ((X'. Y ' ) - ( R ( X , P')P1. Y ) ) dt
for any X . Y E V-' ( p ) .
Formula (10.39) may be integrated by parts just as in the timelike c a
(cf. Remark 10.5) to give the alternative but equivalent definition of the i~
dex form used in Hawking and Ellis (1973, p. 114) and Bolts (1977, p. 110
Explicitly,
376 10 MORSE INDEX THEORY ON LORENTZIAN MANIFOLDS
where a partition a = to < tl < - .. < t k P l < t k = b has been chosen such that
X 1 It,, taTlj is smooth for i = 0: 1 , 2 , . . . , k - 1 and such that
A,, ( X I ) = X' (a f ),
A,, ( X ' ) = -X'(b-),
and
At, (X') = lim X1( t ) - lim X1( t ) t-+t;t t-t;
f o r i = 1 , 2 ,..., k - 1 .
If a is a variation of the null geodesic ,@ such that the neighboring curves
are all null, then all derivatives of the energy functional vanish at s = 0 since
E(a,) = 0 for all s. Thus it is customary to restrict attention to the following
class of variations.
Definition 10.58. (Admisszble Variation) A piecewise smooth variation
cu : [a, b ] x ( - 6 , E) 4 A4 of a piecewise smooth nonspacelike curve /3 : [a, b] -+
M is said to be admissible if all the neighboring curves a, : [a, b ] + M given
by a,(t) = aft, s ) are timelike for each s # 0 in ( - 6 , E ) .
Now suppose that for W E VL(P), there exists an admissible variation
a . [a, b ] x ( - E , E) -+ M with variation vector field a, d / d ~ / ( , , ~ ) = W ( t ) for each
t E [a, b ] . Let E(u,) be the energy of the neighboring curve a , : [a, b] -- M
for each s E ( - E , t). Then d/ds [E(a , ) ] Is=, = 0 since p is a geodesic and
As E(P) = E(ao) = 0. we must have d2/ds2 [E(a , ) ] I,=, > 0. Thus a
necessary condition for W E VL(/3) to be the variation vector field of some
admissible variation a of /3 is that I(TN, W ) 2 0. Note also that if the future
null cut point to P(a) along P comes after P(b). then there are no admissible
proper variations of P (cf. Corollary 4.14 and Section 9.2).
It is immediate from (10.39) that if X E I/dF($') is a vector field of the form
X ( t ) = f (t),B1(t) for any piecewise smooth function : [a, b] -+ R with f (a ) =
f (b) = 0. and Y 6 V&(P) is arbitrary, then I ( X , Y) = 0. Thm the index forrr
I : Vo-(P) x V$($) -7 W is never negative definite and even worse, alwaytys ha
an infinite-dimensional null space. This suggests that the index form should be
projected to G(D) [cf. Ifawking and Ellis (1973. p. 114). Bolts (1977, p. 111)]
Recall that the notation X(@) was introduced for piecewise smooth sections o
G(P), and Xo(P) = {W E X ( P ) : W ( a ) = [P'(a)l, W ( b ) = [P'fb)] 1. Definition 10.58. (Quotient Bundle Index F o n n ) The indea: form
7 : X(p) x X ( p ) 3 R is given by
(10.42) [3(V', W') - 3 @(v, P1)P1, W ) ] dt
where V , W E X(P) and g R, and the covariant differentiation of sectlons of
G(P) are given by (10.31), (10.34), and (10.32). respectively.
Just as for the index form I : V L ( P ) x V L ( p ) 4 IR, formula (10.42) may
be integrated by parts to give the formula
where a partition a = t o < tl < . - . < < tk = 1, has been chosen such that
Vl[t , , t,,l] is smooth for i = 0,1, . . . . k - 1. In particular, if V is a smooth
section of G(P) we obtain
and if V is a Jacobi class in X(P), we have
Also, since vector fields of the form f ( t)P1(t) are in the null spzce of the index
form I . V L ( P ) x Vi(P) -+ R, for any X,Y E vL(,B) with a ( X ) = V and
sr(Y) = W , it follows that
378 10 MORSE INDEX THEORY ON LORENTZIAN MANIFOLCS
where the index on the left-hand side is calculated in X ( P ) and on the right-
hand side in V L ( B ) .
We saw in Section 10.1 that for timelike geodesic segments. the null space of
the index form consists preciseljr of the smooth Jacobi fields vanishing at both
endpoints. We now establish the analogous result for the index form 7 on the
quotient space Xo(Y) for an arbitrary null geodesic segment : la, b ] --+ M.
Theorem 10.60. For piecewise smooth vector classes W E Xo(P) , the
following are equimlent:
(1) W is a smooth Jacobi class in Xo(/3).
( 2 ) I ( W, Z) = 0 for all Z 5 Xo (9).
Proof. First. (1) 5 (2) is clear from formula (10.45). For the purpose
of showing (2) + ( I ) , fix the unique piecewise smooth vector field Y In the
geometric realization V ( P ) for G ( P ) given by (10.28) with r ( Y ) = W and
Y ( a ) = Y ( b ) = 0. Let a = to < t1 < . . - < t k = b be a finite partition of [a , b ]
such that Y / [t3,t3+1] is smooth for J = 0 ,1 , . . . , k - 1. Let p : [a, b ] --+ R be
a smooth function with p(t3) = 0 for each j with 0 5 j 5 i, and p ( t ) > 0
otherwise. Set Z = 7r(p(Y1' + R(Y, /3 ' )Pf) ) E Xo(P). Then from (10.43) we
obtain
Remembering that 3 is positive definite (cf. Remark 10.48), we obtain Wtl(tj+ - R ( W ( t ) , P 1 ( t ) ) P ' ( t ) = [,Bf(t)] except possibly at the t3's. Thus JV / [tl , t,+l] is
a smooth Jacobi class for each j. To show that W fits together at the tJ's to
form a smooth Jacobi class on all of [a, b ] , it is enough to show that the vector
field Y in the geometric realization V ( P ) representing W is a 6' vector field
First observe that since W1'(t) R(w ( t ) . /3'(t))P1(t) = [PI (t)] except possiblj
at the t,'s, we have using (10.43) that
k - 1
0 = T(w, Z ) = r;: 3 ( Z ( t , ) . A , ( W 1 ) ) j=1
for any Z E Xo(j3). Since Y E V(P) , we have ( Y ( t ) , P1( t ) ) = ( Y ( t ) , ~ ( t ) : = 0 for
all t E [a, b ] . Thus ( Y r ( t ) , B / ( t ) ) = ( Y 1 ( t ) , 7 ( t ) ) = 0 for t $ ( to , t l . . . . , t k ) . and
10.3 THE NULL MORSE INDEX THEORY 379
it follows by continuity that AtJ ( Y ' ) E V ( P ( t 3 ) ) for each j = 1.2,. . . , k-1. Let
X E V(P) be a smooth vector field urith X ( a ) = X ( b ) = 0 and X ( t 3 ) = A,, ( Y 1 )
for each j = 1 ,2 , . . . , k - 1. Set Z = T ( X ) E X o ( P ) Then we obtain
which yields At, ( Y 1 ) = 0 for 3 = 1,2, . . . , k - I. Hence Y' is continuous at the
t,'s as required. P
Notice that in the first part of the proof of ( 2 ) + (1) of Theorem 10.60, we
do not know that Y" + R ( Y . f l ) Y E V(P) even though Y E V(O). Thus we
cannot conclude that Y is a Jacobi field except at the t,'s. Thls is precisely
the point In passing to the quotient bundle G ( P ) in which W" + z(W, pP')P' lies in G ( P ) by construction and the induced metric g is positive definite on
G ( P ) x G ( P ) . The aim of the next portion of this section is to prove the important result
that the index form 7 : Xo(P) x Xo(/3) -t R is negative definite if and only if
there are no conjugate points to t = a along ,B in ju, b 1. A proof of this result,
which is implicitly given in Propositions 4.5.11 and 4 5.12 of Hawking and
Ells (1973), is given in complete detail in Bolts (1977. pp. 117-123). Because
the proof of negative definiteness differs considerably from the corresponding
proof (cf. Theorem 10.22) for the timelike geodesics, we briefly outline the
proof in the timelike case in order to clarlfy the differences. Recall that w-e
first showed that arbitrary proper piecewise smooth variations a of a timelike
geodesic segment have timelike neighboring curves a, for sufKciently small s It
was then a conseqnence of the Gauss Lemma and the inverse function theorem
that if c : u, b] --+ M is a future direc:ed timelike geodesic segment without
conjugate points. then for any proper piecewise smooth variation a of cj the
neighboring curves a, satisfy L(a , ) 5 L(c ) for all .s isufficiently small This
implied that the lndex form is negative semidefinite In the absence of conjugate
points. Algebra~callv, we were then able to prove the negative definiteness.
Conversely. if c had a conjugate point in (a, b ] we produced a piecewise smooth
vector field Z E Vg'(c) of zero index csing a nontrivial Jacobi field guaranteed
by the existence of a conjugate point to t = a.
382 10 MORSE INDEX THEORY ON LORENTZIAN MANIFOLDS
For the purpose of the null index theory, however, we need to consider
(1,1) tensor fields not on VL(P) but rather on the quotient bundle G(P). A
(1, l) tensor field 2 = x( t ) : G(P(t)) -+ G(P(t)) is a linear map for each
t E [a, b ] which maps vector classes to vector classes Using the projection
of the covariant derivative to X(P) [cf. equation (10.32)], we may differentiate
piecewise smooth (1,1) tensor fields in G(P(t)) and obtain the formulas
and
provided is nonsingular. Since the projected metric 3 : G(P(-t)) x G(O(t)) +
R is positive definite, we may also define the adjoznt x* = z * ( t ) to the (1,1)
tensor field A(t) for G(P(t)) by the formula
for all vector classes Z, W E G(P(t)) and all t E [a, b ] . We may also define the
composed endomorphism fjz : G(P(t)) --, G(P(t)) by
where 3 is the projected curvature operator given by (10.34). Also the kernel
ker ( X ( t ) ) of the (1.1) tensor field z(t) : G(P(t)) -+ G(P(t)) is the vector
space
-4 (1,1) tensor field x(t) : G(fl(t)) -, G(F(t)) is the zero tensor field, written - A = 0, if z(t)(w) = [P1(t)] for all w E G(P(tf) and all t E [a, b].
With these preliminaries out of the way, we are ready to define Jacobi and
Lagrange tensor fields.
10.3 THE NULL MOXSE INDEX THEORY 383
Definition 10.61. (Jacob2 Tensor Fzeld) A snlooth (1.1) tensor field - A : G(P) G(P) is said to be a Jacobz tensor if 3 satisfies the conditions
and
for all t E [a, b].
Condition (10.56) has the consequence that ~f Y E X(P) is any parallel
vector class along P: then J = A(Y) is a Jacobi class along 8. This follows
since
J" + B(J, O')pl = Z"(Y) + Z A ( y )
= (A" + R;?)(Y) = 0
using Y' = 0 Condition (10 57) has the follow~ng impllcatlon if Yl , Yn-2 -
are linearly Independent parallel sections of X(P). then ~ ( Y I ) . . , A(Yn-2) are
!inearly Independent Jacobi sect~ons in the following sense. If X I . . . , are
real constants such that
for a l l t E [ a , b ] , thenX1 = A 2 =. . ,=X, -2=0 .
The converse of the above construction may be used to show that Jacobi
tensors exist,. Fbr let El. Ez, . . . . En-2 be the spacehke parallel fields spanning
V(P) chosen ?n (10 28). Let z, = .;r(E,) be the corresponding parallel vector - - -
class ~ I I X(P) Also let J 1 , J z , . . . , Jr.-2 be the Jacobi classes along P with
initial conditions 5,(a) = [@'(a)] and TZf (a ) = F,(a) for z = 1.2 , . . . ,n - 2.
A Jacobi tensor 2 satisfying the initial conditions x ( a ) = 0, x1(a ) = Id may
then be defined by requiring that
384 10 MORSE INDEX THEORY ON LORENTZIAN MANIFOLDS
for each i = 1,2. . . . , n - 2 and extending to all G(P) by linearity. Since the - JI3s are Jacobi classes and the z ' s are parallel classes in X(P), it follows that - A satisfies (10.56). To check that 3 satisfies (10.571, suppose that x ( t ) ( ~ ) = - 1 A (tj(?;) = [P1(t)] for some E G(P(t)). Choose the unique v E V(P(t)) with
~ ( v ) = 'ii, and write v = ~rz x,E,(~). Then 'ii = X,F,(t), and we
obtain [P1(t)] = -A(%) = ~:=;2 ~ ~ 5 % ( t ) and
Thus 7 = ~:_;2 ~ ~ 5 % is a Jacobi class in G(P) with J ( t ) = 5'(t) = [P1(t)].
Hence 7 = [p'j. Thus X,Z,(s) = [P1(s)] for all s E [a, b], contradicting - - -
the linear independence of the parallel classes El, Ez, . . . , Therefore
ker(a(s)) n ker(zl(s)) = [@'(a)] for all s E [a, b] as required.
I t is also useful to note the following result.
Lemma 10.62- ker(x(so)) n ker(xl(so)) = [pl(so)] for some so E [a, b] iff - A satisfies condition (10.57).
Proof. Suppose v E ker(X(s0)) n ker(X'(so)), v # [P1(so)]. Let Y be the
parallel class in T(,/i) with Y(so) = u. Then J = Z(Y) is a Jacobi class
satisfying J(so) = A(u) = {@'(so)] and .J1(.so) = ~ ' ( z J ) = !P1(so)]. Hence
J = [ P I ] from which it follows that Y(t) E ker(z(t)) n ker(xl(t)) for all
t ~ [ a , b ] .
The following two lemmas are not d~fficult to obtain from the relationship
between parallel classes, Jacobi classes, and Jacobi tensors indicated above
icf. Bolts (19771, p. 28 and p. 49)j.
Lemma 10.63. The points @ ( t o ) and $(tl) are conjugate along the null
geodesic P : [a, b -+ M iff there exists a Jacobi tensor x : G(P) -+ G(0) with - A(to) = 0: z'(t0) = Id, and ker(Z(t1)) # (p(tl)]).
Lemrna 10.64. Let /3 : [a,b] -, A4 be a null geodesic without eonju-
gate points to t = a. The= there exists a unique smooth (1, l) tensor field - A : G(P) -+ G(P) satisfying the differential equation x" + RZ = 0 with
10 3 THE NULL MORSE INDEX THEORY 385
given boundary conditions A(a) : G(B(n)) - G(/?(a)) and A(b) . G(B(b)) --+
G(a(b)).
Proof, Let J' denote the space of (1.1) tensor fields in G(0) satisfying the
differential equation 2" i- ?iz = 0. A linear map $ : J' -+ L(G(P(a))) x
L(G(P(b))) may then be defined by Q ( l ) = (z(a),A(b)). Since P : [a, b] -, M
has no conjugate points, Qj is injective. For if +(x) = (0.0) and Y is any
parallel vector class in X(P), then J = X(Y) is a Jacobi class with J ( a ) =
[@'(a)] and J(b) = [Fi(b)] so that J = [$'I. Slnce b is an injective h e a r map
and dim J' = ddim[L(G(P(a))) x L(G(P(b))) 1, it follows that rp is surjective. 13
Even though R = R* and (A1)* = ( A * ) ' , the adjoint 3' of a Jacobi - -*
tensor 3 is not necessarily a Jacobi tensor since (zx)* ji R A in general
Rather, ( R z )* = 2 *R. Nonetheless, Jacobi tensors and thelr adjoints have
the following useful property which is conveniently stated using the Tironskian
tensor field [cf. Eschenburg and O'Sullivan (1976, p 226), Bolts (1977. p. 23))
Definition 10.65. (Wronskian) Let 2 and B be two Jacobi tensors - -
along G(@). Then their Wronskian W(A, B) is the ( I l l ) tensor field along
G(0) given by
It follows from the fact that R* = R and equations (10 51) and (10.56) that - -
if 2 and B are any two Jacobi tensor fields along G(,f3), then [ W ( A ; B)!' = 0. - -
Thus W ( A, B ) is a constant tensor field. It is then natural to consider the
following subclass of Jacobi tensors.
Definition 10.65. (Lagraege Tensor Fzekd) X Jacobi tenbur field 2 along G(P) is said to be a Lagrange tensor field if W ( Z , Z ) = 0.
For the proof of Proposition 10.68, W P need the following consequence of
Definition 10.66.
Lemma 10.67. Let 2 be a Jacobi tensor along G(P). If x(so) = 0 for
some 30 E [a, b], then 2 is a Lagrange tensor, and in particular,
386 10 MORSE INDEX TEEORY ON LORENTZIAN MAXIFOLDS
Proof. We know that W ( 2 , X ) is a constant tensor already. But if Z ( s o ) = - -
0: then z*(so) = 0 also, and iletlce W ( x , X ) ( s o ) = 0. Thus W ( A , A ) = 0 as
required.
JVe are now ready to prove the following important proposition
Proposition 10.68. Let ,i3 : [a, b ] -+ M be a null geodesic without conju-
gate points to t = a in (a , b ] . Tl~eu P(W, W ) < 0 for all W E Xo(P), T.I' # [P'].
Proof. Let ';l be the Jacobi tensor along G(P) with initial conditions X ( a ) =
0 and ~ ' ( a ) = Id. Since ,!3 has no conjugate points, Lemma 10 63 implies that
k e r ( z ( t ) ) = ( [ p l ( t ) ] ) for all t E (a, b ]
Now let W E Xo(/3) be arbitrary. Since 2 is nonsingular in (a , b ] and
W ( a ) = ip1(a)], we may find Z E X ( 0 ) with W = x(Z). By the rules for
covariant differentiation of tensor fields. we obtain
and
Now we are ready to calculate ?(w; W ) . First
h
=-i [ g ( 7 i i ( 2 ) , X 1 ( 2 ) ) -k 2 3(Z1(2) , Z(2 ' ) )
+ i j ( Z ( z ' ) . X ( z f ) ) + B ( ~ ~ ' ( Z ) , A ( Z ) ) ] dt
using (10.56) and (10.61). Now
~ ( X " ( Z ) . ~ ( 2 ) ) = ij([Z1(.Z)]'. ~ ( 2 ) ) - i j ( A ' ( Z 1 j , ~ ( 2 ) )
= [g (Z1 ( z ) , Z(z));l - ~ ( Z ' ( Z ) > [ X ( Z ) 1') - g(Z1(z ' ) ,B(z) )
= [T ( 2 ' ( ~ ) , Z ( z j ) ] ~ - g ( A ' ( z ) , Z ( z l ) ) - 3(A1(z),X'(z))
- 3(A1(z1) ,A(zj) .
10.3 THE NULL MORSE INDEX THEORY
Substituting into the above formula for 7 ( ~ : w) then yields
Now the first term vanishes since W ( a ) = :,!3'(a)] and W(b) = [pl(b)]. Thus
we obtain
using (10.60). If Z i ( t ) = 0 for all t , then Z is parallel along 8 and hence
W = X ( Z ) would be a nontrivial Jacobi vector class along /3 with W ( a ) =
lP1(a)] znd W(b) = [/3'(b)]. Since 3 has no conjugate points in (a. l i ] , this is
impossible. Hence from the positive definiteness of 3 and the nonsingularity
of A ( t ) for t E (a , b ] , we obta~n ?(I.I/, W ) < 0 as required.
This proposition has the well-known geometric consequence in general rel-
ativity that if /3 : [a, b ] -+ (M, g) is free of conjugate points, then no "small"
variation of f l gives a timelike curve from p to q [cf. Hawking and Ellis (1973:
p. 115)].
We are now ready to prove the fnllowing theorem
Theorem 10.69. Let O : [a, b ] -, M be a null geodesic segment Then the
following are equivalent.
(1) The segment ,t3 has no conjugate points to t = a in (a, b ] .
( 2 ) f(W, W ) 4 0 for dl W E Xo(P), W # [pi:.
388 10 MORSE INDEX TWEGRY ON LOFtENTZIAN MANIFOLDS
Proof. We have already shown ( 1 ) + (2) in the proof of Proposition 10.68.
To show (2) + ( I ) , we suppose @(a) is conjugate to P(to) with O < t o 5 b
and produce a nontrivial vector class W E Xo(p) with?(W, W ) 2 0. If ,/3(a) is
conjugate to P(tof, we know that there is a nontrivial Jaeobi cl= Z E X(P)
with
Z(a) = /P1(a)] and Z(to) = [Pf(to)].
Set z(t) a < t L t o , W( t ) = ial(tji to t 5 b.
Then
- I(W, W ) = f (w, W)? -t 7(W, w)!',
= -g(Zi , 2)12 s 0 = 0
using (10.451, Z ( a ) = [PJ(a)] , and Z( t0) = [,fj"(to)]. Thus (2) + ( 1 ) holds.
-4s in the Riemannian and timelike cases (cf. Theorem 10.23); Theorem
10.69 has the following consequence which is essential to the proof of the
Morse Index Theorem for null geodesics.
Theorem 10.70 (Maxirnality of Jacobi Classes). Let ,5 : [a, b j --+ M be a null geodesic segment with no conjugate points t o t = a, and let J E X ( 0 ) be any Jacobi class. Then for any vector class Y E X ( P ) with Y f J ,
(10.63) Y ( a ) = J ( a ) , and Y ( b ) = J ( b )
we have
- (10.64) I ( J ; 3) > f (Y, Y).
Proof. This may be established just as in Theorem 10.23 using (10.45) and
Theorem 10.69. U
Since the canonical variation (10.5) of Section 10.1 applied to a vector fie
W E V'(@ is not necessarily an admissible variation of f i in the sense of D
inition 10.58, Theorem 10.69 do- not imply the existence of a timelike c
10.3 T a E NULL MORSE INDEX THEORY 38s
from @(a) to P(b) provided that t = a 1s conjugate to some to E ( a , 6) along
p. We thus give a separate proof of this result m Theorem 10.72 [cf. Hawking
and Ellis (1973, pp. 115-116), Bolts (1977, pp. 117-121); It 1s first helpfu
to derive conditions for a proper varlstion of a null geodeslc to be admissible
For convenience, we will assume that /3 : [O. 11 -+ M so that the formulas lr
the proof of Theorem 10.72 will be simpler.
Now let a : jO. 1) x ( - t , ~ ) -+ M be a piecewise smooth proper variation o
0 such that each neighboring curve a , ( t ) = a(t, s ) 1s future timelike for s # 0
Let d / d t and a i d s denote the coordinate vector fields on [O, I] x ( - E . e ) , anc
put V = cr,(d/as) and T = a , ( d / a t ) . Then T / ( t , o ) = P1( t ) , and Vl[,,,) is
called the vanatoon vector field of the variation CA of ,B. Since a is a proper
variation, V/(o,o) = V/(, ,) = 0. Also,
since g(T . T)j(,,,) < 0 for s # 0 and g(T, T)j(t,o) = g($( t ) , P ' ( t ) ) = 0. Calcu-
lating,
Thus since ,3 is a geodesic, we obtain
Now f (t) = g(V. T)l( t ,o) is a piecewise smooth function with f ( 0 ) = f ( 1 ) = 0.
Hence if f ( t ) # 0 for some t 6 10, 11, there exists a to E (O,1) with f l ( t o ) > 0.
Then
390 10 MORSE INDEX THEORY ON LORENTZIAN MANIFOLDS
in contradiction to (10.65). Hence we obtain as a first necessary condition for
a to be an admissible deformation of ,6 that
for all t E [0, I]. Thus V E V-(0). Consequently
for all t E [O, I]. It then follows from (10.67) that the neighboring curves a, of
the variation will be timelike provided that the variation vector field satisfies
(10.66), (10.67). and the condition that d 2 / d s 2 ( g ( T , T))l(t,O) < -c < 0 for all
t E (0: 1) . As above,
Hence
Using the identities
and
ValatTllt ,) = 'ValatP1lt = 0 ,
10.3 THE NULL MORSE INDEX THEORY
we obtain
This calculation yields the following lemma.
Lemma 10.11. A sufficient condition for the plecewise smooth proper vari-
ation CL. [0,1] x ( - e , ~ ) -+ A4 of the null geodes~c D [0, I] -+ M to he an
adrnlsslble varlatlon [I e , the curves cx,(t) = a(t s ) are tlrnellke for s j O] for
a11 s # 0 suficientlysmall i s that the vanation vector field V ( t ) = ~ ~ . , d / d $ l ( ~ , ~ )
satisfies the condltlons
(10.68) g ( ~ ( t ) , p ' ( t ) ) = 0 for all t E {O,I]:
(10.69) d
- ( g ( T T ) ) / ( t , O ) = 0 for f E [O,l:; and d s
for all t E (O,1) at which V is smooth.
We are now ready to prove the desired result [cf. Hawking and Ellis (1973,
p. 115))
Theorem 10.72. Let P : [0,1] -+ M be a null geodesic. If ,!3(to) is conju-
gate to F(0) along /3 for some to E (0. I ) , then there is a timelike curve from
a(o f tool-1).
Proof. We uiill suppose that to > 0 is the first conjugate point of B(0) along
0. It is enough to show that for some t2 wlth to < ta < 1. there is a future
directed timelike curve from $(O) to $(t2). For then we have B(O) < B(tz) <
392 10 MORSE INDEX THEORY ON LORENTZIAN MANIFOLDS
P(1) whence P(O) < P ( 1 ) Thus there exists a timelike curve from B(0) to
P(1). Since P(to) is conjugate to B(0) dong ,D, there exists a smooth nontrivial
3acobi class W E X(3) with W ( 0 ) = {fl(O)] and W ( t o ) = [$'(to)]. W-e may
write
where I@ is a smooth -vector class along /? with g(%, I&') = 1 and f : [ O , I ] --+ B a smooth function. Since to is the first conjugate point along P, f (0) = f ( to) =
0 , and changing t%' to -T@ if necessary, we may assume that f ( t ) > 0 for
all t E (0,to). Since W is a nontrivial Jacobi class and W(to) = [$'/(to)].
we have Url(to) # [P1(to)]. Thus from the formula W1(to) = f l ( to)I&'(to) + f (to)@'(to) = f'(to)l.i/(to), we obtain fl(to) # 0. Hence we may choose
tl E ( to . 11 such that W ( t ) # $'(t)] and f (f) < O for all t E (to, t l ] .
With tl as above, the idea of the proof 1s now to show that there exists a t2 E
( to , t l ] such that there is an admissible proper variation cu : [0, t2] x ( -e , €1 - M
of ,B I [O, tz] . Then the neighboring curves a, of the variation a will be timelike
curves from P ( O ) to O ( t 2 ) for s $ 0 . This will then imply that F ( O ) < P(1) as
required. To this end, we want to deform W 1 [0, t l ] to a vector class Z 1 [I). t2] - - 11
with 7.j ( 2. Z +a@, ot)@l) > 0 so that if Z E V L ( P ) is an appropriate lift of - Z E X($ / [0, t z ] ) and a is a variation of ,B 1 [O: tz] with variation vector field Z ,
then conditions (10.69) and (10.70) of Lemma 10.71 will be satisfied.
Consider a vector class of the form
with b = - f ( t l )(eatl - I)- ' E R and a > O in R chosen such that
(a2 + glb(h(t) : t E [O, t l ] ) ) > 0,
where
h(;) = g(* + K(I@,B')P', 1,.
10.3 THE NULL lvIORSE INDEX THEORY
Because W is a Jacobi class, we obtain
o = 3 (w", a(w pl)pt , %)
= fIi + 2 ft3 (w,@-) + f g ( W , @) + f 3 (R(*,pl)P'.
= f"+2f1?j(lV'.*)+ fh .
But since g($, @) = a(g(@, I@))' = $(l) ' = 0, we obtain the formllla f" =
- fh. Returning to consideration of the vector class 2, first note that by choice
of the constants a and b, we have z ( 0 ) = [P1(0)] and z ( t l ) = [P1(tl)]. In view - - / I --
of formula (10.70), we wish to have g(Z, Z + R(Z,P1)P1) > 0 also. Setting
r ( t ) = b(eat - 1) + f ( t ) remembering that 7j(lvi/', I%') = 0. and dderentiating
yields ij(Z, Z1' + x (Z , p l )p l ) = r(rU + r h ) = r[beat(n2 + h) - hh + fl ' + f h].
Since f" = - f h. we obtain
Noiv b = - f (tl)(eatl - 1) > O as f ( t l ) < 0. so the expression b(eat[a2 + - - I / - -
h( t )] - h( t ) ) > 0 for all t E [0, t l ] . Thus g(Z , Z + R ( Z , ,LY)O1)It > O provided
r ( t ) > 0. Since f ( t) > 0 for t E (O,to), we have r(t) > 0 for t E (0, to]. By
continuity, there is some t z > to with r ( t ) > 0 for t E [to, t z ) and r ( t z ) = 0. If
t 2 > t l r then in fact t 2 = t l since r ( t l ) = 0 by construction, and we let tz = tl below. If tz < tl , then the vector class Z / 10, tz] will satisfy Z ( t a ) = [Pr(t2)]
- - , I -- since r ( t z ) = 0 and also g ( Z , Z + R ( Z . P')F1)lt > 0 for all t E (0, t a ) We now
work with ,B 1 [O, t2] .
Let 2 E V L ( P I [0, t z ] ) satisfy a ( 2 ) - = z. Since z(0) = [,LY(O)] and Z( t2 ) =
[pt ( t2)] , we have Z(0) = pP1(0) and Z(t2) = XP1(t-,) for some constants p. X E
R. Set Z = 2 - p,8' + [ ( j i - X)/t2]t01. Then Z(0) = Z ( t a ) = 0 and 7i(Z) = 2. Consequently
for all t E (0 , t 2 ) . Chowe a constant E > O SO that
(10.73) e < glb {g (Z1 ' ( t ) + R(z( t ) .P1( t ) )P ' f t ) . Z(t ) ) : t G [?, 21)
394 10 MORSE INDEX THEORY ON LORENTZIAN MANIFOLDS
which is possible in view of (10.72). Now define a function p : [O: tP] -+ R by
- €t o i t i 2 ,
€ ( t - 2 ) h < t < & 4 - - 4 ,
~ ( t 2 - t ) 9 5L5t2.
We now have a given vector field Z E V&(P / [0, t z ] ) and a given function
,G : (0.t2] -+ R. Recall that we had fixed a pseudo-orthonormal frame field
El. E2,. . . . En-?. 7, @' for ,6 with (? ,p i ) = -1 in (10.28). We now need to
find a proper variation a : [O, tz] x ( - E , E ) --+ M of ,B / [O, t2] satisfying the initial
conditions
and
for all t E [O, t2]. Thus we wish to specify the first and second derivatives of
the curves s -+ a ( t , s) for each t E [O, tzj. The existence of such a deformation
is guaranteed by the theory of differential equations applied to (10.74) and
(10.75) written out in terms of a Fermi coordinate system for the geodesic
defined by the pseudo-orthonormal frame El. E2, . . . , En-2, 7, pl.
Given the proper variation cu of ,O 1 [O, tz] satisfying (10.74) and (10.75) arid
setting T = a,a/dt and V = a,d/ds as above, it follows that
Hence
ost<g, d - (g(Va/asV, p') + Q(V, V')) /(,,o) = ~ ' ( t ) = dt
? s t < ? , y<t<tz
Thus in vlew of (10.73), the variation a of P I [O. ta] satisfies condition (10.70)
of Lemma 10.71. Applying Lemma 10 71, we find that this variation produces
timelike curves a, from P(0) to P(t2) for smail s f 0 as required. U
10.3 THE NULL MORSE INDEX THEORY 395
Corollary 10.73. The null cut point of ,B : [O, a ) -r M comes at or before
the first nul l conjugate point.
We are at last ready to turn to the proof of the T\/iorse Index Theorem for
null geodesics The proof parallels that of the Timelike Morse Index Theorem.
Theorem 10.27. In view of Theorem 10.69, the index ind(0) and the extended
index Indo(@) of p with respect to the index form 7 : Xo(P) x Xo(P) -+ R
should be defined as follows.
Definition 10.7'4. (Index and Extended index) The zndex Ind(P) and
extended znde.c Indo(P) of P with respect to the index form -f : Xo(O) xXo(0 ) --+
R are given by
Ind(p) = lub {dim A : A is a subspace of Xo(p)
and 1 A x A is positive definite )
and
Indo(Oj = lub {dim A : A is a subspace of Xo(P)
and f 1 A x A is positive semidefinite ) ,
respectively.
As a preliminary step toward establishing the null index theorem, we need
the following lemma.
Lemma 10.75. If ,l3 / [5 , t ] is free of conjugate points, then give11 ary v f G(P(s) ) and iij E G(P( t ) ) , there is a unique Jacobi class Z E X(P) with
Z(5) = ij dnd Z(t) = a.
Proof. Let v E V(P(s)) and zu 6 V(P(1)) satisfy ~ ( v ) = Ti and ~ ( w ) = E
By the nonconjugacy hypothesis, there is a unique Jacohi field J 6 VL(/?)
with .7(.9) = a and J ( t ) = a,. Then Z = ~ ( 5 ) is a Jacobi class in X ( 3 ) with
Z(s) = 5 and Z ( t ) = E.
Suppose now that Zl is a second Jacobi class in X(B) with Z l ( s ) = 5
and Zl(t) = E. By Lemma 10.51, there 1s a Jacobi field JI E VL(fi) wlth
7i(JI) = ZI. Sirice Zl(s) = B and Zl ( t ) = a, ~t follows that Jl (s) = v + clfl (s)
398 10 MORSE INDEX THEORY ON LOWNTZIAN MANIFOLDS
contradiction. Hence Ind'(0) < dimB as required, completing the proof of the
proposition. ti
Now that we have obtained Proposition 10.76, it i,i straightforward to see
that the proof of Theorem 10.27 of Section 10.1 may be applied to the index
form f : G(P) x G(P) -t R and the projected positive definite metric ?j to yield
the equalities
and
Since dimjt(/3) = dim Ji(/3) by Corollary 10.55, we have thus established the
following Morse Index Theorem for null geodesics.
Theorem 10.77. Let @ : [a, b] -t M be a null geodesjc in an arbitrary
space-time Let 7 : Xo(0) x Xo(P) -+ lft be the index form on plecewise
smooth sections of the quotient bundle G(P) defined in (10.42). Then 0 hhas only finitely inany conjugate points, and the index Ind(P) and extended index
Indo (P) off : Xo (P) x Xo (P) -t R are related to the geodesic index of the null
geodesic P by the formuias
Ind(P) = dim J t ( @
t € ( a , b )
and
where &(a) denotes the vector space of jacobi fields Y along P with Y ( a ) =
U ( t ) = 0.
Extensions of Theorem 10.77 to the focal case of a null geodesic segment
perpendicular to spacelike endmanifolds have been given irr Ehrlich and Kim
(1989a,b), Studies of spacelike conjugate points have recently been made in
Helfer (1934a,b).
CH.4PTER 11
SOME RESULTS I N GLOBAL LORENTZIAN GEOMETRY
In Chapter I1 we apply the techniques of the preceding chapters to obtain
Lorentzian analogues of two remarkable results in global Riemannian geometry.
The first, the Bonnet-Myers Diameter Theorem, asserts that if a complete Rie-
mannian manifold AT has everywhere positwe Rlccl curvature bounded away
from dero, then N is compact, has finite dlameter. and has finlte fundamen-
tal group. The second result, the Hadamard-Cartan Theorem, states that if
a corilplete Riernannlan manifold has everywhere nonpositlve sectional curva-
ture, then its universal cowring manifold is diffeomorphic to IRr" and thus the
higher hoinotopy groups r,(Ai, *) = (e) for z 2 2. In addition. the un~versal
covering space with the pullback Riemannian metric has the property that any
two points may be joined by exactlj one geodesic, up to reparametrization.
In Section 11.1 we consider the Loreritzian a~lalogue of the Bonnet-Myers
Theorem and in so doing, study the timelike diameter of space-times The
tzmelake dzameter diam(M, g ) of a space-time (M, y) is given by
diam(M. g) = sup(d(p, q) : p, q E A l ) .
Classes of space-times with finlte timellke dlameter, lncludlng the "Wheeler
universes," have been studied in general relativltg [cf. Tipler (1977c p. 500)j.
If a complete Riemannian manifold has finite dlameter, ~t is compact by the
Nopf-Rillow Theorem. Even so, all geodesics have znfinzte length as a result of
the metric completeness. But for spitce-times (M, g ) since L(y) 5 d(p. q) for
all future directed nonspacelike curtes 7 from p to q. evenj timelike geodesic
must satisfy L(r) < diam(M, g). Thus if a space-tlme ( M . g ) has finite timelike
diameter, all timelike geodesics have finite length and hence are incorilplete.
In particular, a space-time (M.g) with finite timelike diameter is timelike
geodesically incomplete.
3 99
400 11 SOME RESULTS IN GLOBAL LORENTZIAX GEOMETRY
Since we have adopted the signature convention (-, +, . . . , +) for the Lor-
entzian metric instead of (+, -, . - . , -), curvature conditions of positive (re-
spectively, negative) sectional curvature for Riemannian manifolds trw-slate
as curvature conditions of negative (respectively, positive) timelike sectional
curvature for Lorentzian manifo!ds.
Using the timelike index theory developed in Section 10.1, we obtain the
fol!owing Lorentzian analogue of the Bonnet-Myers Theorem for complete Rie-
mannian manifolds. Let ( M , g) be a globally hyperbolic space-time with either
(1) all nonspacelike Ricci curvatures positive and bounded away from zero, or
(2) all timelike sectional curvatures negative and bounded away from zero.
Then ( M , g) has finite timelike diameter.
In Section 11.2 we give Lorentman versions of two well-known comparison
theorems in Eemannian geometry, the index comparison theorem and the
Rauch Comparison Theorem. Using the latter of these two results, we are
able to give an easy proof (Corollary 11.12) of the basic fact that in a space-
time with everywhere nonnegative timelike sectional curvatures, the differen-
tial expp* of the exponential map,
is norm nondecreasing on nonsparelike tangent vectors.
In Section 11.3 u-e consider two analogues of the Nadamard-Cartan Theo-
1c space- rem. The first of these is for future one-connected globally hyperbol:
times. The space-time (M.g) is said to be future one-connected if for any
p, q E M with p < q, any two smooth. future directed timelike curves with
endpoints p and q are homotopic through smooth. future directed timelike
curves with endpoints p and q. Using the hlorse theory of the timelike path
space C(,.,) from Sectlon 10.2, it may be shown that if (M,g) is a future
one-connected globally hyperbolic space-time with no nonspacelike conjugate
points, then given any p, q E M with p << q, there is exactly one future directed
timelike geodesic segment (up to parametrization) from p to q (cf. Theorem
11.16).
A Lorentzian manifold is said to be geodesically connected if each pair of
distinct p o i ~ t s is joined by at least one geodesic segment. Global hyperbolic-
11.1 THE TIMELIKE DIAMETER 401
ity guarantees the existence of at least one (maximal length) geodesic segment
joining any two causally related points. However, it does not yield any informa-
tion about points which fail to be causally related. Furthermore, geodesic com-
pleteness does not imply geodesic connectedness for Lorentzian manifolds. In
fact, even compact space-times may fail to be geodesically connected (cf. Ex-
ample 11.23). SufKcient conditions for geodesic connectedness are given in
Proposition 11.22 in terms of disprisonment, pseudoconvexity, and a lack of
conjugate points along all geodesics. In Theorem 11.25 we find these conditions
actually suffice for our second version of the Kadamard-Cartan Theorem This
result yields sufficient conditions for the exponential map at any fixed point to
be a diffeomorphism from its domain in the tangent space onto the manifold
M.
11.1 The Timelike Diameter
Motivated by the concept of the diameter of a complete Riemannian man-
ifold, the following analogue has been considered for arbitrary space-times
[cf. Beem and Ehrlich (1979c, Section 9)].
Definition I f .I. (Timelike Dzameter) The tzmelzke dzameter of the
space-time (M, g), denoted by diam(M, g), is defined to be
diam(M, g) = supid@, q) : p, q G M ) .
A similar concept has been used by Tipler (1977a. p. 17) m studying singu-
iarity theory in general relativity. Physically. the timel~ke diameter represents
the supremum of possible proper times any particle could possibiy experience
in the given space-time. A space-time of finite timelike diemeter is singular
(recall Definition 6.3) in a striking way.
Remark 11.2. If diarn(M, g) < m, then all timelike geodesics have length
less than or equal to diam(M, g ) end are thus incomplete.
Proof. Suppose that c . (a, b) -+ M is a timelike geodehic which satisfies
L(c) > diam(N,g). We may then find s , t E (a; b), 5 < t , such that
402 11 SOME RESULTS IN GLOBAL LORENTZI.4N GEOMETRY
But then
d(c(.s), e ( t ) ) 2 L(c / :s, t 1) > diam(M. g)
which is impossible.
From a physlcal point of view. the most interesting space-times of finite
timelike diameter are the Wheeler universes [cf. Tipler (1977c, p. 500)]. In par-
ticular, the "closed" Fhedmann cosmological models are examples of Wheeler
universes.
For a complete Riemann~an manifold ( N . go), the diameter is finite if the
manifold is compact. In this case. we may always find two points of Ai whose
distance realizes the diameter. On the other hand, for space-times with finite
timelike diameter, the diameter is never achieved.
Proposition 11.3. Let (34,g) be an arbitrary space-time. and suppose
that there exist p ,q E M such that d(p, q) = diam(M, g). Then d(p. q) = oo.
Proof. Suppose d ( p , q ) = diam(.%f:g) < m, Let q' E If ( q ) be arbitrary.
Then
d(p , 9') L d(p: 9 ) + 4% p') > d(p, 9) = diam(M, gl
in contradiction. C
Recalling that globally hyperbolic spacetimes satisfy the finite distance
condition (cf. Definition 4.6), we have the following corollary to Proposition
11.3.
Corollary 11.4. (1) The timelike diameter is never realized by any pair of
points in a space-time of finite timelike diameter.
(2) The timelike diameter is never achieved in a globally hyperbolic spacetime
TVe now prove the Lorentzian analogue (Theorem 11.9) of Bonnet's Theo-
rem and Myers' Theorem for complete Riemannian manifolds [cf. Cheeger and
Ebin (1975. pp. 27-28)], Similar results have been given by Avez (seminar lec-
ture), Flaherty (unpublished), Uhlenbeck (1975, Theorem 5.4 and Corollary
5 4 , and Beem and Ehrlich (1979c, Theorem 9.5). Also, Theorem 11.9 is con-
tained implicitly in stronger results using the Raycl~audhuri equatiori needed
11.1 T H E TIMELIKE DIAMETER 403
for singularity theory in general relativity [cf. Section 12.2 or Hawking and
Ellis (1973. Section 4.4)].
Definition 11.5. (Tzmelzke Two-Plane) A tzmellke two-plane 0 is a two-
dirnenqional subspace of T,M for some p € M which is spanned by a spacelike
tangent vector and a timelike tangent vector.
Recall that the sectional curvature K(o) of the timelike two-plane 0 may
be calculated by choos~ng a basis {v. w) for u consisting of a timelike and a
spacelike tangent vector and setting
Remark 11.6. Rather than considering all sectional curvatures. zt is es-
sential to restrict attention to timelike sectional curvatures for the following
reason. If (M, g) is a space-time of dimension n 1 3 and the sectional cur-
vature function of ( M , g ) is either bounded from above for all nonsingular
two-planes or bounded from below for all nonsingular two-planes, then (M, g)
has constant sectional curvature [Kulkarni (1979)'. Here a two-plane a 1s said
to be nonszngular ~f g(v, v)g(w, w) - [g(v,m)I2 # 0 for some (and hence for
any) basis {v. w) for a. On the other hand, space-times with all tzmelzke sec-
tional curvatures K ( a ) 5 -k2 or a11 timel~ke sectional curvatures K(u) 2 k2
exist. But Harris (1982a) has shown that if all timelike sectional curvatures
are bounded both from above and below, then (iM,g) has corlstant sectional
curvature. Thus no obvious analogue for the Lorentzian sectional curvature
exists for pinched Riemannian rnanifoids [cf. Cheeger and Ebin (1975. p 118)
for Riemannian pinching].
With the signature convention (-: i-, . . . . +) used here for Lorentzian met-
rics, curvature conditions in Riemannian geometry for complete Riemann-
ian manifolds of positive (respectively, negative) sectional curvature tend to
correspond to theorems for globally hyperbolic space-times of negative (re-
spectively, positive) sectional curvature. On the other hand, if we change
this signature convention to (+, -, . . . , -) by setting (M, g) = (M, -g) , then
K(3) = -K(g). Thus Riemannian theorems for pos~tive (respectively. nega-
404 11 SOME RESULTS IN GLOBAL LORENTZlAN GEOMETRY
tive) sectional curvature correspond to Lorentzian theorems for positive (re-
spectively, negative) sectional curvature for (M,tj) [cf. for instance Flaherty
(1975a, pp. 395-396) where the convention (+, -, . . . : -) is used]. But whether
g or ij is chosen as the Lorentzian metric for M , Ricfg) = Ric(6).
It is convenient to isolate part of the proof of Theorem 11.9 in the folfotving
proposition. &call the notation V&(c) from Definition 10.1.
Proposition 11.7. Let (M,g) be an arbitrary space-time of dimension
n 2 2. Suppose that (M, g) satisfies either the curvature condition
(1) Every timelike plane 0 has sectional curvature K ( a ) < -k i 0, or the curvature condition
(2) Ric(g)(v, v) 1 (n - 1)k > O for aI I unit timelike tangent vectors v E
T M .
Then if c : [D, b ] -+ M is any timelike geodesic with L(c) 2 TI&, the geodesic
segment c has a pair of conjugate points.
Prooj. Since curvature condition (1) implies curvature condition (2) by tak-
ing the trace, we will prove that condition (2) implies the desired conclus~on.
For convenience, we will suppose that c : [0, L] --, A4 is parametrized as a unit
speed tinlelike geodesic with length L. Set En(t) = ct(t) for all t E [O, L]. and
let (El, E2, . . . , En-l) be n - 1 spacelike parallel vector fields along c such
that {El (t). Ez(t), . . . , En(t)) fornls a Lorentzian orthonormal basis of T'(l)lM for each t E (0, L]. Set liV,(t) = sin(xt/L)E,(t), so that VnJ, E q ( c ) . Using
equation (10.3) of Definition 10.4, we obtain
Hence
If Ric(c'(t),ct(t)) 2 (n - l)k for all t E [O. L] and L 2 Ti/&, we find that
I(WL, Ws) > 0. Hence I(W,,W,) 2 0 for some i E ( I ,? , . . . ,n - 1). On
11.1 THE TIMELIKE DIAMETER 405
the other band, if c 1 10, Lj is free of conjugate points, then I(W,. W,f < O for
each i by Theorem 10.22. Hence c has a pair of conjugate points if L > T/&,
as required. 13
A slight variant of Proposition 11.7 may also be proved similarly using the
Timelike Morse Index Theorem (Theorem 10.27).
Proposition 11.8. Let (M,g) be an arbitrary space-time of dimension
n satisfying either (or both) of the curvature conditions of Proposition 11.7.
If c : [a, b ] -+ M is any timelike geodesic with L(c) > 7r/&, then t = a is
conjugate along c to some to E (a, b), and hence c is not maximal.
Proof. Using L(c) > n/& and the same vector field ti/; as in the proof of
Proposition 11.7, we obtain this time that
n-1
CI(W%,W%) > 0. %=I
Hence I(Ur2, I/Yd) > O for some i. Thus Ind(c) > O. By the Timelike hIorse
Index Theorem (Theorem 10.27), dim Jt(c) # 0 for some t E (a. 6) . This
completes the proof.
Now we are ready to give the Lorentzian analogue of the Bonnet Myers
Diameter Theorem for complete Riemannian manifolds.
Theorem 11.9. Let (M, g) be a globally hyperbolic space-time of dimen-
sion n satisfying either of the following curmture conditions:
(1) K(o) < -k < 0 for all timelike sectional curvatures K ( o ) .
(2) Ric(%, a ) 2 (n - l)k > O for all unit timelike vectors v E T M .
Then diam(M, g) I 7i./&.
Pmof. Suppose that &am(M,g) > n/&. We may then find p , q E M
with d(p , q) > 7i/& by defin~tion of diam(M,g). S~nce (M,g) 1s globally
hyperbolic, there exists a maximal tlrnellke geodesic segment c : 10, 11 -- M
with c(O) = p, c(1) = q. But as L(c) = d ( p , q ) > n/&, the geodmic segment
c is not maximal by Proposition 11.8, in contradiction. U
It is clear that an analogue of Proposition 11.8 could be obtained for null
geodesics using the null index theory developed in Section 10.3. However. a
406 I1 SOME RESULTS IN GLOBAL LORENTZIAN GEOMETRY
stronger result may be obtained using the Raychaudhuri effect [cf. Hawking
and Ellis (1973, p. 101)]. Thus we refer the reader to Section 12.2 for a dis-
cussion of these results rather than pursuing this analogy any further here
[cf. also Harrls (1982a)l. In addition to extensive results in the general relativ-
ity literature along the lines of Proposition 11.8, obtained using Raychaudhuri
techniques. disconjugacy theory of O.D.E.'s has also been applied in this set-
ting (cf. Tipler (1978), Galloway (1979), Kupeli (19861, just to single out a
few of many possible references).
11.2 Lorentzian Comparison Theorems
For use in Section 11.3 as well as for their own intrinsic interest, we now
present the timelike analogues of two important tools in global Riemannian
geometry: the index comparison theorem [Gromoll, Klingenberg, and Meyer
(1975, p. 174)] and the Rauch Comparison Theorem [Gromoll, Klingenberg,
and Meyer (1975, p. 178) or Cheeger and Ebin (1975, p. 29)]. The results in this
section, except for Corollary 11.12, have been published in Beem and Ehrlich
(1979c, Section 9). Actually, there are two versions of the b u c h Comparison
Theorem, often called Rauch Theorem I and Rauch Theorem 11, that are useful
in global Riemannian geometry [cf. Cheeger and Ebin (1975, Theorems 1.28
and 1.29, respectively)]. The result (Theorem 11.11) given in this section is
the Lorentzian analogue of Rauch Theorem I. Harris (1979, 19S2a) has given
proofs of Lorentzian analogues for both Rauch Theorems I and I1 and using
Rauch Theorem I1 has given a Lorentzian version of Toponogov's Comparison
Theorem [cf. Cheeger and Ebln (1975, p. 42)j for timelike geodesic triangles in
certain classes of space-times. Using this result, Karrls (1979, 1982a) has aiso
obtained a Lorentzian analogue of Toponogov's Diameter Theorem [cf. Cheeger
and Ebin (1975, p. 110) and Appendix A].
In the rest of this section, let (MI, gl) and (A42,92) be arbitrary space-times
with dim n/r, 5 dim &if2. Also let c, : [O, L] -+ M,, z = 1,2, be unit speed future
directed timelike geodesic segments. Throughout this section we will denote
both the index form on ~ ' ( c l ) arid V1(cz) by I. Also, during the proofs we
will denote both Lorentzian metrics gl and g2 by ( , ). The index form I for
11.2 LORENTZIAN COMPARISON THEOREMS 407
a timelike geodesic c and the index Indic) and extended index Indo(c) of c are
defined in Section 10.1, equation (10.1) and Definition 19 24. respectivel?..
TVe first need to define an isomorphism
4 : vl(cl) -+ v ~ ( c ~ )
so that g2(4X(t). q X ( t ) ) = g l ( X ( t ) , X ( t ) ) for all t E 10. L] This may be done
following the usual parallel translation constructron in Riemannian geometry
We first define an rsometry
4t : Tc, :t)fif~ + Tc,(t)Al,
as follows. Let
Pt : Tcl( t )M~ -+ T,,(G)MI
denote the Lorent7ian inner product-preserving isomorphism of parallel trans-
lation along rl . Explicitly, given u € TC,(,)IMI, let Y be the unique parailel
field along cl with Y(t) = v, and set Pt(v) = Y(0) . S~milarly. let
Qt : Tc,(,)hf2 + TC2(0)M2
denote parallel translation along cz. Cnoose an injective Lorentzian inner
product-preserving linear map
i : (T,,(o).MI. l,,(o)) -+ (Tcs(0)1~2. g21cz(o))
where i(cll(0)) = czt(O). Then the map
Qt : (TC,(tpfl. 9llc,(t)) + (Z2, t )n / f , , g21c2!t))
given by Ot = Q,' o i o Pt 1s an isometry since parallel translation preserves
the Lorentzian structures. We may then define the map
4 : vL(cl) + vL (4 as follows. Given X E VL(cl). define OX E vL(c2) by ((OXj(t) = Qt(X(t)).
It follows as in the Riemannian proof that (QX)' = q ( X 1 ) , where the first
covariant differentiation is in M2 and the second is in hif~
Let 6 2 , t ( ~ z ) denote the set of ail timelike planes 0 containing c,'(t) for
i = 1.2. There is then an induced map
4t : Gz,t(cl) -+ G2,t(c2)
defined as follows. If (T E GZ, t (~ l ) we may write g = span(v.cl1(t)) where v
is spacelike. Put &(v) = span{&(u), czl(t)) E G2 t(c2)
408 11 SOME RESULTS IN GLOBAL LOREXTZIAN GEOMETRY
We are now ready to state the timelike version of the index comparison
theorem.
Theorem 11.10 (Timelike Index Comparison Theorem).
Let ( M I , 91) and (M2.92) be space-times with dim MI I dim M2, m d let
cl : [0, p] + MI and c2 : [O,@] -+ M2 be unit speed, future d~rected cimelike
geodesics. Suppose for a21 t with 0 5 t 5 ,O and dl timelike planes a E G2,t(c1)
that the sectional curvature KMl (a) 2 %(dim) . Then for any X E v1(cl)
we have
(1) I(X,X) I I(OX, 4x1; (2) Ind(c1) 5 Ind(c2); and
(3) Indo(s) 5 Indo(c2).
Proof. Recalling that (cl', el') = -1 and (X, cl') = 0, we obtain
Similar formulas hold for QX and czl. Thus
P {-(XI, XI) + ( R ( X , cll)cl', X ) ] dt = I ( X , X). CI
We may now obtain the more powerful Timelike Rauch Comparison The-
orem using the timelike index comparison theorem. Recall Erst that since
c, : [0, L] -+ Mi is a timelike geodaic segment for each i , the vector fields in
VL(c,) are all spacelike vector fields.
Theorem 11.11 (Timelike RaucP1 Comparison Theorem).
Let (&, gl) and (M2,gz) be space-times with dim MI 5 dim A&. Let cl :
[O, L] -- -MI and e2 : [O,L] + M2 be future directed, timelike, unit speed
geodesic segments. Suppose for all t E [O, L] and my a E Gs,t(cl) that
11.2 LORENTZIAN COMPARISON THEOREMS '
Let Yl E VL(c:) and Yz E VL(c2) be Jacobi fields on -211 and respective
satisfying the initial conditions
and
rf cz ha? no conjugate points t o t = 0 in (0) L), then
for all t E (0, L]. In particular. cl has no conjugate points to t = 0 in (0. L )
Proof. This may be given along the lines of Gromoll, Klingenberg, a1
Meyer (1975, pp. 130-161), except for the proof of their inequality (7), p. 1€
which must be modified as follows. Let Z 6 Vi(cs) be the unique Jacobi fie
along cz with Z(0) = 0 and Z(to) = cpYl(to), $ as above. We must show th;
To this end, let cg = CI 110, to] and c4 = c2 / 10. to]. Then
(the above inequality by the t i~el ike index comparison theorem [Thcore 11.1&(1)])
(the above inequal~ty by the rnaximality of Jacobi fields with respect to t: index form in the absence of conjugate points [Theorem 10.231)
410 I1 SOME RESULTS IN GLOBAL LORENTZIAN GEOMETRY
It may also be assumed m Theorein 11.11 that c2 has no conjugate points
in [O, L]. Then cl would also ha%e no conjugate points in (0, L] by Theorem
11.10-(3).
For Riemannian manifolds of nonpositive sectional curvature, by ~qu ipp~ng
the tangent space with the "Aat metrlc" (cf Definition 10.17) it may be shown
thar, the exponentla1 map does not decrease the length of tangent ~ectors
[cf. Blshop and Crittenaen (1964, p. 178, Theorem 2 (i)) for a precise state-
ment]. A slmple proof of this fact may be given uslng the R a ~ c h Comparison
Theorem and conlpdring Jacob1 fields on the given Riemannian manifold to
those in R". We will now use the Timelike Rauch Comparison Theorem to
prove the analogous ~esul t for space-times of nonnegative tlmelike sectional
curvature [cf. Flaherty (1975a- p. 39771. Intuitively. Corollary 11.12 below ex-
presses the fact that if all timelike sectional curvatures of (M,g) are pos~tive,
:hen future directed t~melike geodesics emanating from a given point of lliP
spread apart faster than "corresponding" geodesics in M~nkowski space-time.
Recall that the canonical isomorphis~n r, has been defined in Sect~on 10.1,
Definition 10.15.
Corollary 11.12. Let (M.g) be a space-time with everywhere nonn
tive timelike sectional curvature, and let v E T,M be a given future direct
timelike tangent vector with g(v, z) = -1 Then for any future directed no
spacelike tangent vector zu 6 T'M, the vector b = ~ ( w ) E T, (T,M) satisfi
the mequality
9fexpp, b, exp,. 0) 2 g(w, w) = ( (4 b)).
Proof. We first prove the inequality for b = T,(w) E T,(T,llf) w ~ t h g(v, w)
0 by applying the Timelike Rauch Comparison Theorem with (Ml,gl)
(iC.1, g) and (Mz, gz) identified with Minkowski space-time (P?, go) with n
dimM. Setting cl(t) = exp, tv, let Yl E VL(c) be the unique Jacobi field wi
Yl(0) = 0, YI1(0) = w By Proposition 10.16, we have Yl(l) = exp,. b.
Now let cz : 10. I] -+ Ry be an arbitrary unit speed timelike geodesic, a
choose 5 E N(cz(0)) with g o ( F , ~ > = gl(w,w). Let & E VL(c2) be t
umque Jacobi field with Yz(0) = 0 and V2'(0) = 3. Then Yz(t) = t
11.3 LOREKTZIAN HADAMARD-CARTAN THEOREMS 411
where Pt denote the Lorentzian parallel translation along cz from c2(0) to
cz(t). Applying Theorem 11.11, we obtain
gl(expp. b. exp,* b) = gi(K(1). &(1))
L gofY2(1).Yz(l))
= 5Q(PI (El, PI @)I
= go (727. G) = gl ( w , W) = ((b, b ) )
as required.
Returning to the general case, we may decompose ,w = w1 + w2 where
wl = Xv for some X > 0 and g(v, wz) = 0. Set b, = r,(wz) for i = 1 , 2 SO that
b = bl + 6% Wc now calculate
Applying the Gauss Lemma (Theorem 10.18) to the first two terms, we
obtain
as ((b1,bz)) = g(wl. wz) = 0. Now applying the first part of the proof to the
last term, we have
as required. 0
11.3 Lorentzian Nadarnard-Cartan Theorems
We first prove a basic result linking conjtlgate points and timelike sectional
curvat3xe [cf. Flaherty (1975a, Proposition 2.lj-here Flaherty uses the signa-
ture convention (+, -, . . . , -) for the Lorentzian metric so that his sectional
curvature condition has the opposite sign from ours].
412 11 SOME RESULTS IN GLOB.4L LORENTZIAN GEOMETRY
Proposit ion 11.13. Let (M, g) be a space-time with everywhere nonneg-
ative timelike sectional curmtures. Then no nonspacelike geodesic has any
conjugate points.
Proof. First let c : 10,a) -, iM be an arbitrary future directed unit speed
timelike geodesic. Recall from Corollary 10.10 that if X is a Jaeobi field along
c with X(O) = X(t0) = O for some to E (O,a), then X E VL(c). Thus we
may restrict our attention to Jacobi fields J E vL(c) with J(O) = 0. Since
J E VL(c) and c is a timelike geodesic, J' E VL(c) also. Consider the smooth
function f (t) = g ( J ( t ) , J t ( t ) ) . Differentiating yields
for all t t [O, a) . Now if J(t0) = 0 for some to E [&a), then f (0) = f (to) = 0.
Thus as j is nondecreasing. f ( t ) = 0 for all t € [O, to]. Hence 0 = f'(0) =
g(J'(0). J1(0)) as J (0 ) = 0. from which we conciude that J1(0) = 0. Therefore
J = 0, and no t o f [O, a ) is conjugate to t = 0 along c.
We now treat the case that f l : [0, a ) -+ M is a null geodesic using the null
index form. Let t o E [O, a ) be arbitrary. We will show that 7 : Xo(P [O, to]) x
Xo(P 1 (0, to]) -+ R is negative definite. Hence no s E (0, to] is conjugate to
t = 0 along j3 by Theorem 10.69.
If W E Xo(P) is a smooth parallel vector class, then W = [P'] since W(0) =
[,B1(0)]. Thus if W E Xo(B) is not a smooth parallel vector class, we have
g(W1(s) , W 1 ( s ) ) > O for some s E ( @ t o ) . Also i j (Wr(t) . W 1 ( t ) ) 2 O for all t E
[O, to] since 3 is positive definite. Now consider 3 @ ( ~ ( t ) , j3'(t))Dt(t), W( t ) )
for any fixed t E [O,to]. If W(t ) = [/3'(t)]. then g @ ( ~ ( t ) , ~ ' ( t ) ) f l ' ( t ) W ( t ) ) =
0. Otherwise, we rnay find a spacelike tangent vector w perpendicular to P1(t ) with ~ ( w ) = W ( t ) . Then, using equation (10.35) of Section 10.3 we have
3 (-iE(w(t), p'(t))pl ( t ) , W ( t ) ) = g (R(w , P( t ) )Pt ( t ) , w). Since w is spaceliie,
we may find a sequence of t i d i k e two-plaw 5, = (wn,v,) with w, spacelike,
0, timelike, g(w,,u,) = 0, w, -+ w , and v, -r pl(t) , so that a, -+ (w.P1(t)) .
11.3 LORENTZIAN HADAMARPCARTAN THEOREMS 413
We then obtain by continuity
3 Cfi(w(t>:P1(t))P1(t),w(t)) = 9 (R(w,P1(t))P' t t ) . w)
= lim g(R(wn, vn)vn, w,) i"M
= lim K(wn,vm)g(w,,wn)g(vm.vn) L 0 n-m
since g(w,, w,) > 0, g(v,, v,) < 0. and K(v,, w,) 2 0 for each n. Thus in
either case, 3 @(W( t ) , Pt( t))p '( t) , TV(t)) < 0. Hence, provided W # [@'] we
have - t o
I(W, W ) = Lo [-3wJ, wi) +z(x(w ~ l ) p l , w)ll, dt <
as required.
Motivated by a standard definition in global Riemannian geometry, we make
the following definition.
Definition 11.14. The space-time (M, g) is said to have no future timelzke
conjugate poznts if for any future directed timelike geodesic c : 10, a) 4 ( M , g),
no nontrivial Jacobi field in VL(c) vanishes more than once
In view of Lemma 10.46, similar definitions may be formulated for space-
times with no future null conjugate points or no future nonspacelike conjugate
points. Proposition 11.13 guarantees that if (M,g) is a space-time with ev-
erywhere nonnegative timelike sectional curvature, then ( M , g) has no future
nonspacelike conjugate points.
Lorentzian manifolds with nonnegative timelike sectional curvature or with
no future timelike conjugate points may be characterized in terms of the be-
havior of their Jacobi fields. A similar characterization applies to Riemannian
manifolds [cf. O'Sullivan (1974, Proposition 4)]
Proposition 11.18. i
(If (M,g) has everywhere nonnegative timelike sectiorid curvature iff P
for every Jacobi field Y E VL(c) along any future d~rected timelike
geodesic c.
414 11 SOME RESC'LTS IV GLOBAL LORENTZIAN GEOMETRY
(2) (M, 9) has no future t~rneljke conjugate pornts iff g(Y(t), Y(t)) > 0 for
all t > 0 where Y E VL(c) is any nontrivial Jacobi Geld with Y(0) = 0
along any future directed timellke geodesic c.
Proof. (1) Assume that (Id. g) has everywhere nonnegative timel~ke sec-
tional curvature. Let Y E VL(c) be a Jacobi fieid along the u n ~ t speed timel~ke
geodesic c. Then Y' E VL(c) also, and we obtain
d 2 -(g(Y, Y)) = 2g(Y1, Y') - 2g(R(Y, c1)c', Y) dt2
= 2g(Y1 ,Y ' )+2g(Y ,Y)K(Y ,c ' ) 2 0
Conversely, let {v,u;) be future timelike and spacelike tangent vectors,
respect~vely, spanning an arbitrary timelike two-plane with g(v, v) = - 1.
g(w,u~) = 1 and g(v, w) = 0 Let c(t) = exp(lv) and let Y E VL(c) be
the .Tacohi field with initial conditions Y(0) = w and Y'(0) = 0. Then we have
by hypothesis,
d2 0 5 9 1 = -29 (R(Y(0) c1(0))c1(O),Y(Oi)
t=O
= -29 (R(w, v)v, w) = 2K(v, w)
since the first term in the differentiation vanishes as Y'(0) = 0. Thus K(v, w) 2 0 as required
(2) This is clear from Defiilition 11.14. 0
Using the timelike index theory of Sections 10.1 and 10.2, we now gi
the following version of a Lorentzian Radamard-Cartan Theorem for global1
hyperbolic space-tlmes This proof is similar to the Morse theory proof of t
Hadamard-Cartan Theorem for complete Riemannian manifolds [cf. Lililn
(1963. p. 102)] Recall that a spacetime 1s said to be future one-connect
(Definition 10.28) if any two smooth, future directed timelike curves from p t
q are homotopic through (smooth) future directed timelike curves w ~ t h fixe
endpo~nts p and q.
Theorem 11.16. Let (M, g) be a future one-connected globally hyperbo
space-time with no future nonspacelike conjugate points. Then given
11.3 LORENTZIAK HADAMARD-CARTAN THEOREMS 415
p, q E M with p << q, there is exactly one future directed timelike geodes~c (up
to reparmetrization) from p to q.
Proof Since (M g) is globally hyperbolic, there exists a maximal f2~ture
directed timelike geodesic from p to q. Since there are no future nonspacelike
conjugate points, any future directed geodesic from p to q has index 0 by
Theorem 10.27. Thus the timelike path space C(p,,) has the homotopy type
of a CW-complex with a cell of dimension 0, i.e., a point, for each future
directed timelike geodesic from p to q On the other hand, since hf is future
one-connected, C(,,,) is connected and hence consists of a single point. Thus
there is at most one future directed timelike geodesic from p to q.
d similar result was obtained by Uhlenbeck (1975, Theorem 5 3) for globally
l~yperbolic spacetimes satisfying a metric growth condition [Uhlenbeck (1975.
p. 72) ] and the curvature condition g (R(v. U)W, 0) 5 0 for all future dlrected
null hectors v and vectors w with g(v, 2 ~ ) = 0 at every point of M . Namely, M
can be covered by a space which is topologically Mlnkowski (3.e.. Euclidean)
space.
Flaherty (1975a, p. 398) has also shown that if ()If, g) 1s future 1-connected,
future nonspacelike carnplete, and has everywhere nonnegative timelike sec-
tional curvatures. then the exporlential map exp, regularly embeds the future
cone in TpM at each point p into M. To obtain this result, Flaherty used
a lifting argument to show that urider these hypotheses, if v. w E T,M are
any two future directed timelike tangent vectors with expp v = exp, w, then
v = w. Thus future onoconnected, future nonspacelike co~rlplete space-times
with nonnegative timelike sectional curvatures satisfy the conclusion of Theo-
rem 11.16. On the other hand, Flaherty showed (1975b, p. 200) that any fu~ure
one-connected, future nonspacelike complete space time with everywhere non-
negative timelike sectional curvatures is also globally hyperbolic [cf. Galloway
(1986a, 1989b)j.
h c a l l that in Section 7.4 we applied disprisonment and pseudoconvcxity
to nonspacelike geodesics to obtain sufficient conditions for stability of non-
spacelike geodesic incompleteness (nonspacelike d~sprlsonment suffces) and
the stability of nonspacelike geodesic completeness (nonspacelike disprison-
416 11 SOME RESULTS IN GLOBAL LORENTZIAN GEOMETRY
rnent and nonspacelike pseudoconvexity taken together suffice). In order to get
a Hadamard-Cartan Theorem that applies to the entire space-time, we will
apply these two conditions to the set of all geodesics-not just to nonspace-
like geodesics [cf. Beem and Parker (1989)j. A manifold will be geodesically
disprisoning if each end of every inextendible geodesic fails to be imprisoned
in a compact set. The manifold will be geodesically pseudoconvex if for each
compact set K the convex hull, using geodesic segments joining points of K,
iies in a larger compact set. More precisely,
Definition 11.17. ( Geodesacally Disprasonzng) The spacetime (M. g)
is geodesically ckisprisonzng if, for each inextendible geodesic c : (a, b) - M and any fixed to E (a , b), the images of each of the two maps c I (a, to] and
c I [to, b) fail to have compact closure.
Definition 11.18. ( Geodesically Pseudoconvex) The space-time (M, g)
is geodesically pseudoconvex if for each compact subset K of M there is a
compact set N such that each geodesic segment c : [a, b] -+ M with c(a), c(b) E
K satisfies c([a ,b] ) C W .
By itself, Definition 11.17 allows for the possibility that either (or both) ends
of a given geodesic may be pwtially imprisoned (cf. Definition 7.29). However,
the next lemma shows that if ( M , g ) satisfies both of the above conditions,
then partial imprisonment cannot occur.
Lemma 11.19. Let (M, g) be both geodesirally disprisoning and geodesi-
calIy pseudoconvex. If c : (a. b) -+ M is any inextendible geodesir of M and
K is any compact subset of M . then there are parameter values sl . sz with
a < 51 < s~ < b such that c(t) # K for all a < t < sl a d all s~g < t < b.
Proof. Assume the bmma is false. Then without loss of generality we may
assume there is some geodesic c: (a, b) -+ M and a sequence {tk) satisfying
a < tl < . . . < tk < . . < b, tk + b-, and c ( t k ) E: I( for some compact subset
K of M . Let Ii be a cornpact set such that any geodesic segment of ( M , g )
with endpoints in K lies in W . Then the image of the map c 1 itl, b) lies in the
compact set N, in contradiction to the disprisonment assumption. 0
11.3 LORENTZIAN HAD.4MAtLD-CARTAN THEOREMS 417
Lemma 11.20. Let (M, g) be both geodesically disprisoning and geodesl-
cally pseudoconvex. Let {p,) and (T,) be sequences converging to p and T ,
respectively. If each pair {p,, T,) is joined by a geodesic h, then here is a ge-
odesic c joining p and r . Furthermore, if each c, is nonspizcelike (respectively,
null). then c may be chosen to be nonspacelike (respectively, null).
Proof, Let h be an auxlliarjr Riemannian metric on M . Let K be a compact
set containing p and r as well as all {p,) and {T,). Fbr each n, let c, : [O, b,) -4
M be a geodesic for the metric g which satisfies c,(O) = p,, %(a,) = r,, and
which is not extendible to b,. We may assume without loss of generality that
h(cn l (0 ) , cnJ(0)) = 1. Since h is positive definite, the sphere bundle over K
with respect to h is compact. Consequently, there 1s a subsequence (m) of
(n) and a vector w E T,M with ~ ' ( 0 ) -+ w. Let c: [0, b) -+ M be the unique
geodesic with c'(0) = u: and such that c is not extendible to t = b. Let H be a compact set such that all geodesic segments with endpoints in K lie in
H . Since (M, g ) is disprisoning, there is some 0 < T < b with C(T) $ H. The
construction of W then implies c(t) g' K for all r 5 t < b. Since c,(T) -+ c(T),
it follows that O < a , < T for all suffic~ently large rn. Using the fact that
[O, T ] is compact, we find there is a limit point a of the sequence {a,). Let
{k) be a subsequence of {m) with ak -+ a. Then ck(ak) -+ c ( a ) , and it follows
that c(a) = r = lirn r k . Hence c is a geodesic containing both p and r. If each
c, is nonspacelike (respectively, null), then cl(0) = lim ~ ' ( 0 ) shows c IS also
nonspacelike (respectively, null). U
Definition 11.21. ( Geodesacally Connected) The spacetime (M, g ) is
geodesically connected if for each pair of distinct points p and q there is at
least one geodesic segment joining p to q.
The Nopf-anow Theorem [cf. Hopf and Rinow (1931)] guarantees the geu
desic connectedness of any complete Riemannian man~fold. However, complete
Lorentzian manifolds may fail to be geodesically connected. In fact, for com-
plete Lorentzian manifolds, even if r is in the causal future of p, it may happen
that there is no geodesic from p to r (cf. Figure 6 1). Of course, global hyper-
bolicity yields a (maximal length) geodesic segment joining any two causally
related points. However, geodesic cornpletenms does not imply global hyper-
418 11 SOME Rl3SULTS IN GLOBAL LORENTZlAN GEOMETRY
bolicity. Furthermore. global hyperbolicity does not yield the existence of
geodesic segments between points that far1 to be causally related The next
proposition gives sufficient conditions for the existence of a geodesic segment
between anj two points The hypotheses include disprisonment, pseudocon-
vesity, and a lack of conjugate points but do not require geodesic completeness.
Proposit ion 11.22. Let ( M , g ) be geodesically disprisoning and geodesi-
cally pseudoconvex. If (M, g ) has no conjugate points, t h ~ n (M, g ) is geod~si-
cally connected.
Proof Let p and r be distlnct points of M, and construct an arbitrary curve
a : [O, I ] + M with a ( 0 ) = p and a ( 1 ) = r . Let .r = supis E [O,11 : there
IS a geodesic segment from p to a( t ) for all 0 5 t 5 s). The fact that p lles
in a convex normal neighborhood yields r 3 0. Lemma 11.20 implles that
there is a geodesic segment c. 10, l] --t M wlth c(0) = p and c (1 ) = a(r) Let
X = c l (0 ) Since ( M , g ) has no conjugate points, the exponential map exp,
takes a sufficiently small open neighborhood of X onto an open neighborhood
of c(1) = a ( r ) . Thus, there exist geodesic segments from p to all points of
a sufficiently near ~(7). It follows that T = 1, and hence there is a geodesic
segment from p to T , as desired. O
Compact Riemannian manifolds are always geodeslcally connected. In con-
trast, the next example shows that compact Lorentzian manifolds may fail to
be geodesically connected.
Exsunple 11.23. (Compact space-tzmes need not be geodeszcally connecte
~ e t M = S b SS' = ((t,s) .Oi s 5 l a n d -47; < t I 4 n ) w i t h t h e u s u
identifications Let g = (- cost)dt2 + (2 sint)dtds + ( c o s t ) d s v h ~ s is
Lorentzian metric, and X = cos(tl2) a/& - sin(tl2) B/ds is a unit t ~ m
vector field which tlme orients hf. Using light cones, it is not hard to c
that all geodesics starting on the circle t = 0 must lie in the set -5.rr/2 < t 7x/2. It follows that ( M , g) 1s a compact space-tlme which is not geodesical
connected.
11.3 LORENTZIAN HXDAMARD-CARTAN THEOREMS 41s
Let G be a group of homeomorpl.lisrns acting on the manifold 111. The group
G is said to act freely if f ( p ) = p for some p E M irriplies that j is the identity
e of 6. If G acts freely, then e is the only element of G that has any fixed
points. A discrete group G is said to act properly dascontznzlously jcf Boothby
(1986), Wolf (1974)] if the following hold:
( 1 ) Each p E M has a neighborhood U such that the set { f E G : f ( U ) n U # 0) is finite, and
(2) If f (p) # q for all f E G, then there are neighborhoods V and W of p
and q, respectively, such that j (V) n W = 0 for all f E 6.
Group actions arise naturally in the study of co~rering spaces of manifolds.
Let M be a smooth manifold. A covenng space of iM 1s a tnple (E, T, iM) where
E is a smooth manifold and n : E + M IS a smooth map of E onto il.1 which
is evenly covered [cf. O'Neill (1953)l. Here is sald to be evenly covered l f for
eachp E M there is a neighborhood U of p such that T - ' ( U ) consists of dlsjolnt
open sets (W,) in E and .rr / W, is a diffeomorphisrn of W, onto U for each a.
If E is simply connected, then (E. n, M ) is the unzversal coverzng space of &if.
A covering map T is a lo~a l diffeomorphism, but not all local diffeomorphisms
are covering maps because a local diffeomorphism is not necessarill an even
cover. For example, let M = S' = {(x, y) : ;c2 + y2 = 1) = { z E C : 121 = 1) .
The map f . ( 0 . 4 ~ ) -+ S' defined by f (6 ) = r" is d local diffeomorphism
but fails to be a covering map. The inverse image of e'" (1 ,O) is the single
number 27r. Other polnts have two preimages, and the Inverse image of aniall
connected open set U about the point (1,O) in S1 is three open intervals, one
of which contains the number 27r. Thc map f restricted to the interval of
f-l(U) containing 2n is a diffeomorphism onto U , but f restricted to either
of the other two intervals is not a diffeomorphism onto U. The following topological result will be of use in the proof of Theorem 11 25.
Lemma 11.24. If T : Rn --+ M is the universal covenng space of the
rnanrfold IM, then the fundamental group i71(.44) of M is either tr~vtal or
infinite. If ?rl (M) is trivial, then the covering map .ir :s one-to-one.
Proof The fundamentai group 7r l ( iZ / [ ) of hf acts freely and properly dis-
continuously on the total space Rn of the covering [cf. Kobayashi and Nom~zu
420 11 SOME RESULTS IN GLOBAL LORENTZIAN GEOMETRY 11.3 LOREKTZIAN WADAMARD-CARTAX THEOREMS 421
(1963, p. 61)]. If ax (M) 1s finite, then it is either trivial or else has an element
of prime order. However, a result of Smith (1941, p. 3) yields that any homeo-
morphism of Bn having prime order must fix some point of Rn. But this would
contradict the free action of ~ 1 j i t l ) . Thus, the fundamental group is infinite or
has exactly one element. If the fundamental group is trivial, then each p E -M has exactly one preimage and hence the covering is one-to-one. U
The next theorem is a Lorentzian version of the Hadamard-Cartan Theorem
and is actually valid for any (connected) manifold with an &ne connection
[cf. Beem and Parker (1989)l. The usual completeness assumption used in the
Riemannian version [cf. Kobayashi (1961)] is replaced by disprisonment and
pseudoconvexity assumptions. In some sense, the pseudoconvexity condition
is a type of "internal completeness" assumption which plays a role similar to
global hyperbollcity (cf. Section 7.4). Since global hyperbolicity is a causality
assuimption, it only provides a certain amount of "control" over nonspacelike
geodesics. It should also be recalled that global hyperbolicity, which was used
in the hypothesis of our previous version of a Lorentzian Hadamard-Cartan
Theorem, Theorem 11.16 earlier in this section, implies nonspaceltke geodesic
pseudoconvexity (cf. Proposition 7.36).
Theorem 11.25. Let (M, g) be geodesically disprisoning m d geodesically
pseudoconvex If (M, g) has no conjugate points, then for each point p E M ,
the exponential map exp, : D -4 M is a diffeomorphism from the domain
D T,M of exp, onto M. Consequently, M is diffeomorphic to Rn, M is
geodesically connected, and any two distinct points of M determine exactly
one geodesic.
Proof. The map expp is onto M by Proposition 11.22. Furthermore, the
exponential map at p must be at least a local diffeomorphism from its domain
D onto M since no conjugate points exist. In order to show that expp is a
diffeomorphism. we will first show that each r E M has a finite number of
preimages Secondly, the map expi, will be shown to be a covering map, and
finally, Lemma 11.24 will yield that exp, is one-to-one.
To show that a fixed point r of M has a finite number of preimages in D, iue
begin by choosing a compact set K of M containing p and with r in the interior
of K. Then there is an open neighborhood U(r) of T with r E U(r) c K. Let
H be a compact set such that any geodesic segment wlth endpoints in K lies
in W . Pseudoconvexity guarantees the existence of N. Choose an auxiliary
Riemannian metric h, and let Sh(p) = (V E T,M h(v, u) = 1). Let .I; E Sh(p)
be arbitrary. If c, : [0 ,b) -+ M is the geodesic of g wltn c,'((O = a. then
disprisoriment yields c,(r) $? H for some T. It follows that c,(t) @ K for
all r 5 t < b. Using the continuous dependence of geodesics on the initla1
tangent vector, it follows that there is a neighborhood N(v) of v in Sh(p) such
that each geodesic with initial tangent vector lying in this neighborhood must
have 7 in its domain and rrlust have image points not in K for all t > T . In
part~cular, if c, happens to intersect U ( r ) . then the intersection must occur
for pasameter values not greater than T. Let r(v) = {w E T?M : w =
a v with v E Sh(p) and 0 5 cr 5 r ) . The set I'(v) is compact and lies in the
domain of the local diffeomorphism exp,. Consequently, each point q E U ( r )
has only a finite number of preimages in I'(v). We cover the compact set Sh (p)
with a finite number of neighborhoods N(vi), . . . iV(v,) of the above type for
corresponding vectors vl , . . . . v,. Thus, we obtain a finite number of compact
sets I'(vl). . . . . r(v,) each having a finite number of preimages of any fixed
point q E U(r). Let I' = I'(vl) U. -UI'(z,). By construction, all preimages of
each q E U(r) lie in the compact set I'. Hence we find that r E M has a finite
number of preimages in D 5 T,M.
Assume that exp, 1s not a covering map, There must be some r E M such
that for each open neighborhood U of r , the set expil(U) fails to consisr, of
components each of which is diffeornorphlc to U under expp. Let X, U ( r ) , H.
and I? be as above. Let r have k preimages u l , . . . . uk. Slnce exp, is a local
diffeomorphism, there is an open neighborhood V c U(r) of r diffeomorphic
to an open ball. and corresponding disjoint open neighborhoods T.PPi, . . . , in T,M containing the individual prelrriages of 7 , such that exp, / is a
digeomorphism onto V for each 2. It follows that exp;' (V) consists of more
than Wl U T.Y2 U U K. and this must be -,rue no matter Iiow sinall V 1s
chosen Consequently, there is a sequence r, - r such that the nuriiber of
preimages of each r, is larger than k . This yields a sequence (w,) in T p M
42% 11 SOME RESULTS IN GLOBAL LORENTZIAN GEOMETRY
with exp,(w,) = T, such that no subsequence of (w,) converges to one of the
preimages ul, . . . , u k of r. Since each w, lies in the compact set I?, there is a
subsequence {w3 1 converging to some w in r. Thus. w is in the domain of expP,
and expp(w) = limexp,(w,) = limr, = r , which shows w must be one of the
k preimages of T, in contradiction. We conclude that expp must be a covering
map. The (star shaped) domain D of exp, is some open set in Tpn/l, and D
must be homeomorphic to W". Thus exp, . D --t M is a universal covering
and the number of elements in (M) is equal to the number of preimages of
any point r E M . Lemma 11.24 now yields that T ~ ( M ) is trivial and hence
that exp, : D + A4 is a diffeomorphism. as desired.
- Let a Lorentzian manifold ( M , g ) be given, and assume n : M 4 A4 is
a covering of M by z. One may obtain an induced metric ij on z such
that .ir is a local isometry [cf. Wolf [1974), pp. 41-42]. The geodesics of li;?
project to geodesics of M , and geodesics of M lift to geodesics of ;i?. It may
easily 'happen that the cover is geodesically disprisoning even if the original
base manifold ( M , g) fails to be disprisoning. For example, one could have
Af = S1 x 5" = { ( t , x ) : 0 5 t 5 1, 0 5 5: 5 1) with the usual identifications,
given the flat metric g = -dt2+dz2. Then (x,?j) is the usual two-dimensional
hPvlinkowski spacetlme.
Clearly, Theorem 11.25 may be applied to the cover to obtain information
on the base manifold M.
Corollary 11.26. Let ( M , g ) have a covering space n : --+ 11.1 with
induced metric ?j = 7i'g on z. If this covering space (%: 3) has no conjugate
points, is geodesically pseudoconvex, and is geodesically disprisoning, then
both (=, 3) and (M, g) are geodesically connected. Furthermore, if p is any
point of M , then the exponential map exp, is a covering map.
Proof. Theorem 11.25 yields the geodesic connectedness of the cover To
show that M is geodesically connected, first let p and q be two points of M .
Lift them to points jj and Ti, respect~vely, in z. Then the geodesic segment
from p to Fj projects to a geodesic segment from p to q. Thus the geodesic
connectivity of the cover implies the geodesic connectivity of the base.
11.3 LORENTZIAN RXDAMARD-CARTAX THEOREMS 423
It remains to show that exp, is a covering map. Select a point p E sr - ' (p ) .
Let D, 2 T,M be the domain of exp, and D y 5 TF % be the domain of
expp. The composition a o exp-, , Dy + M is a covering map since exp-, is
a diffeomorphism onto M and .;; is a covering map w:th domain ;i?. Recall
that n is an isometry of any sufficiently smali neighborhood W' of P to a
corresponding ne~ghborhood U of p. Also, each geodesic through p projects to
a geodesic through p, and each geodesic though p lifts (uniquely) to a geodesic
through p. It follows that the differential of the map (n / W)-' at p yields a
diffeomorphism f : D, --+ D F such that expp = K 0 expp o f . Consequently.
exp, : Dp --+ M is a covering map, as desired. C;
Versions of Proposit~on 11.22 and Theorem 11.25 for certain classes of sprays
on manifolds have been given in Del Riego and Parker (1995).
CHAPTER 12
SXNGULARPTIES
A common assumption made in studying Riemannian manifolds is that the
spaces under consideration are Cauchy complete or. equivalently, geodesically
complete. This assumption seems reasonable since a large number of important
Riemannian manifolds are complete.
The situation for Lorentzian manifolds 1s quite different. A large number of
the more important Lorentzian manifolds used as models in general reiativiry
fail to be geodesically complete. Also, the problem of completeness is further
complicated by the fact, observed in earlier chapters, that there are a number
of inequivalent forms of completeness for Lorentzian manifolds,
In this chapter we will be concerned wwlth establ~shing theorems which guar-
antee the nonspacelike geodesic incompleteness of a large class of space-tlmes.
These spacetimes contain at least one nonspacelike geodesic which is both
inextendible and incomplete. Such a geodesic has an endpoint in the causal
boundary &A4 which may be thought of as being outside the space-time but
not a t infinity. For example, if y is a future inextendibie and future incomplete
timelike geodesic which has jj E d,M as a future endpoint, then y corresponds
to the path of a "freely falling" test particle which falls to the edge of the
universe (at p ) in finite time.
It has been known for a long t m e in general relativity that a number of
important space-times are nonspacelike incompiete. Nonetheless, this incom-
pleteness was thought to be caused by the symmetries of these models. Thus
it was felt that nonspacelike completeness w s a reasonable assumption for
physically realistic space-times. The argument for this assumption was based
on physical intuition which was evidently unjustified with hindsight [cf. Tipler,
Clarke, and Ellis (1980, Chapter 4)).
426 12 SINGULARITIES
If (M, g) is an inextendible space-time which has an inextendible nonspace-
like geodesic which is incomplete, then ( M , g) is said to have a singularity. The
purpose of this chapter is to establ~sh several singularity (i.e.. incompleteness)
theorems.
Before beginning our study of singularity theory, we pause to explain why.
from a mathema~ical viewpoint. this theory works for all space-times of di-
mension n > 3 but not for spacetimes of dimension two. It ts simply the
fact noted at the beginning of Section 10.3 that no null geodesic in a two-
dimensional space-time contains any conjugate points. Yet a key argument in
proving singularity theorems is showlng that certain curvature conditions force
every complete nonspacelike geodesic to contain a pax of conjugate points.
Familiarity with the notations and some of the basic properties of Jacobi
fields treated in Sections 10.1 and 10.3 will be assunied in this chapter.
12.1 Jacobi Tens~rs
As we saw in Section 10.3 (Definition 10.61 ff.), Jacobi terisors provide
a convenient way of studying conjugate points. Given a timelike geodesic
segment c : [a, bj -4 M, let N(c(t)) denote the (n - 1)-dimensional subspace of
Tc(t$4 consisting of tangent vectors orthogonal to cJ(t) a s in Definition 10.1.
A (1,l) tensor field A(t) on VL(c) is a linear map
for each t 6 [a, b]. Further. a composit~ endomorphism R.4(t) : N(c(t)) 4
Nfcft)) may be defined by
The adjoint A*(t) of A(t) is defined by requiring that
for all v, w E N(c(t)).
A smooth (1,l) tensor field A(t) on VL(c) is said to be a Jacobz tensorjeld
if
A'/ + RA = 0
12.1 JACOB1 TENSORS 427
and
ker(A(t)) nker(A1(t)) = ( 0 )
for all t E [a, b ] If Y is a parallel vector field along c and -4 1s a J ~ c o b ~ tensor
on VL(c). then the vector field J = A ( Y ) satisfies the d~fferentlal equation
J" - R(J,cJ)c' = 0 and hence 1s a Jacobi field The condition ker(A(t)) 17
ker(ill(t)) = (0) for ail t E [a. b] guarantees that ~f Y 1s ;t nonzero parallel
field along c, then J = A(Y) IS a nontrivial Jacobi field. Suppose that A is a
Jacob] tensor on VL(c) with A(a) = 0. If A(tn)(v) = 0 for some to E (a, b ]
and 0 # v E N(c(to)), then letting Y be the unique parallel field along c with
Y( to ) = v. we find that J = A(Y) is a Jacobi field with J ( a ) = J(t0) = 0.
A Jacobi tensor A IS sald to be a Lagrange tensor field ~f
for all t 6 [a, b]. As in the proof of Lemma 10.67, ~t may be shown that a Jacobi
tensor field A is a Lagrange tensor field if A(to) = 0 for some to E [a. b ] .
Remark 12.1. Let c : ja, b] --. M be a unit speed timelike geodesic, and
suppose El. E2,. . . , En is a parallelly propagated orthonormal basis along c
with En = GI. Then N(c(t)) is the span of El, E2,. . . , En-lr and each Jacobi
vector field J along c which is everywhere orthogonal to c' may be expressed in
terms of El. ET, . . . ,En-]. Thus J may be represented as a column vector with
(n - 1) components. Using this representation. let J, = J,(t) be the column
vector corresponding to the Jacobi field J along c which satisfies J ( t o ) = 0
and J 1 ( f n ) ='E,(to) Let
be the (n - 1) x (n - 1) matrix with J,( t ) for the 7th coli~mn Thic: matrix
A(t) is a representation of a Lagrange tensor field along c Using this same
basis El, Ez, . . . , E,-i, the adjolnt A* (t) is represented by the transpose of
A@). The space of Jacob1 fields whlch vanlsh at to and which have derivatives
orthogonal to c' at to may be identifed wiih the span of the columns of A. Tnus
conjugate points of c(to) along c are exactly the points where det A(t) = 0.
Hence det A(t) has isolated zeroes on the interval [a, b] . Also, the multiplicity
of a conjugate point t = t1 to to along c is just the nullity of A(tl) : iV(c(tl)) -,
N(c(t1)).
The fact that &to) = O and A1(to) = E is essential to the above discussion.
Lagrange tensors along a timelike geodesic c : J + ( M , g) may be constructed
which are singular at distinct to, t l E J , yet c has no conjugate points. Fbr
example, let ( M , g) be IR3 with the Lorentzian metric ds2 = -dz2 + dy2 + dz2
and let c(t) = (t, 0,O). Let El = d/By and E2 = d/Bz. Then if A is the Jxobi
tensor along c with the matrix representation
with respect to El 0 c and E2 o c, we have A' = E and A* = A so that
(A1)*A-A*A1 = 0, and A is a Lagrange tensor. Evidently, A(o)(E~(c(o))) = O
and ~ ( l ) ( E ~ ( c ( l ) ) ) = 0. But c has no conjugate points since (IE3,ds2) is
Minkowski &space.
W-e now define the expansion, vorticity, and shear of a Jacobi tensor A along
the timelike geodesic c : [a, b] + M. As before, E = E(t) will represent the
(1, l) tensor field on vi(c) such that E(t) = Id : N(c(t)) -+ Ar(c(t)) for each
t. Note that this definition of B from A parallels the passage in O.D.E. theory
from the Jacobi equation to the associated Riccati equation.
Definition 12.2. (Expanston. Vortzczty. and Shear Tensors) Let A be
a Jacobi tensor field along a timelike geodesic, and set B = A'A-I at points
where A-I is defined.
(1) The expansion B is defined by
6 = tr(B).
(2) The vorticzty tensor w is defined by
1 w = - (B - B*).
2
(3) The shear tensor 0 is defined by
Using an orthonormal basis of parallel fields for VL(c) and matrix algebra,
it may be shown that
e = tr(AIA-'1 = (det A)-'(det A)'
Thus if A is a Jacobi tensor field with A(&) = 0, A1(to) = E . and IB(t)l -i oo
as t + t l , then detA(tl) = 0 and t = tl is conjugate to to along c.
We now calculate the derivative of B = AIA-' using ( -4 - I ) ' = -A-lAIA-'.
First we have
Using 6' = tr(B) and B = w + o + [O/(n - 1)]E, we obtain
where we have used tr(w) = tr(a) = tr(w0) = 0. Using the orthonormal basis
El, Ezl . . . , En along c with En = cl, we find that
= Ric(cl, c'j.
and - 1 82 6 = - t r ( 3 ) - t r (z2) - t r ( ~ ' ) - -
n - 2' We may calculate t r ( x ) as follows. Let V ( P ) denote the geometric realiza-
tion for G(P) constructed as in (40.28) of Section 10.3: and let (Yl,. . . , Yn-2)
be an orthonormal basis for V(P) at every point of ,B. Extend (Y', . . . , Yn-*) to an orthonormal basis {YI, Yz, . . . , Y,) along P , where Y, is timelike and
,B1 = (Yn-l + y,)/h. Then we have
using the basic properties of the curvature tensor. Consequently, we obtain
This yields the Raychaudhuri equation for Jacobi tensors along null geodesics:
The same reasoning as in the proof of Lemma 12.3 shows we may slmplify - * - I
the above equation when ;? is a Lagrange tensor field ji.e.. when A A =
z'*z). Lemma 12.6. If A is a Lagrange tensor field, then the vorticity tensor Zj
vanishes along PI.
We thus obtain the vorticzty-free Raychaudhuri equation for Lagrange tensor
fields along null geodesics:
12.2 GENERIC AND TIMELIKE CONVERGENCE CONDITIONS 433
12.2 The Generic a n d Tirnelike Convergence Conditions
111 this section we show that if (1Z.l.g) is a space-time of dimemion at
least three which satisfies the generic and tirilelike convergence conditions,
then every complete nonspacelike geodesic contains a pair of conjugate points
[cf. Hawking and Penrose (1970. p. 539)j. The timelike arid null cases are
handled separately. Similar treatments of the ma:erial in this section may be
found in Bolts (1977), Hawking and Ellis (1973, pp. 96-101), and Eschenburg
and 03ullivan (1976).
We may state the definitions of the generic condition and the timelike con-
vergence condition for our purposes in this section ES follows (cf. Propositions
2.7 and 2.11).
Definition 12.7. (Generic Condition) A timelike geodesic c . (n,b) --t
( M , g ) is said to satisfy the generic wndition if there exists some to E (a , b)
such that the curvature endomorphism
is not ~dentlcally zero. A null geodesic /3 : (a, b) --+ ( M . g ) is said to satisfy
the genenc condition if there exists some t o E (a , b) such that the curvature
endomorphism
of the quotient space G(P(to)) is not identically zero. The space-time (M. g) is
said to satisfy the genenc condztaon if each inextendible nonsgacelike geodeslc
satisfies this condition.
In Section 2.5 we have shown that this formuiation of the generic condition
is equivalent to the usual defin~tion given in general relativity; namely. a non-
spacelike geodesic c with tangent vector W satisfie8 the generic condition if
here is some point of c at which
w ~ w ~ u I [ , R ~ ~ ~ ~ ~ ~ % f 0.
434 12 SINGULARITIES
Definition 12.8. (Temeizke Convergence Condztzon) A space-tlme sat-
isfies the tzme13e contergence condatzon ~f Ricjv, v) > O for all nonspacelike
tangent vectors v E TA4.
By continuity, the curvature condition of Definition 12.8 is equivalent to
the tzrnelzke conveqence condztzon of Hawking and Ellis (1973, p 95) that
&c(v,v) > O for all timehke v E TM. Hawking and Ellis (1973, p 95) also
call the curvature cond~tlon h c ( w , w) 2 O for all null w E TM the null
convergence condatzon. In Hawking and Ellis (1973, p 89), a Four-dimensional
space-time (1M.g) with energy-momentum tensor T (cf. Section 2.6) is said to
satisfy the weak energy condatzon if T(v , v) > O for all tlmelike v E T M . If
the Einstein equations hold for the four-d~mens~ona! space-time (M, g) and T
with cosmological constant A, then the cond~tion Bc(v,v) 2 O for all timel~ke
c E TM implies that
T(v, t ( y - :_&) 0(". v)
for all timelike v E TM. Hence in Hawking and Ellis (1973, p. 95). the four-
dimensional space-time (M,g) and energy-momentum tensor T are said to
satisfy the strong energy condztzon if T(v, v) > (trT/S)g(v,u) for all timelike
v E TM. When dimM = 4 and A = 0, this nlay be seen to be equivalent
to the condition R~c('u, v) 2 0 for all timelike z E TM (cf. Sectio11 2 6). In
Hawking and Penrose (1970, p 533), the condition Ric(v, v) 2 0 for d l unit
timelike vectors v E T M is called the energy condztaon. In Frankel (1979) and
Lee (1975), the term "strong energy condition'' is provided the same definition
which we have given to "timelike convergence condition" in Definition 12.
A discussion of the physical interpretat~on of these curvature conditions in
general relativity may he found in Hawking and Ellis (1973. Section 4.3)
As we have just noted above, if ( M g) satisfies the timelike convergence co
dttion or the strong energy condition, then (M, g) satisfies the null convergen
condition. In view of several rigidity theorems associated wlth curvature co
dltlons In R~emannian geometry [cf Cheeger and Ebin (1975. pp. v and v~
1s natural to consrder the impllcatlons of the curvature condition Rc(w. wj
for all null vectors w 6 T M . Applying linear algebraic arguments to each ta
12.2 GENERIC AND TIMELIKE CONVERGENCE CONDITIONS 435
gent space. Dajczer and Komizu (198Oa) have obtained the rigidity result that
if dim M > 3 and Ric(w, w) = O for ail null vectors ui E TM, then (Ad, g)
is Einstein, i.e., Ric = Xg for some constant X E W. Thus if ( M , g ) is not
Einstein, there are some nonzero null Ricci curvatures. Suppose further that
(M, g ) is globally hyperbolic with a smooth globally hyperbolic time function
h : &I - R such that for some Cauchy hypersurface S = h-'(to), all null Ritci
curvatures satisfy Ric(g)(w, w) > O ~f 7i(w) E S. If ( M , g ) also satisfies the
null convergence condition, then M admits a metric gl globally conformal to
g such that the globally hyperbolic space-time (M, gl) satisfies the curvature
condition Ric(gl)(w. w) > 0 for all null vectors w E TA4 [cf. Beem and Ehrlich
(1978, p. 174, Theorem 7.l)j.
An essential step in proving that any complete timelike geodesic in a space-
time satisfying the generic and timelike convergence conditions contains a pair
of conjugate points is the following proposition.
Proposition 12.9. Let c : J -t (M, g) be an ~nextendible timelike geodesic
satisfying Ric(cl(t), cl(t)) > 0 for all t E J. Lct A be a Lagrange tensor field
along c. Suppose that the expansion B(t) = tr(A1(t)A-'(t)) has a negative
[respectively, positive] value B1 = B(tl) at ti E J Then det A(t) = 0 for some
t in the interval from t j to t l - (n - l)/B1 [respectively, some t in the interval
from tl - (n - I)/& to tlj provided that t E J.
Proof. Since Q = (detA)'(detA)-', we have detA(t0) = 0 provided that
101 -+ oo as t -+ to. Thus we need only show that I@( -+ oa on the above
intervals. Put n - 1
s1= -. 81
The \orticity-free hychaudhuri equation (12.2) for timelike geodesics and the
condition Ric(cl: c') > 0 yield the inequality
In the case that < 0, integrating this inequality from tl to t > t l we obtain
436 12 SINGULARITIES
for t E [tl. t l - s l ) . Hence /O(t)l becomes infinite for some t E ( t l , t l - sl
provided that c(t) is defined. In the case that B1 > 0, we obtain for t
(tl - sl , t l i that n - 1
l + si - t l , Hence lB(t)l again becomes irliinite for some t E [tl - sl.tl) provided that c(t
is defined. D
We now show that a timelike geodesic in a space-time that satisfies t
timellke convergence condition and the generic condition must either be
complete or else have a pair of conjugate points.
Proposition 12.10. Let (M,g) be an arbitrary space-time of d~men
n _> 2. Suppose that c : R -r (M,g) is a complete timelike geodesic wh~
satisfies Itic(c1(t). cl(t)) 2 0 for dl t E B. If R(,,c'(tl))cl(tl) : N(c(tl))
N(c(tl)) is not zero for sorne t l E W, then c has a pair of conjugate point
We first prove four lemmas which are needed for the proof of Propositio
12.10 [cf. BGIts (1977, pp. 30-37)].
Lemma 12.11. Let c : [a, b] -r (M, g) be a timelike geodes~c w ~ t
conjugate points Then there is a unique (I, 1) tensor field A on VL(c)
satisfies the diflerexltial equation A1'+RA = 0 with given boundary co
A(a) and A(b).
Proof. Let S be the vector space of (1,1) tensor fields A on VL(c)
A" + RA = 0, and let L(N(c(t))) denote the set of linear endomorphisms
N(c(t)). Define a linear transformation 4 S -+ L(N(c(a))) x L(N(c(b)))
@(A) = (A(a), .4(b)).
Since d i m s = dim(L(N(c(a)))) + dim(L(N(c(b)))) = 2(n - I ) ~ , ~t IS o
necessary to show that $ is injective in order to prove that Q is an isomorph
and establish the existence of a unique solution A. Assume that $(
(A(a), A(b)) = (0,O). If Y(t) is any parallel vector field along c, then J ( t
A(t)Y (t) is a Jacobi field -with J(a) = J(b) = 0. Thus J = 0. However,
Y( t ) was an arbitrary parallel field, this implies that A(t) = 0, which sh
is injective and establishes the lemma. CI
12.2 GENERIC AND TIMELIKE CONVERGENCE CONDITIONS 437
Now let c . [ t l , w ) -+ (M,g ) be a timelike geodesic without conjugate
points and fix s E (t l ,oo), Then by Lemma 12.11, there exists a unique (1,1)
tensor field on VL(c). which we will denote by D,, satisfying the differential
equation D," + RDs = 0 with initial conditions D,(tl) = E and D,(s) = 0.
As D,(tl) = E, we have ker(D,(tl)) nker(DBi(tl)) = 10). Thus L), is a Jacobi
tensor field (cf. Lemma 10.62). Abo, since D,(s) = 0, it follows that D, is a
Lagrange tensor field. It will also be shown dur~ng the course of the proof of
Lemma 12.12 that if A is the Lagrange tensor field on VL(c) with A(tl) = 0
and A1(tl) = E, then Cls1(s) = -(A*)-'(s)
Lemma 12.12. Let c : [tl, oo) -+ M be a timelike geodesic without con-
jugate points. Let A be the unique Lagrange tensor on VL(c) with A(tl) = 0
and A1(tl) = E. Then for each s E (tl, eo) the Lagrange tensor D, on VL(c)
with Ds(tl) = E and D,(s) = 0 satisfies the equation
for all t E ( t s] . Thus D, ( t ) is nonsingular for t E (t 1, s) .
Proof. Set X ( t ) = A(t) S;(A*A)-'(T)~T. It suffices to show that XI' + RX = 0. X(s ) = D,(s) = 0, and X'(s) = D,'(s).
We first check that X" r RX = 0. Differentiating, we obtain
Hence
438 12 SINGULARITTES
But since -4 1s a nonsinguiar Lagrange tensor, (A')' = A*rllA-I, so that
(A*)-l(A*)'(A*)-l = A'A-l(A")-l , and we obtain
We then have
xl'(t) + R ( t ) X ( t ) = (Atl( t) + ~ ( t ) A ( t ) ] / ' (A*A) -~ ( I ) dr = o t
since A1'(t) + R( t )A( t ) = 0. Thus X satisfies the Jacobi differential equatio
Setting t = s, we obtain
x ( s )= ~ ( s ) l s ( ~ * ~ ) - l ( I ) r l r = o
and
s
@*A)-I(.) d~ - A ( s ) ( A * A ) - l ( ~ ) = -(A*)-'(s).
Thus it remalns to check that Dal(s) = -(A*)- '(s) . But ustng R* = R. w
obtain
[(A*)'D, - A*D,'jl = (A*)I1D, + (A*)'DSJ - (A")'Dsl - A'D,"
= (A f ) "D3 - A* Dgl'
= -A*R8DS +A* RD, = 0.
Thus (A*)'D, - A"D,' is parallel along c. -4t t = t1, the initial condltlo
A(t1) = 0 and A'(t1) = E for A mpiy that Ae( t l ) = 0 and (Ae ) ' ( t l )
(Ar )* ( t l ) = E. Hence as D,(t l) = E, we obtain
((A*)'D, - A*DS1)( t l ) = E.
Hence ((A*)'D, - A'DS1)(t) = E for all t. Setting t = A, we have
E = ((A*)'D, - A*Dal)(s) = - (A"Dar)(s )
which implies that Dal(s) = -(A")- ' (s) = X1(s ) . Therefore since D, and
both satisfy A" + RA = 0 and have the same valum and first derivativ
t = s , the tensors must agree for all t.
12.2 GENERIC AND TIMELIKE CONVERGENCE CONDITIONS 439
Finally, the nonsingularity of D,(t) for t E ( t l , s) foliows from the formula
since (A*A)- ' ( t ) is a positive definite, self-adjoint sensor field for all t > tl . C3
Note that while the integral representation of the Lagrange tensor D, along
c satisfying D, ( t l ) = E and D,(s) = 0 given in Le~nrr~a 12.12 was proven only
for t E ( t l , s]. if c is defined for all t E R and has 110 conjugate points, then
D,(t) is defined for all t E R.
We now show that if c : [a .m) -+ (M,g) is a timelike geodesic without
conjugate points. then the above tensor fields D, converge to a Lagrange tensor
field D as s -+ oo. This construction parallels the construction of stable Jacobi
fields in certain classes of complete Riemannian manifolds without conjugate
points [cf. Eschenburg and O'Sullivan (1976, pp. 227 ff ), Green (1958). E. Ropf
(1948, p. 48)).
Lemma 12.13. Let c : la,co) -+ (1W,g) be a timelike geodesic without
conjugate points. For it1 > a and s E [a. x) - ( t l ) , let D, be the Lagrange
tensor field along c determined by D,(tl) = E and D,(s) = 0 Then D(t ) =
lim,,, D,(t) is a Lagrange tensor field. Furthermore, D(t ) is nonsingular for
ail t with tl < t < m.
Proof. CVe first show that Dal( t l ) has a self-adjoint lir~lit ds s --+ x. Since
D, is a Lagrange tensor, (D,'*D,)(tl) = (D,*D,')(tl). Using D,(l l) = E, we
obtain DS1*(tl) = Dal( t l ) . Thus the limit of DS1(tl) must be a self-adjoint
linear map which we will denote by Dr( t l ) . N ( c ( t l ) ) -+ N ( c ( t l ) ) if it exists.
Consequently, we need only show that for each y E N ( c ( t t ) ) , the value of
g(Dsl(t l)y, y) converges to some value g(D1(tl)y, y).
We ~vil! show that the function s - g(DS1(t1)y, y ) is monotone increasing
for all s with tl < s < oo and is bounded from above by g(Da1( t l )y ,y) to
establish the existence of this limit To this end. assume that tl < r < s
Then by TJernma 12 12 we have
440 12 SINGULARITIES
Thus for t E (tl, s) we obtain
where A is the Lagrange tensor field along c satisfying A ( t l ) = 0; A1(t l ) = E,
arid Y ( t ) is the parallel vector field along c with Y ( t l ) = y. Thus for t with
t l < t < T , lt follows that
is given by
Letting t --+ t l and using Y ( t l ) = y and A1(tl) = 27, we then have
u ( o s l ( t ~ ) a 9 ) - g(~ . ' ( t i ) y , 9 ) = g ( [ l g ( ~ * ~ ) - l ( T ) d i ] ti)) Y ( ~ I ) . 1 Since Y is parallel along c. it may be checked by choosing an orthonormal
basis of parallel fields for VL(c) that
Since (A*A)-' = A-~A*-' , this may be written as
which must be positive because (A*)-"(T)Y(T) is a spacelike vector In N(c(
for each T E [r, s]. Thus
g(D8"tl)XA v ) - g(DT1(tl1v. y) > 0,
and the map s -+ g(DS1(tl)y, y) is monotone for all s > t l as required.
12.2 GENERIC AND TIMELIKE CONVERGEKCE CONDITIONS 441
We now show that g(DS1(tl)y, y) < g(Dal(t l)y, y) for all s > t : and any
y E N(c( t l ) ) . Again let Y be the unique pardlei field along c with Y( t l ) = y.
Let J be the piecewise smooth Jacobi field along c / [a. s] given by
DG(t)Y ( t ) for a s t < t l . J ( t ) =
Ds ( t ) Y ( t ) fort1 < t < s.
Also let J, = J 1 [a, t l ] and J, = J l [ t l .s] . Then J ( a ) = J ( s ) = 0 , and J is
well-defined at t = t l since D,(tl) = D,(tl) = E. Using the index form I for
cI [a. s] given in Definition 10.4 of Section 10.1, we obtain
where we have used formula (10.2) of Definition 10,4 and D, ( t ! ) = D, ( t l ) = E.
Since J ( a ) = J ( s ) = 0 and c has no conjugate points in [a, cc). we have
I ( J , J) < 0 by Theorem 10.22. Thus
for all s > t l ; we conclude that the self-adjoint tensor D1(t l ) = lim DS1(t l ) s- -+a
exists.
Now we define D(t ) by setting D(t ) equal to the unique Jacob! tensor along c
which satisfies D(t1) = E and D1(t l ) = lim,,, D,'(tl). Since D ( t ) and D,(t)
both satisfy the differential equation A" + RA = 0 and the initial conditions
of D, approach the initial conditions of D a s s --t w, it follows that D ( t ) =
lim,,, D,(t) and D1(t) = lim,,, DS1(t ) for all t E [a, cc). This inlplies that
the limit D(t) of the Lagrange tensors D,(t) must also be a L- &grange :ensor.
The last statement of the lemma now follows using the representation
w(t) = ~ ( t ) J ~ - ( A * A ) - ~ ( ~ ) d7
and the fact that (A*A)-I ( t ) is a positive definite self-ad~o~nt tensor field for
a l l t> t l . El
442 12 SINGULARITIES
Now divide the Lagrange tensors with A(t1) = E along a complete timelike
geodesic c : (-m, 400) -+ ( M , g ) with Ric(cl, c') 2 0 and R( . . c'(tl))c1(tI) # 0 for some tl E R into two classes L+ and L- as follows [cf. Bolts (1977. p. 36).
Hawking and Ellis (1973, p. 98)], Put
L+ = (A : A is a Lagrange tensor with A(tl) = E and
q t l ) = tr(A'(t1)) Z 0)
and
L- = ( A : A is a Lagrange tengor with A(tl) = E and
B(tl) = tr(A1(tl)) 5 0)
Lemma 12.14. Let c : R -+ (34, g) be a complete timel~ke geodesic such
that Ec(cl , c') 2 0 and R( . , c'(t1))c1(tl) # 0 for some t l E R. Then each
A E L- satisfies detA(t) = 0 for sorrle t > t l , and each A E L+ satisfies
detA(t) = 0 for some t < t i .
Proof if A 6 L-, then B(tl) = tr(AJ(t~)A-'(ti)) = tr(A1(ti)) < 0. Using
the vorticity-free Raychaudhuri equation (12.2) for timelike geodesics with
Ric(ct, c') 2 0 and tr(02) 2 0, we find #( t ) < 0 for all t. Thus B(t) < 0 for
all t 1 t l . If 6(to) < 0 for some to > t i , then the result for A fo l lo~~s from
Proposition 12.9. Assume therefore that B(t) = 0 for t > tl. This implies
B1(t) = 0 for t 2 t l which yields tr(a2) = 0. Hence a = 0 for t > tl since
cr is self-adjoint. CTsing B = 0 and the self-adjointness of B, ure thus have
B = o = 0 which by equation (12.1) implies that = -B2 - B' = 0 for
t 2 t l . in contradiction to R(tl) $ 0.
If A E L+, the proof is similar. U
We now come to the
Proof of Proposztzon 12.10. Let c . R -+ (M,g) be a complete timelike
geodesic with Ric(ci(t), c1(t)) > 0 for all t E R and with R( - ,c'(tl))cl(tl) # 0
for some t1 E R. Suppose that c has no conjugate points. Then let D =
lirn,,, D, be the Lagrange tensor field on Vi(c) with D(tl) = E constructed
12.2 GENERIC AND TIMELIKE CONVERGENCE CONDITIONS 443
in Lemma 12.13 Since c I [tl, m) has no conjugate polnts D(t) is nonsingular
for all t 2 t l . Thus D $ L- by Lemma 12.14. Hence D E L- and moreover,
tr D1(tl) > 0 as D (t L-. Since Di(tlf = lirn,,, DSi(tl) there exists an
s > tl such that tr(DBfjtl)) > 0. Hence by Lemma 12.14, there exist a tz < tl and a nonzero tangent vector v E iV(c(t2)) such that D,(tz)(v) = 0. Recall
also from the proof of Lemma 12.12 that D,(s) = 0 but DS1(s) = -(A*)-'(s)
is nonsingular, Therefore, ~f we let Y E vi(c) be the unique parallel field
dong c with Y(t2) = 'u, then J = DD,(Y) is a nontrivial Jacobi field along c
with Y( t2 ) = Y(s) = 0, In contradiction.
Corolary 12.15. Let ( M , g ) be a space-time of dimension n 2 2 which
satisfies the timelike convergence condition and the generic condizion. Then
each timelike geodesic of (M,g) 1s either incomplete or e!se has a pair of con-
jugate points.
We now consider the existence of conjugate points on null geodesics. The
methods and results for null geodesics are much the same as for timelike
geodesics except that it is now necessary to assume that dimitl 2 3 since
dim G(P) = dim M - 2 and null geodesics in two-dimensional space-times are
free of conjugate points First, using the vorticity-free Fhychaudhuri equation
(12.5) for null geodesics, the following analogue of Proposition 12.9 may be
established with the same type of reasoning in the timel~ke case
Proposition 12.16. Let (M. g) be an arbitrary space-time of dimension
7~ > 3. Suppose that ,b : J -+ ( M . g) 1s an inexteridible null geodesic satisfying
Ric(pl(t), pl(t)) > 0 for all t 6 J. Let 2 be a Lagrange tensor field dong P such 1
that the expansion 8(t) = t rV1( t )A- ( t ) ) = /detz(t)]-'\detz(t)]' has the
negative [respectively, positive] vdue 8i = 8( t l ) at tl E J. Then detx( t ) = 0
for some t in the i n t e r d from t l to tl - (n- 2)/81 [respectrvely. some t in the
interval from tl - (n - 2)/iJ1 to ti] provided that t 5 J.
The null analogue of Proposition 12.10 may also be established for all space- - - - times of dimension n 2 3 using A, B, 8, Zr, etc. in place of the corresponding
A, B, 8. a, etc. used in the proof for timelike geodesics.
444 12 SINGULANTIES
Proposition 12.17. Let /3 . R 4 (M,g) be a complete null geodesic with
Ric(/?'(t), ,f3'(t)) 2 0 for all i E IW. If dimM 2 3 and if z(. , p(t))P'(t) :
G(P(t)) -, G(P(tf) is nonzero for some tl E R, then ,f3 has a pmr of conjugate
points.
Combining this result with Corollary 12.15, we obtain tine following theo-
rem.
Theorem 12.18. Let ( IM, g) be a space-time of dimension n 2 3 which
satisfies the timelike convergence cond~tion and the generic condition. Then
each nonspacelike geodesic in (1W, g) is e~ther ~ncomplete or else has a pair of
conjugate points. Thus every nonspacelike geodesic in (M, g) without conju-
gate points is incomplete.
The material presented in this section may also be treated within the frame
work of conjugate points and oscillation theory in ordinary differential equa
tions [cf. Tipler (1977d, 1978), Chicone and Ehrlich (1980), Ehrlich and Kim
(1994)l. In this approach, the Raychaudhuri equation is transformed by a
change of variables to the differential equation
zU(t) + F(t)x(t) = 0
with
F( t ) = A[Ric(y1(t), yl(t)) + 2a2(t)j
where rn = n - 1 if y is timelike and rn = n - 2 if y is null.
12.3 Focal Points
The concept of a conjugate point along a geodesic can be generalized to t
notion of a focal point of a submanifold. Let H be a nondegenerate subrnani
of the space-time (M,g) . At each p E H the tangent space TFW may
naturally identified with the vectors of T,M which are tangent to H at
The normal space T k H consists of all vectors orthogonal to H at p. Sinc
H is nondegenerate, T k H n T,H = 10,) for all p E W . We -will denote t
exponential map restricted to the norma! bundle TiH by expi. Then
vector X E T k H is said to be a focal poznt of H if (expi), is singular at
12.3 FOCAL POINTS 445
The corresponding point expi(x) of M is said t,o be a focal point o j H along
the geodesic segment expi(tX). When W is a single point, then T,IH = T,M,
and a focal point is just an ordinary conjugate point.
Focal points may also be defined using Jacobi fields and the second funda-
mental form [cf. Bishop and Crittenden (1964, p. 225)j. This approach will
be used in this section following the treatment given in Bolts (1977). Jacobi
fields are used to measure the separation (or deviation) of nearby geodesics.
For example, when a point q 1s conjugate to p along a geodesic c, geodesics
which start a t p with initla1 tangent close to c' at p wlll tend to focus a t q up to
second order. They need not actually pass through q but must pass close to q.
In studying submanifolds one may take a congruence of geodesics orthogonal
to the submanifold and use Jacobi fields to measure the separation of geodesics
in this congruence. If p is a focal point along a geodesic c which is orthogonal
to the submanifold H, then some geodesics close to c and orthogonal to H
tend to focus at p. This is illustrated for the Euclidean plane with the usual
positive definite metric in Figure 12.1 and for Lorentzian manifolds in Figure
In Section 3.5 we defined the second fundamental form S, : TpW x T p H -t R in the direction n, the second fundamental form S T ~ U x T,H x T,H -t R
and the second fundamental form operator L, : T,W - TpH (cf. Defini-
tion 3.48). Recall that, S(n, x, y) = S,(x, y) = S,(y, x) and g(L,(x), y) =
S,(z, y) = g(VXY, n) for n E T i H and z, y E TFH. where X and Y are local
vector field extensions of z, y.
In this section we will primarily be concerned w ~ t h the operator L, : TpH -$
T,W. Note that a vector field 7 which is orthogonal to H at all points of H
defines a (1,1) tensor field Lq on H. We will first consider spacelike hypersur-
ces. If the timelike normal vector field 7j on H sat~sfies g(g,7) = -1, then
we may calculate L, as follows.
Lemma 12.19. Let H be a spacelike hypersurface with unit timelike nor-
mal field 7. Ifx is a vector tangent to W , then L,(x) = -V,v
Pmoj. Since g ( ~ , 9 ) = -1, we have 0 = X(~(V, 7)) = 2g(0,9, 7 ) which
bows that V,q is tangent to W . Now if Y is any vector field tangent to
12 SINGCLARITIES 1
FIGURE 12.1. A curve in the Euclidean plane has its centers of
curvature as its focal points. The osculating circle to the curve y
at t = t o is shown. The segment from $to) to the center p of the
osculating circle contains x In its mterior. The point y lies beyond
p on the ray from ?( to ) through p. For some ~nterval to - €1 < t < t o + €1, the closest point of y to x is ?(to). On the other hand. for
some interval to - €2 < t < t o T €2, the farthest pomt of y(t) from
y is ?(to). Furthennore, the straight lines which are orthogonal to
y near $to) tend to focus at p.
H, then g ( ~ , Y ) = 0. Thus O = x(g(q,Y)) = g(V,q, Y ) $ g(7, V,Y
that g(V,Y, q) = g(-V,q, Y ) . Consequently, g(L,(x), Y) = g(V,Y, q )
g(-V,q,Y). Since Y was arbitrary, the result follows O
Given a unit normal field 77 for the spacelike hypersurface N, the collectio
of unit speed timelike g ~ d ~ i c s orthogonal to W with initial direction
12.3 FOCAL POINTS 447
at q E H determines a congruence of timelike geodesics. Let c he a timelike
geodesic of this congruence intersecting N at q. and let J denote the variation
vector field along c of a one-parameter subfam~ly of the congruence. Then
J is a Jacob1 field which measures the rate of separation of geodesics in the
one-parameter subfamily from c. Since the geodesics of the congruence are
all orthogonal to N, it may be shown using Lemma 1219 that J satisfies the
initial condition
This suggests the following definition of a focal point to a spacelike hypersur-
face in terms of Jacobi fields.
Definition 12.20. (Focal Point on a Timelike Geodesic) Let c be a time-
like geodesic which is orthogonal to the spacelike hypersurfate N at q. A point
p on c is said to be a Jocal po~nt of H along c if there is a nontrivial Jacobi
field J along c such that J is orthogonal to c', vanishes at p, and satisfies
J ' = -L,J at q,
Suppose that A is a Jacob1 tensor along the timelike geodesic c which sat-
isfies A = E and A' = -L A = -L, at the point q, where c intersects the r )
epacelike Iiypersurface N. Then prior to the first focal point, every Jacob1 field
J orthogonal to c which satisfies J' = -L, J at q may be expressed as J = AY
where Y = Y( t ) is a parallel vector field along c whlch is orthogonal to c.
Since there are n - 1 linearly independent parallel vector fields orthogonal to
c, there is an (n - 1)-dimensional vector space of Jacobi fields along c which
satisfy J' = -L,J at q. We now show that such a Jacob1 tensor A satisfying
A = E and A' = -L, at q is in fact a lagrange tensor field.
Lernrna 12.21. Suppose that A is a Jacobi teilsor field along the timelike
geodesic c . Let c be orthogonal co W at ti, and let ZV be the second funds-
mental form operator on H . If A(t l ) = E and A1(t1) = -L,A(tl), then A is
a Lagrange tensor field.
P~oof . The second fundamental form S, at 77 E T'H 1s symmetric, which
implies that L, is self-adjoint at q since g(L , (x ) , y) = SV( zp g) = S7(g; x ) =
12 SINGULARITIES 12.3 FOCAL POINTS 449
g(L,(?~),s). Thus at t l we have that At(tl) = -L,A(tl) = -L, is self-
adjoint. Hence AS'(tl) = A1(tlj. Using A(tl j = A*(tl) = E, it follows that
A1((t)*A(tl) = A*'(t1).4(tl) = A* (tl)A1jtl). Thus A is a Lagrange tensor field
as required. C
The tensor B = A'A-', expansion 0, and shear o of Lagrange tensors
A satisfying the conditions of Lemma 12.21 may be defined as In Sectlon
12.1, Definition 12.2. As before, the expansion 0 of the iagrange tensor A
along c satisfies the vorticity-free hychaudhuri equation (12.2) for timelike
geodesics. We now establish the following analogue of Proposition 12.9 for
spacelike hypersurfaces.
Proposit ion 12.22. Let (M, g) be an arbitrary space-time of di~nension
n 2 2. Stlppose that c : J --+ ( M , g) is an inextcndible timelike geodesic which
satisfies Ric(cl(t), c'(t)) 2 0 for all t E J and is orthogonal to the spacelike hy-
persurface H at q = c(tlf. If - tr(L,) has the negative [respectively, positive]
value 81 at q: then there is a focal point c ( t ) to W for t in thc interval from
tl to tl - (n - I)/&, [respectively, in the interval from tl - (n - l ) /d i to tl ]
provided that t E J.
Proof, By Lemma 12.21, the Jacobi tensor field A along c with initial con-
ditions A(tl) = E and A1(tl) = -L,(,) is a Lagrange tensor field. Also,
= 8(tl) = - tr(L). Hence by Proposition 12.9 the tensor A is singular on
the interval from t l to t l - (n- I)/@!. Thus the result follows from the remarks
following Definition 12.20.
Fbr the purpose of studying focal points to subman~folds. it will be helpful
to have the second variation formuia for the arc length functional at h a ~ d We
thus give a derivation of the first and second variation formulas for complete-
ness. Consider a piecewise smooth variation a : [a, b ] x (-€, 6) 4 ( M , g) of a
piecewise smooth timelike curve c : [a, b] -+ (M, g). Kerice a(t , 0) = c(t) for
all t E [a. b ] , and there is a finite partition a = to < t l < . . < tk-l < tk = b
such that a 1 [t,-l.t,] x (-6, E ) is a smooth variation of c 1 [t,-t, tE ] for each
i = 1.2 , . . . , k (cf. Definition 10.6). We will also assume that the neighboring
curves a, = a ( . , s ) : [a, b ] -+ (?M,g) are timelike for all s with -c < s < E
FIGURE 12.2. A spacelike submanifold H in a Lorentzian manifold
( M . g ) is shown. Were p is a focal point of H along the geodesic c.
The geodesic segment from p to q contains s in its interior, and 4~
lies beyond p on the geodesic c. All nonspacelike curves "close" to
c[q, x] which join a point of H near q to s have length less than or
equal to c[q, XI. On the other hand, the farthest point of H near q
to y is not q. There are points close to q on H which may be joined
to y by timelike curves longer than c[q, gj. Furthermore, there is at
least one curve y on H through q such that a family of geodesics
orthogonal to W with respect to the given Lorentzian metric and
starting on y near q tend to focus at p up to second order.
450 12 SINGULARITIES
(cf. Lemma 10.7). As in Section 10.1, for t with t,...~ < t < t,, we define the
variation vector field V of a: along c by
V ( t ) = (a It,-I, t,] x (-6. E ) ) , - --n ---
Also, set
At , (Y1) = Yi(t:) - Y ' ( t t ) for i = 1,2,. . . , k - 1,
( Y = Y ) and At , (Y t ) = Yt ( t$ )
for any piecewise smooth vector field Y ( t ) along c which is smooth on each
subinterval t , ) .
As in Section 10.1, we will use L(s ) = L(cr,) to denote the length of the
curve t I-+ a ( t , sj. The fir& variation formula for the arc length functional may
then be derived as usual.
Proposition 12.23. Let c . [a, b ] -, (M,g) be a unit speed timelike curve
which is piecewise smooth. I f a: : [a, b ] x ( - 6 . 6 ) -* ( M , g) is a variation of c
through timelike curves, then
Pmof. If L, : (-E, E ) -+ R denotes the arc length runctlon or cu \t.-l,t.j
{s). then L(s) = ~ f = ~ L,(s) and
Thus
12.3 FOCAL POINTS 451
On the other hand, since
and
it follows that
Thus we have
which implies that
as required. C1
If c : [a, b ] - hf is a timelike geodesic, then c" = V,fcl = 0 and A,, (c') = O
for all z = 1,2,. . . , k - 1. Thus for a timellke geodesic, the first variation
formula simplifies as follo~.~s.
Corollary 12.24. If c : [a. b ] -, (&I, g ) is a unit speed timehke geodesic
segment and cu : [a, b ] x ( - E , E ) -+ (hd, g ) IS a variatian of c, chen
Now let H be a spacelike hypersurface and assume that c : /a, bi i M is
unit speed timelike curve with c(a) E H In studying focal points of the
manifold H , attention may be restricted to variations a + [a, b ] x ( - E , E ) -+
g) of c which start on H [i.e., a(a, s) E H for all s E (-E, E ) ] and which end
c(b) [i.e., a(b, s ) = c(b) for all s E ( - E , E ) ] . For these variations, V ( b ) = 0
nd V ( a ) is tangent to H at c(a). Proposition 12.23 then yields the following
rst variation formula:
452 12 SINGULAWTIES 12 3 FOCAL POINTS 453
Given a spacelike hypersurface H without boundary in (M,gf , fix a point miation o f c which is smooth on each s e ~ (t,-l, t,) x (-LC) for the part~tion
q E M - H. Consider the collect~on (possibly empty) of all timelike curv a = to < t l < . < tk = b of [a. b ] . Let V(t) = c r , d / d ~ l ( ~ , ~ ) denote the
which join some point of H to q. If this collection contains a longest curve variation vector field of a along c, and set
c : [a, b ] -+ (M.9). then c must be a smooth timeiike geodesic by the usual N ( t ) = V(t) + g(V(t),c'(t))c'(t) arguments that timehke geodesics locally maximize arc length and using the d d first variational formula to see that c has no corners. Thus assume that the + g a,-,a,- a, =.*%I ,,,, ( a. t ) $ 1 ,,,, ,
unit speed timelike geodesic c : [a, b] + (M, g ) is the longest curve from H to
with a(a , s) E H and a(b, s ) = q for all s with - E 5 s 5 E , then using V(b) =
and Corollary 12.24 we have k
+ ) : g ( ~ ( t * ) , At , (N1) ) - 9(V2?,as~, ct)jL. z=o
Lt(0) = g(V(a), c'(a)).
On the other hand, Lt(0) = 0 since c is of maximal length from H to q. Th -5 ( a*-,a*- ; ;)I [-:g ( v ~ / & (@* $1 . Ll* dt.
V ( a ) is orthogonal to cl(a). Since variations a as above may be constructed
with V(a) E T,(,)H arbitrary, it follows that c must be orthogonal to H
e(a). Thus we have obtained the following standard result. Note also that
is necessary for H to have no boundary in order to be sure that the extre
c is perpendicular to N at c(a).
Proposition 12.25. Let H be a spacelike hypersurface without bound
q = c ( b ) $ H which is of maximal length among all timelike curves from H t
q. Then c is a timelike geodesic segment, and c is orthogonal to H at c(a). 9(Va/asValdt~*a/as, a * a / a ) - 9(Va/ata*alas. oa/asa*a/at) [-g(a,d/dt, a,d/&);1/2
g(Valata*a/as, a,a/at)g(Va,dta*a/as, a*d /d t ) have discontinuitia in its derivative at the t-parameter values {tl , tz, . . . ,
[-g(a,d/&, a,d/&)J1/2g(cy,a/dt. cy,d/at) a t which cr may fail to be smooth. Thus the normal component N = a,d
for ( t , s ) E (t,-*, t,) x (-6. E) Furthermore, we have g(a,B/ds,a,d/bt)a,B/dt of V along c may also fail to be smooth at th
terms of N and V [cf. Bolts (1977, pp. 86-90), Hawking and Ellis
p. 108)].
Proposition 12.26. Let c : [a, b ] -, (M.g) be a unit speed tim
odesic segment, and let cr : [a. b ] x (-6, e) + (M, g) be a piece
since c is a geodesic and g(N, a,d/at)l(,,o) = 0 for all t E (taml, t,). Using
these last two calculations, we then obtain
Thus since
it follows from these calculations that
12.3 FOCAL POINTS 455
Substituting this expression for g(Valat~,d/ds, ValaBa,d/dt)l(,,o) in (12.8)
and using g(a,d/dt, ~ , a / d t ) j ~ , , ~ ) = -1, we obtain
a ~ o / a ~ v a / a ~ a : a, a*;) /lt ,oj - g ( v a / a t ~ . vaIBt-v) / ( t , O ) .
This yields
L,"(Oj =
a V~/~aVt?/na*- . 0 s a*:) 1 - g ( V B / C ~ N , vajal") ( i ,o)
(t 0)
Also
a
since
Since c(t) = a(t, 0) is a geodesic,
a a d a - at = ~g (Oalasa*z, *.at
! t , O )
Therefore,
The equations
456 12 SINGULARITIES 12.3 FOCAL POINTS 457
and V(t ) = a,B/8sI(t,o) together with L = xFE1 L, yield
since V = nT - g(V, cl)c'. Thus
where
Proposition 12.26 has the following consequence.
Corollary 12.27. Let H be a spacelike hypersurface, and assume tl
c : [a, b] -+ (M. g) is a unit speed timelike geodesic which is orthogonal to
at p = c(a) E H Suppose that a : [a, bj x ( - E , E ) 4 (M,g) is a variation
c such that ~ ( n , s ) 6 N and a(b, s) = q = c(b) for all s with - E < s 5 E .
V = a,d/dsj;t,o) and N = V + g(V , cl)c', then
where L,, is the second fundamental form operator of H at p.
Proof. In view of Proposition 12.26 and the equation
it is only necessary to show that -g(Va/asV, c1)j: is equal to g(L,,(Ar), N). To this end, we first note that a(b, s) = q implies that a , B / d ~ / ( ~ , , ) = O for
all s which yields g(Vala,V.cl)lb = 0. Also a (a , s ) E H for all s with - E < s < E implies that a,d/8sj(,,,) is tangential to N for all s , and hence ,&'(a) =
ry,d/f/a~l~,,~~. Let y(s) = a ( a , s) for all s with - E < s < 6. Extend the vector
N(a) E T p H to a local vector field X along N with X o - ( s ) = a*df8s/(,,,)
for all s with - E < s < E . Then
by Definition 3.48. Also let 77 be a unit normal field to H near p with 71(p) =
cl(a). Then we have
But g(a,b/as,.rl o a)l(,,,) = 0 for all s since a,a/bs)(,,,) is tangential to H . Thus we obtain
as required. Here we have used the fact that since XIp = a,d/dslf,,o),
XI,(g(X, 77)) may be calculated as
458 12 SINGULARITIES
In view of Corollary 12.27, the index IH(V, V ) of a vector field V along a
t~melike geodesic orthogonal to a spacelike hypersurface should be defined as
follows [cf. Bolts (1977, p. 94)].
Definition 12.28. (Spacelake Nypersurjace Index Form) Let c : [a, b ] -i
(M.9) be a unit speed timelike geodesic which is orthogonal to a spacehke
hypersurface N at c(a). Assume that Z is a piecewise smooth vector fie1
along c which is orthogonal to c. if Z(a) # 0 and Z(b) = 0, then the zndex
Z *zth respect to H is given by
TH ( z , z ) = I ( z . z ) + g(Lcl(Z), Z)la
where
It-1
where a partition it,) of (a, b] is chosen such that Z is differentiable except at
the t,'s.
Let a : [a, b ] x ( - E , E ) -+ ( M , g) be a variation of the timelike geodesic c, and
assume the variation vector field V = a , d / d ~ j ( ~ , ~ ) satisfies the conditions of
Definition 12.28. Then equation (10.4) of Sect~on 10.1 together with Coroliary
12.27 yield L1'(0) = .fH(V,v). This may also be written
L* * (V) = IH (br, V) ,
thogonai to a spacehke hypersurface H falls to maximize arc !ength to N a
the first focal point. Recall that V L ( c ) consists of plecewlse smooth vect
fieids along c which are orthogonal to c.
Proposition 12.29. Let c . la, b ] -+ M be a unit speed timelike geo
12.3 FOCAL POINTS 459
and Z(b) = 0 such that I H ( Z , Z ) > 0. Consequentl~; there are tlmelike curves
from H to c(b) which are longer than c.
Proof, By hypothesis there exists a nontrivial Jacobi field JI along c with
J1 orthogonal to c. Jl ( t k ) = 0 and Jli(a) = -L,,(,) Jl (a ) . Define a piecewise
smooth Jacobi field J along c hy
Since Atb(J1) # 0 , we may construct a smooth vector field V orthogonal to
c such that V'(a) = V ( a ) = V(b) = 0 and g(V(tk), A t = ( J f ) ) = -1. Defiae a
vector field Z in V L ( c ) by
Then ZJ(a) = -Lct(,)Z(a), and the index I N ( Z . 2) is given by
IH(Z, Z ) = I ( Z , Z ) +g(Lc/(Z) , Z)ja
= I ( Z , Z ) + g ( ~ , l ( r - l ~ - r V ) , r - ' ~ - rV)la
= 1(Z, 2) + r-'g(~,t J , J ) 1, = I ( Z , Z ) + r - 2 g ( - ~ ' , J ) I ~ = T - ~ I ( J , J ) + r21(V, V ) - 2i(J. V ) + T - ~ ~ ( - J' J ) 1, = r 2 i ( v V ) - 2I(J, V ) .
Here we have used
Since J is a piecewise smooth Jacobi field, equation (10.4) of Section 10.1
implies that
I ( J , V ) = g(V(tk) , &,(J1)) = -1.
Consequently,
I H ( Z , Z ) = T ~ I ( V . V ] + 2
460 12 SINGULARITIES
which shows that for sufficiently small T # 0 the index satisfies
IH(Z, Z) > 0.
This last inequality and the condition Z1(a) = -LC,(,) (Z(a) ) imply that there
exist small variations of c with varlation vector field Z which join W to c(b)
and have length greater than c. Cl
We now turn our attention to focal points along null geodesics which are
orthogonal to (n - 2)-dimensional spacelike submanifolds. If W is a spacelike
(n - 2)-dimensional submanifold, then the induced metric on TpH is positive
definite and the induced metric on T;H is a two-dimensional Minkowski metric
for each p E H . Thus there are exactly two null lines through the origin in
T ~ H . Since the time orientation of (M, g) induces a time-orientation on T ~ H , there are thus two well-defined future null directions in T i H for each p E H.
Locally, we may then choose a smooth pseudo-orthonormd basis of vector
fields El, E2,. . , E, on N such that En-1 and En are future directed
null vectors in T k H for p E H. That is,
9(Et, E J ) = 6,, i f l S z , j < n - 2 ,
g(E*. En-I) = g ( E , En) = 0 i f I < i < n - 2 ,
9(En, En) = g(En-1, En-1) = 0,
and g(13n,En-l)=-1.
The null vector fields E,-l and En defined locally on N give rise to second
fundamental forms LE,-& and LE,, respectively, which are locally defined
(I, 1) tensor fields on H.
If p : /a. b ] --, ( M , g ) is a future directed null geodesic with ,b"(a) = E
[or @(a) = En-* (p)] at p E H, the tangent vectors El ( p ) , Ez(p), . . . . En ( p
p may then be parallel translated along P to give a pseudr>-orthonormal b
along /3 whlch will also be denoted by El, E2, . . . , En. The set of vectors no
to ,BJ(t) is thus the space N(P( t ) ) spanned by El, E2,. . . , En-2, P', and we
form the quotier~r, space G(P(t)) = N(B( t ) ) / [p( t ) ] with corresponding
bundle G(P) as in Section 10.3. If 7(- : N(P) -+ G(P) denotes the pr
12.3 FOCAL POINTS 461
map, then r T ~ H : TpH -+ G(P(a)) is a vector space isomorphism. Hence we
may project the second fundamental form operator LEs . T,W --+ T,H to an
operator ZE, : G(P(a)) + G(F(a)) by setting
- LE, = n o LE, (T / TpN)-I for 2 = n - 1, n
Let a : [a, b] x ( - E , E ) --, ( M , g) be a variation of the null geodesic ,O such that
a(a, s) 6 H for all s with -E < s < E . If V = a*d/ds and W = a,6/&, we
will also require that W(a , s ) = E,(a(a,s)) for all s with - E < s < E. Thus
the neighboring curve a ( . , s) starts at a(a, s) on H perpendicular to W and
has as initial direction the null vector En(a(a, s)) for all s.
We now calculate V 1 ( a ) .
Lemma 12.30. Let V = a,d/ds be tangential to N for f = n and s
arbitrary as above. Then V1(a) = -Lp(a) (V(a) ) + XP'(a) for some constant
X E R.
Prooj. Using [V. W ] = [cu,d/ds, a,b/dt] = cu, [a/&, d/6t] = 0, we have
VVW = VwV = V p V at t = a. s = 0.
On the other hand, W(a . s) = E?,(a(a, s ) ) and g(E,,. En) = 0 yield 0 =
V(g(W, W ) ) = 2g(VvW, W ) = 2g(DwV, W ) = 2g(Vp1V, P') at t = a. s = 0.
Thus
Vp(,,V E N(O(a)) = T p H 9 [P1(a)].
Since Lp,(,)(V(a)) E Tp(,)H and 'CiP(,)V E N(,B(a)), ~t thus suffices to
ow that
for every y E T9(,]H to establish the result. To calculate g(Lai( , ) (V(a)) .~) ,
extend y to a local vector field Y along the curve s -+ @(a. s) that 1s tangent
462 12 SINGULAWTIES
to N Then we have
9(Lg~(a)(V(a)) , l l ! = 9 ( ~ V Y I ( , , , ) ? B ' ( ~ ) )
= 9(VvY . Wlj(,,o)
- - a. a/ (dY. W ) ) - g!Y, vv W ) l b 0 ) 8.3 (a,,)
= 0 - g(Y. VV) I ( a , o ) = -.Y(Y~ v~' (a)V)
where we have used a*3/Dbs\(,,o)(g(Y, W ) ) = 0 since g(Y, W) = g(Y. En) = 0
along the curve s + a(a , s ) . O
If V is to be a Jacobi field measuring the separation of a congruence of nu1
geodesics perpendicular to H , then the last result implies that the vector class - V = n(V) along ,l3 should satisfy the Initial condition
- I V (a) = -Z,p(,)V(a)
at p = P(a) E H. Here zp, = x o Lp, o (n / T,H)-' as above This rnotivat
the following definition of focal point along a null geodeslc perpendicular to
{cf. Hawking and Ellis (1973, p. 102), Bolts (1977, p. 58)].
Definition 12.31. (Focal Poznt on a Null Geodesgc) Let H be a spacelik
submanifold of dimension (n - 2) , and suppose that 13 . [a, b ] - (M, g) is
null geodesic orthogonal to H at p = P ( a ) Then to E (a , b ] is said to be
focal poznt of N along ,l3 if there is a nontrivial smooth Jacobi class 7 in G(P
such that ; f t ( a ) = -Zpt(,) J ( a ) and ; f ( t o ) = [BJ( to)] .
We noted above that a t~melike geodesic orthogonal to a spacellke hy
surface falls to be the longest nonspacehke curve to the hypersurface a
the first focal point (cf. Proposition 12.29). A simllar result holds for a n
geodesic which is orthogonal to an (n - 2)-dimensional spacel~ke submanlfol
[cf. Hawking and Ellis (1973, p. 116), Bdlts (1977, p. 123)]. We sta
result as the following propohition.
12.3 FOCAL POINTS 463
Proposition 12.32. Let N be a spacelike submanifofd of ( M , g ) of dimen-
sion n - 2, and let p : [a, b ] -+ ( M I g) be a null geodesic orthogonal a t $(a) to
H . If to E (a, b) is a focal point of H along P, then there is a timelike curve
from N to P(b). Thus P does not maximize the distance to N after the first
focal point.
A simple example of a focal point is shown in Figure 12.3.
Using the same type of reasoning as in Proposition 12.22, the following
result may also be established.
Proposition 12.33. Let (hf, g) be a space-time of dimension n 2 3, and
let H be a spacelike submanifold of dimension 12 - 2. Suppose that P : J -+
(M. g) is an inextcndible null geodesic which is orthogonal to H at p = P(tl )
and satisfies the curvature condition Ric(p1(t).,8'(t)) 2 0 for all t E J. If
- tr(Lp.(t,)) has the negative Ircspectively. positive] value fI1 at p, then there is
a focal point to to H alongp in the parameter interval from t l to t l - (n- 2)/01
[respectively, in the parameter interval from t l - (n - 2)/Q1 to t l ] provided
that to E J
A particularly important case occurs when H is a compact spacelike (n-2)-
dimensional submanifold which satisfies the condition (tr LEn) (tr LE,-l) > 0
at each point [cf. Hawking and Ellis (1973. p. 262)) Here if (el, e2,. . . , en-z)
is an orthonormal basis for T,H, then t r LEn may be calculated as
Definition 12.34. (Clo.qea' lhpped Surface) Suppose that N is a com-
pact spacelike submanifold withoiit boundary of ( M , g ) of dimension n - 2.
Let En and En-I be linearly independent future directed null vector fields on
N as above. Assume that L1 and L2 are the second fundamental forms on H
corresponding to En and En-', respectively. Then N is said to be a closed
ce if t r LI and trL2 are bosh elther always pos~tive or always
FIGURE 12.3. Let (M, g ) be three-dimensional Minkowski space-
time with the usual metric ds2 = -dt2+dz2+dy2, Let H be a circle
of radius a in the z-y plane. Then H is a spacelike submanifold
of codimension two. For each q E N , there are exactly two future
directed null geodesics and ,B2 through q which are orthogonal
to H . The focal point of M along Dl is pl = (Q,O, a), and the focal
point along /?z is pz = (0, 0, -a). The point y1 lies on beyond p l .
All of H - {q) is contained in the chronological paqt of g1. Thus
there are timelike curves from points on H arbitrarily close to q to
yl, and pl[q. 5111 does not realize the distance from N to yl.
12.3 FOCAL POINTS 46
A related concept is that of a trapped set jcf. Hawking and Penrose (1970
pp. 534-537):. Recall that the future horisrnos of a bet A is defined by E + ( A ) =
J+ ( A ) - I+ (A) .
Definition 12.35. (Trapped St) A nonempty achronal set A is said tc
befutz~re [respectively, past] trapped if E+(A) [respectively, E-(A)] is compact
A trapped set is a set which is either future trapped or past trapped.
In general, a closed trapped surface need not be a trapped set and vice versa
However, in Proposition 12.45 we will show that under certaln condlt~ons, tht
existence of a closed trapped surface implies either null incompleteness or thf
existence of a trapped set. In establishing Proposition 12.45. we will need tht
following corollary to Proposition 12.33.
Corollary 12.36. Let ( M , g ) be a space-tjmo of dimension n 2 3 whicf
satisfies the condition Ric(v,v) > 0 for all null vectors u E TM. If (M,g:
contains a closed trapped surface H, then either ( I ) or (2) or both is true:
(1) At least one of the sets E f ( H ) or E - ( H ) i s compact.
(2) ( M , g) is nu12 incomplete.
Proof. Assume that ( M , g) is ilull complete and that t r L1 > 0 and tr L2 > G for all q E W . Consider all future directed null geodesics which start at some
point of H and have initial direction either or E, at this point. Each
such geodesic contains a geodesic segment which goes from a point q 6 H to
a first focal point p to W. Using Proposition 12.33 and the cornpactness of H ,
it follows that the union of all such null geodesic segments from I1 to a focal
point is contained in a compact set K consisting of null geodesic segrnents
starting on N. Now if T E E i ( H ) , then T can be joined to H by a past
directed null geodesic but not by a past directed timelike curve. Thus r E K.
Hence E + ( H ) 5 K. To show that E i ( H ) is closed let {x,) be s sequence of
points of E f ( H ) . This sequence has a limit point x E K From the definition
of K we have x E J+(H) . i f .G E I f ( N ) , then the open set I + ( H ) must
contain some elements of the sequence {z,), c~ntradicting x, E E + ( N ) for
all n. Thus x 4 I f ( H ) which yields z E E+(H) . This shows that E + ( H ) is a
closed subset of the compact set K and hence is compact.
-3 12 SINGULARITIES d
FIGURE 12.4. The future Cauchy development D+(S) of S con-
sists of all points p such that every past inextendible nonspacelike
curve through p intersects S .
If we assume that (AM, g) is null complete and that t r L1 < Q and tr L2 < 0
for all q E N, then similar arguments show E - ( H ) is compact This establishes
the corollary.
We now define the Cauchy developments of a closed subset S of (
[cf. Hawking and Ellis (1973, pp. 201-205))
Definition 12.37. (Cazlchy Development) Let S be a closed subset o
(M,g) . The future [respectively, past] Cauchy development Di ( S ) [respec
tively, D - ( S ) ] consists of all points p E M such that every past [respective
future] inextendible nonspacelike curwe through p intersects S. The Cauc
deuelopment of S is D ( S ) = D t ( S ) J D - ( S ) (cf. Figure 12.4).
Closed achronal sets have played an important role in causality the
and singularity theory in general relativity. They have the follo
erty [cf. Hawking and Ellis (1973, pp. 209, 268), Hawking and Penr
p 537)).
Proposit ion 12.38. If S is a closed achronal set in the space-time (
then In t (D(S ) ) = D ( S ) - d D ( S ) is globally hyperbolic.
12.4 THE EXISTENCE OF SINGULARITIES 467
12.4 The Existence of Singularities
In this section we give the proofs of several singularity theorems in general
relativity. In part~cular, we prove the maln theorem of Hawking and Penrose
(1970, p. 533). Our approach is somewhat different from Hawking and Penrose
(1970) in that we show ( M , g) is causally disconnected if :t contains a trapped
set and then apply Theorem 6.3 of Beem and Ehrlich (1979a, p. 172).
The basic technique in proving nonspacelike incornpletenws is to first use
physical or geometric assumptions on (M, g ) to construct an inextendible non-
spacelike geodesic which is maximal and h ~ n c e contains no conjugate points.
If ( M , g ) has dimension n > 3 and satisfies the generic condition and time-
like convergence condition, this geodesic must then be Incomplete by Theorem
12.18.
We first show that a chronological space-time with a sufficient number of
conjugate points is strongly causal [cf Hawking and Penrose (1970, p 536),
Lerner (1972, p 41)]
Proposit ion 12.39. If (M, g ) is a chmnological space-time such that each
inextendible null geodesic has apair of conjugatepoints, then ( M , g ) is strongly
causal.
Proof. Assume that strong causality falls at p E M . Let U be a convex
normal neighborhood of p, and let Vk C U be a sequence of neighborhoods
which converge top. Since strong causality fails at p, for each k there is a future
directed nonspacelike curve yk which starts in V', leaves U , and returns to
Vk. Using Proposition 3.31, one may obtain an inextendible nonspacel~ke limit
curve y through p of the sequence ( y k ) . No two points of y are chronologically
elated since otherwise one could obtain a closed tirrlelike curve, contradicting
he fact that (1%, g) is chronological. Thus y is a null geodesic, This yields a
contradiction since by hypothesis each null geodesic has conjugate po~nts and
thus contains points which may be joined by timelike curves. O
Proposition 12.39 and Theorem 12 18 then imply the followifig result.
468 12 SINGULARlTIES
Proposit ion 12.40. Let ( M , g) be a &rono2ogical space-time of dimen-
sion n 2: 3 which satisfies the generic condition and the timelike convergence
condition. Then (M, g) is either strongly causal or null incomplete.
We may now prove the following singularity theorem. The concept of a
causally disconnected space-time has been given in Section 8.3, Definition
8.11.
Theorem 12.41. Let ( M , g) be a chronological space-time of dimension
n 2 3 which is causally disconnected. If (1W.g) satisfies the generic condition
and the timelike convergence condition, then ( M . g) is nonspacelike incomplete.
Pmf, Assume all nonspacelike geodesics of ( M , g) are complete. By Propo-
sition 12.40. the space-tlme ( M , g ) is strongly causal, and by Theorem 12.18
every nonspacelike geodesic has conjugate points. On the other hand. Theo-
rem 8.13 yields the existence of an inextendible maximal nonspacelike geodesic.
But then this geodesic has no conjugate points, in contradiction. O
Recall that the fut.are [respectively, past] horisrnos of a subset S of ( M ,
is given by E f ( S ) = J+(S) - I f ( S ) [respectively, E - ( S ) = J - ( S ) - 1-(S)
The achronal set S is said to be fit'are trapped [respectively, past trapped]
E+(S) [respectively, E - ( S ) ] is compact. We now give conditions under which
the existence of a trapped set irnpl~es causal disconnection.
Proposition 12-42. Let ( M , g ) be a chronological spacetime of dimension
n 2 3 such that each inextendible null geodesic has a pair of conjugate points.
If ( M , g) contains a future [respectively, past] trapped set S , then ( M . g ) is
causally disconnected by E f (S) [respectively, E- (S)]
Proof First, Proposition 12 39 shows that ( M , g) is strongly causal. If S
assumed to be fi~ture trapped, then Corollary 8.16 yields a frlture inextendibl
timelike curve y in the future Cauchy development D f ( E f (S)). Extend
to a future and past inextendible timelike curve in ( M , g ) , still denoted by
Then y Intersects the achronal set E f ( S ) in exactly one point r. As in
proof of Proposition 8.18, we choose two sequences ( p , ) and (q,) on 7 w
diverge to infinity and satisfy p, << T < qg, for each n. To show that {p
12.4 THE EXISTENCE OF SINGULARITIES 465
(q,). and E 4 ( S ) causally disconnect ( M , g ) , we must show that for each n
every nompacelike curve X : [O, 11 --+ M with X(0) = p, and X(1) = q, meets
E+(S). Given A, extend X to a past inextendible curve 3; by traversing y uy
to p, and then traversing X from p, to q,. As q, E D f ( E + ( S ) ) , the curve - X must intersect E+(S) . Since 7 meets E+(S) only at T , it follows that > intersects E+(S) . This establishes the proposition if S is future trapped. P similar argument n a y be used if S is past trapped 0
Theorem 8.13, Proposition 12.39, and Proposition 12.42 now imply the mair
theorem of Hawking and Penrose (1970, p. 538).
Theorem 12.43. No spacetirne (M, y) of dirnensioi~ n 2 3 can satisfy a1
of the following three require~nents together:
(1) (M, g) contains no closed timelike curves.
(2) Every inextendible nonspacelike geodesic in (M. g) contains a pair o
conjugate points.
(3) There exists a future trapped or past trapped set S in (M, g).
This result of Hawking and Penrose implies the following result which k
more similar to Theorem 12.41.
Theorem 12.44. Let ( M , g ) be a chronologicd space-time of dimensior
n 2 3 which satisfies the generic condition and the timelike convergence condi
tion. If ( M , g ) contains a trapped set, then ( M . g) is nonspacelike incomplete
Recall that a closed trapped surface is a compact spacelike submanifold o
dimension n - 2 for which the trace of both null second fundamental forms L j
and La is either always positive o: always negative (cf Definition 12.34).
Proposition 12.45. Let ( M , g ) be a strongly ca7zsaJ space-time of dimen
sion n 2 3 which satisfies the condition %c(v. v) 2 0 for all null vector,
v E T M . I f (M,g) contains a closed trapped surface N, then at least one o
(1) or (2) is true:
(1) ( M , g) contains a trapped set.
(2 ) ( M , g) is null incomplete.
470 12 SINGULARITIES
Proof. Assume that ( M , g ) IS null complete. Corollary 12.36 then lmpiies
that one of E f (H) or E - ( H ) is compact. We consider the case that E f ( H )
is compact and define S by S = E f ( H ) R H. We will show that S is a futxre
trapped set. Notice that S is achronal since E f (HI is achronal and S is
compact as the intersection of two compact sets.
Since E + ( H ) = J + ( H ) - I L ( H ) , the set S will be nonempty if and only
if H contains some points which are not in I+(N). But ~f H were contained
in I f ( H ) , there would be a finite cover of the compact set H by open sets
I+(pl ) . I+(p2), . . . , l+(p,) with all p, E N. I-Iowever, this would ~mply the
existence of a closed tirnelike curve in (M,g ) (cf. the proof of Proposition
3.10) which would contradict the strong causality of (M, g). Hence S # 0.
In order to show that S is a future trapped set, it is sufficient to prove that
E + ( S ) = E+(H). We will demonstrate this by showing that I A ( S ) = I+(H)
and J + ( S ) = J f ( H ) . To this end, cover the compact set H with a finite
number of open sets Ul. U2.. . . , Uk of (M, g) such that each U, is a convex
normal neighborhood and no nonspacelike curve which leaves U, ever returns.
Since H is a spacelike submanifold, we may also assume that each U, rt H is
achronal by choosing the U, sufficiently small. Clearly, I i ( S ) 5 I+(H) since
S E H. To show I f (H) 5 I f ( S ) , assume that q E I + ( H ) - I+(§) . Then there
exists pl E N with pi << q. Now pl E U,(l) n H for some index i (1 ) . Since
q $ I+(S ) , we have pl S and hence pl $ EC(H) . Thus there exists p2 E H
with p2 << pl. Since U,(l) n H is achronal, pa $ U,(l). Now pz E ? N for some i ( 2 ) # i(l). Again q $ I f ( S ) yields p2 $ E+(N) . Thus there exists
p~ E N with p3 << p2. Furthermore. by construction of the sets U, we have
p~ @ U,(l) U Thus p3 E f l H for some i ( 3 ) different from z ( 1 ) a
i (2) . Continuing in thm fashion, we obtain an infinite sequence pl, ~ 2 . ~ 3
in H with corresponding sets U,(*). U+). . . such that i ( j i ) # i(3z 31 # ja This contradicts the finiteness of the number of sets U, of the giv
cover. Hence I C ( S ) = I f ( H ) . I t remains to show that J+ (S) = J+ ( H
J 4 ( S ) C_ J t ( H ) as S C_ H. Thus assume that q E J+(H) - .T+(S). Th
q $ I + ( S ) = I+(H) , and hence there is a future directed null curve R
some point p E H to the point q. Since p E N and :, 6 I f ( H ) as p <
12.4 THE EXISTENCE OF SINGULARITIES 471
q $ If IN), we have p E Ef ( I f ) Thus p E E-(H) ? N = S which yleldi
q E Jf ( S ) , In contradiction. Thls shows that S 1s a future trapped set.
A similar argument shows that d (M.g) is nuil complete and ~f E - ( H ) i g
compact, then S = E - ( H ) n H IS a past trapped set. Thus the proposition ic
established. 5
It is possible for a trapped set to consist of just a single point. One way
this may arise is as follows [cf. Hawking and Penrose (1970, p. 5431, Hawking
and Ellis (1973. pp. 266-267)]. Let ( M , y) be a space-time of dimension n 2 2
which satisfies the curvature condition Ric(v, c) 2: 0 for all null vectors u E
TM. Suppose that there exists a point p such that on every future directed null
geodesic @ : [O,a) -t ( M , g ) with P(0) = p, the expansion 8 of the Lagrange
tensor fieid Z? on G(P) with x ( 0 ) = 0 and z ' ( 0 ) = E becomes negative for
some t i > 0. Intuitively, each future directed null geodesic emanating from T;
has a point /3(tI) in the future of p at which all future directed null geodesics
are reconverging. Thus provided that, the given null geodesic P can be extended
to the parameter value t l - (n -2 ) /8 ( t i ) . ,D has a future null conjugate point to
t = tl along P. Hence @(t) E I+(p) for all t > tl - ( n - 2) /B( t l ) . Since the set
of null directions at p is a compact set, it follows that E+(p) = J+(p) - I+(p)
is compact provided that ( M , g ) is null geodesically complete. Thus we have
obtained the following result.
Proposition 12.46. Let (M, g) be a space-time of dimension n 2 3 with
Ric(v. v) > 0 for all null vectors 2: E TM. Suppose that there exists a point
p E M such that on every future directed null geodesic P from p = P(O), the expansion $ of the Lagrange tensor field 2 on G(O) with ';i(O) = 0 and - I .4 ( 0 ) = E becomes negatlve for some t l > 0. Then at least one of (1) or (2)
holds:
(1 ) ( p ) is a trapped set, i.e., E+(p) = J+(p) - I+@) is compact;
( 2 ) (A{, gj is null incomplete
We now consider the case of z compact connected spacelike hypersurface S
without boundary in a space-time (M, g). If S is achronal, then E f ( S ) = S S is a trapped set. On the other hand, it may happen that S is not
472 12 SIKGULARiTIES
achronal. In fact, an example may easily be constructed of a compact spacelike
hypersurface S with S 2 I C ( S ) and hence E+(S) = @ (cf. Figure 12.5).
A compact spacelike hypersurface S which is not achronal may be used to
construct an achronal compact spacdike hypersurface in a covering manifold
of M [cf. Geroch (1970), Hawking (1967, p. 194), O'Neill (1983)j. Thus if
a space-time has a compact spacelie hypersurface, then there is a covering
manifold of the given space-time which has a trapped set. But in proving
the nonspacelike incompleten~s of (M, g), we may work with covering mani-
folds just as well as with (M , g) since ( M , g) is nonspacelike incomplete if and
only if each covering manifold of (M, g) equipped with the pullback metric IS
nonspacelike incomplete.
These observations on covering spaces together with Theorem 12.44, Propo-
sition 12.45, and Proposition 12.46 yield the following theorem [cf. Rawkin
and Penrose (1970. p. 544)].
Theorem 12.47. Let (M, g) be a chronological space-time of dimension
n 2 3 which satisfies the generic condition and the timelike convergence co
dition. Then the space-time (M, g) is nonspaceIike incomplete if any of th
following three conditions is satisfied:
(1) ( M , g) has a closed trapped surface.
(2) (M, g) has a point p such that each null geodesic s tar t~ng a t p is recon
verging somewhere in the future (or past) of p.
(3) (M, g) has a compact spacelike hypersurface.
Remark 12.48. Conditions (1) and (2) of Theorem 12.47 are reason
cosmological assumptions. Robertson-Walker space-tirnes with physical1
sonable energy momentturn tensors, positive energy density, and A = 0
closed trapped sxrfaces [cf. Hawkin
(1). There is some astronomical evi
p. 355)j.
12.5 Smooth Boundaries
In this section we consider the relationship between causal disconnec
nompacelike geodesic incompleteness, and points of the causal boundary
12.5 SA4OOTH BOUNDARIES
FIGURE 12.5. Let M = 5% S1 be given the Lorentzian metric
ds2 = dBI2 - dBz2. The set S = {(01, 0) : 1 9 ~ E S1) is a compact
spacelike submanifold of codimension one such that E+(S) = 0. Here S is not a trapped set because - it is not achronal. However.
(M, g) has a covering manifold ( M , i j ) which contains a trapped set
#? which is diffeomorphic to S.
474 12 SINGULARTTIES
(cf. Section 6.4) at which the boundary is differentiable. Many of the more im-
portant spacetimes studied in general relativity have causal boundaries which
are differentiable at a large number of points. For example, the differentiable
part of the causal boundary of Minkowski spacetime consists of JC and 3-
(cf. Figure 5.4). Since these sets correspond to null hypersmfaces, it is tllus
natural to call the points of J+ and 5- null boundary points. Penrose (1968)
has used conformal methods to study smooth boundary points of Minkowski
space-time and other space-times.
In general, we consider a space-time ( M , g) with causal boundary d,M and
let M* denote the causal completion A4 U DcM of (M,g). Placlng various
causality conditions on ( M , g ) (e.g., stable causality) ensures that this com-
pletion may be given a Hausdorff topology such that the original topology on
A 4 agrees with the topology induced on iM as a subset of Mycf . Hawking and
Ellis (1973, pp. 220-2211, Rube (1988). Szabados (1988), or Section 6.41
Assume that j j EcM and let U L = Ut( j j ) be a neighborhood of p in A4
Denote by (U, g) the metric g restricted to the set
A confomak representation of U8(ji) will be a space-time (M'. g') and a hom
omorphic embedding f : U* -t M' such that:
(1) f I U is a smooth map; and
(2) There is a smooth function R : U -+ R such that R > 0 and fig = f *gl
on U (Figure 12.6).
If the conformal representation f : U* -t MI maps U* to a smooth manifo
with boundary, then we will say that jj is a smooth bozlnda9-g poznt.
Definition 12.49. (Smooth Causal Boundary Point) Let U* (p) have
smooth conformal representation f : U* + M' such that f ( U ) is a smo
manifold with a smooth boundary d( f ( U ) ) in M'. Then the point
to be a smooth spacelike (respectively. null, tzmebzke) boundary poznt
corresponding boundary
hypersurface in (MI. 9').
a( f ( U ) ) is a spacelike (respectively, null,
12.5 SMOOTH BOUNDARIES 45
FIGURE 12.6. The space-time ( M , g ) has a causal boundary point
j?, and U*(B) is a neighborhood of p in the causal co~npletion M' =
MU a,M of (M. g). The homeomorphic embedding j : U * --+ M'
is a smooth conformal map on U = Um(p) n M .
If -y : [a, 5) --+ U is a curve in M such that ~ ( t ) --t p E M* - Li as t -+ b, ther
the curve y is said to have the boundary point ,5 E M* as an endpoint, Also i
$j is a smooth spacelike boundary point. then it is not hard to find a compact
set K in M such that all inextendible nonspacelike curves with one endpoint at
must intersect K In fact, K may be chosen as a compact, achronal spacelikg
ypersurface with boundary (cf. Figures 12.7 and 12.8) Furthermore, giver
any neighborhood U*(p) of j3 in M f . the compact set K may be chosen to lit
= u * ( p ) n M .
12 SINGULARITIES
Theorem 12.51. Let ( M . g) be a chronological space-time of dimension
n 2 3 which satisfies the generic condition and the timelike convergence con-
dition. If (M, g) has a smooth spacelike boundary point, then (M, g) is non-
spacelike incomplete. CHAPTER 13
Proof. This follows from Lemma 12.50 and Theorem 12.41. GRAVITATIONAL PLANE WAVE SPACCTIMES
Notice that if (141, g) has one smooth spacelike (respectively, null, timelike)
boundary point, then (M, g) has uncountably many smooth spacelike (respec-
tively, null, timelike) boundary points. For if jj 6 d,hf corresponds to the It is well known and already mentioned in earlier chapters that for general
point z E d( f ( U ) ) under the given conformal representation J : U* -+ M' of space-times, geodesic connectedness and gcodcsic completeness are not well Definition 12.49, then points y f L?(J(U)) close to x wili dso represent smooth
related. Indeed, in Figure 6.1 we have given an example of an elementary spacelike (respectively, null, timelike) boundary points in a,M under the given space-time which is geodesically complete yet have indicated in this figure a conformal representation and d(f (U)) has dimension n - 1 in M'. Hence M point q in I+(p) which is not joined to p by any geodesic whatsoever) let alone contains uncountably many smooth spacelike boundary points. Thus using
by a maximal timelike geodesic. More dramatically, in this same example a the fact that a causally disconnecting set K may be chosen arbitrarily close to whole open neighborhood U about q exists, with U contained in I + ( p ) . yet any smooth spacelike boundary point jj and the result that a limit of maximal
there is no geodesic from p to any point of U . Hence: despite thc fact that curves is maximal in a strongly causal space-time, we may also establish the this space-time is geodesically complete, the exponential map from p fails to following result. map onto open subsets of IT(p). In Section 11.3, geodesic connectedness was
Theorem 12.52. Let (M, g) be a strongly causal space-time of dimension explored from a general viewpoint and related to geodesic pseudoconvexity
n 2 3. Assume that (iM, g) has a smooth spacelike boundary point and sat- and geodesic disprisonment.
isfies the generic condition and the timelike convergence condition. Then for In this chapter we shall instead consider from the viewpoint of this book,
each smooth spacelike boundary point p, there is an incomplete nonspacelike and in particular as an illustration of the nonspacelike cut locus defined in
geodesic which is inextendible and has jj as an endpoint. Thus (iM, g) has an Chapter 9, a class of spacetimes which originated ns wt,rophysical examples
tlncountable number of incomplete, inextendible, nonspacelike geodesics. yet which display a novel, very non-ltiemannian geodesic behavior. Penrose
Pmof. First, it follows from the preceding remarks that (M,g) contains (1965a) was apparently sufficiently impressed with the fact that the focusing
an incomplete nonspacelike geodesic for each smooth nonspacelike boundary behavior of future null geodesics issuing from appropriately chosen points p
point. But we have just noted that if (M, g) contains one spacelike boundary precluded this class of space-times from being globally hyperbolic that he
point, it contains uncountably many spacelike boundary points. chose the title "A remarkable property of plane waves in general relativity" for
(1965a). In a series of articles by Ehrlich and Emch (1992a,b 1993), a detailed
investigation of the behavior of all geodesics issuing from appropriately chosen
p was conducted, and in so doing it was possible to place this class of space-
times precisely in the standard causality ladder as being causally continuous
yet not causally simple. Also, the future nonspacelike cut locus was determined
480 13 GWITATIONAL PLANE WAVE SPACE-TIMES
and seen to coincide with the first future nonspacelike conjugate locus of p.
Unfortunately. since this class of space-time5 was known in advance to fail to
be globally hyperbolic, the theorems of Chapter 9 were not applicable here,
only the basic concepts and definitions.
In Chapters 35 and 36 of Misner, Thorne, and Wheeler (1973), extensive
discussions of gavitational waves as small scale ripples in the shape of space-
time rolling across space-time is given by analogy with water waves a? smali
ripples in the shape of the ocean's surface rolling across the ocean. These
gravitational ripples are supposed to be produced by sources like binary stars,
supernovae, or the gravitational collapse of a star to form a black hole. The
exact plane wave solutions then arise as simplified models for this phenom-
enon, compromising between reality and complexity to paraphrase Misner,
Thorne, and Wheeler (1973). We wili refer the reader to t h s source rather
than discussing the physics of these models in this chapter.
13.1 The Metric, Geodesics, and Curvature
This class of spacetimes, in the form in which we will describe them below,
seems to originate with N. Brinkmann (1925) and sterns from the work of
Einstein jcf. Einstein and Rosen (1937)) These metrics were brought to the
renewed attention of the general relativity community in the late 1950's by
I. Robinson [cf. Bondi, Pirani, and Robinson (1959) for a discussion of this
history]. Let M = B4 with global coordinates (y, z, v, u). Heuristically, we
may think of starting with the usual Midowski coordinates and then setting
I I (13.1) u = - ( t -
45 a) and v = - - ( t + x ) . JZ
In these coordinates, the Minkowski metric 7 has the form
We also make a choice of time orientation which will be convenieni in the
sequel, so that B/Bv is past directed null at all points. Now the class of "plane
fronted waves" may be defined to be those metrics for M = B4 with global
coordinates (y, z , V , u) which may be written in the form
13.1 THE iMETRiC, GEODESICS, AND CUR\.'ATURE 45 1
where H(y, z, u) has an appropr~ate degree of regularity and is nonvanishing
somewhere. The global coordinate u : M -+ R plays an important role in the
differential geometry of this c l a s of space-times and is even more important for
the subclass of gravitational plane waves ~ncroduced below in (13.9). The plane
fronted metrics (13 3) have the interesting property that all have vanishing
scalar curvature but need not be Rcci fiat unless
No general statement can be made about the geodesic completeness of this
class of spacetimes apart from the geodesics lying in the null hyperplanes
On these hyperplanes, the ambient metric (13.3) is degenerate, so these are
not "Lorentzian submanifolds" in the sense of O'Neill (1983) or of our pre-
vious Definition 3.47. Denoting y, Z ,V ,U by 1 ,2 ,3 .4 respectively, the only
non-vanishing Chfistoffel symbols are
and
By direct calculation, it may be verified that any geodesic e : (a , b) --t M
satisfies ( ~ 4 0 c)"(t) = 0. Then in the case that, 24 0 e( t ) = uo for all t , one
obtains that the geodesics in the nail hyperplanes F(710) are precisely stra~ght
lines lying in P(uo) of the form
which are ma~lifestly complete.
These geodesics are either spacelike or null; we will reserve the notation
482 13 GRAVITATIONAL PLANE WAVE SPACETIMES
for the member of the class of future d~rected null geodesics lying in P ( u o )
which passes through the point (yo, zc,O, uo). An aspect of the differential
geometry of plane fronted waves related to this class of null geodesics is the
fact that the vector field gradu = 'Ciu = d/Dv is always a global parallel vector
field, independent of the particular form of W(y, z , u).
Already, it may be seen that the global coordinate u : M -+ R has many
aspects of a time function. First. we set things up in analogy with (13.1) so
that is past directed null. Thus it follows that u is strictly increas~ng
along any smooth future directed timelike curve c( t ) , for
as cl(t) is future timelike and d/dv is past directed nonspacelike. Hence, any
plane fronted wave is chronological. &rthermore, this coordinate function
u is strictly increasing along any smooth future causal geodesic except for
those null geodesics q ( s ) of the form (13.6). For writing an arbitrary geodesic
as ~ ( s ) = ( ~ ( s ) , z ( s ) , ~ ( s ) , u ( s ) ) , the geodesic differential equation reduces as
noted above to ul ' (s) = 0 for this last component. Hence, if u is nonconstant,
it may be rescaled to be u(s ) = s , and
Thus u is indeed strictly increasing along all smooth null geodesics except for
those of the form (13.6) lying in one of the null planes P(uO). Recall that
a chronological space-time which fails to be causal contains a null gcodesic
/? : [0,1] -+ M with P(0) = P(1). But such a segment cannot lie in one of the
planE% P(u0) hecame all null geodesics in these planes are injective by direct
solution of the geodesic PD.E.; hence P must have the form (13.8), but then
the form of the iast coordinate precludes P(0) = p(1) from occurring Thm
the nature of the global coordinate u and the form of the geodesics in the null
planes P(uO) ensure that all members of the class of plane franted waves are
causal space-time.
In Eardley, Isenberg, Illarsden, and Moncrief (1986), it was shown that a
space-time which is a non-trivial vacuum solution of the Einstein equations
and which admits at least one non-homothetic conformal Killing vector field
must be a plane fronted wave.
13.1 THE METRIC, GEODESICS, AND CURVATURE 48:
Definition 13.1. (Gmv~tat ional Plane Wave) The manifold M = R4 together with a plane fronted metric g = q + H(u, y, z)du2 for which H takes
the quadratic form
or in matrix notation
with f ( u ) or g ( u ) not vanishing identically, is said to be a gravatational plane
wave. The wave may he said to be polarezed if g ( u ) = 0. The wave is commonly
termed a sandwa'cli wave if both functions f and g have compact support in u.
I t is interesting that as a consequence of the quadratic nature of (13.9),
or equivalently (13.10), the same linear second order system of differential
equations
arises in studying the Kllhng fields, the Jacobi fields, and the geodesics of
these space-times. C Ahlbrandt has commented that this is to be expected
for the last two of these objects by the classical calculus of variations. given
the quadratic nature of the metric In (13.9) and hence of the assoc~ated art
length functional. Also, all gravitat~onal plane waves are geodesically complete
and satisfy Rie = 0 but are not flat since e~ther f or g 1s assumed to be
nonvanishing. Thus, dny geodesic conjugacy whlch arlses may be attributed
to the shear term rather then the Ricci curvature in equation (12.2) of Sectlon
12.1. It may be checked directly that any future nonspacelike geodesic, other
than one of the complete null geodesics of the form (13.6). experiences some
nonzero timelike sectional curvatures and hence saiisfies the generic cond1t:on
(2.40).
We now turn to a more careful discussion of the geodesics of a plane wave
space-time which are transverse to the null planes P ( u o ) . As previously noted,
such geodesics may be parametrized with U ( S ) = S. or equivalently, as
484 13 GR4VIT-ATIONAL PLANE WAVE SPACE-TIMES
Kow in the case of the gravitational wave, the ChristoRel symbols simplify
to
I = 5 [f '(u) ty2 - z2) + 2g1(u)yz] .
r;* = ril = -rid = ~ ( Z L ) Y I g ( ~ ) ~ ,
and I?:* = I?& = = g(u)y - f (u)z.
Hence, the y and z components of the geodesic P.D E read
(13.13) y"(u) = f(u)y +g(u)z and
(13.14) ~ " ( u ) = - f (u)z
as claimed above. Hence, it 1s ~mrned~ate that if f and g are continuous,
then different~able solutions to (13.13) and (13.14) exist for all values of the
parameter u. Rather than using the geodesic differentlal equatlon to solve
v(u), it is convenient to give the following indirect argument. Suppose we wish
to find the geodesic c(t) with g(cl, c') = A. As cl(u) = (yl(u), zl(u), v'(u) I),
this condition is expressed as
x = q(cf, c') = (7Jf)2 4- (z1)2 + 2vt(u) . 1 + [f (u)(g2 - z2) + 2g(u)yz] - 1. I
or
x = (v1)2 + (z1)2 + (f (u)y + g(t4)z)y + (g(u)y - f(u)z) 2 + 22').
Using (13.13) and (13.14), t h s last equatlon becomes
X = (y')2 + (zf)2 + yl1g + zllz + 22'
so that vl(u) = [-(yyl)' - (zzl)' + A] Integration produces the equation
1 (13.15) v(u) = - [-y(u)y'(u) - z(u)zJ(u) -Xu + C]
2
which shows that slnce y(u) and z(u) are d~fferentiable for all values of u, v
may be calculated for all values of u by employing (13 15). T h ~ s calculatlo
together with our previous knowledge about the geodesics of the null
planes P(zbo), gives an elementary direct proof of the geodesic completen
13.1 THE METRIC. GEODESICS, AND CURVATURE 485
for this class of metrics; an alternate proof may be given based on a knowledge
of the isometry group of this class of space-times. Equaiions (13.12)-(13.14)
reveal that an important first step in understanding the geodesic behavior of
these space-times is studying the system (13.11).
Thorough studies of the Killing vector fields and related isometry groups
of these space-times have been given in Ehlers and Kundt (1962), Krarner,
Stephani, MacCallum, Herit, and Schmutzer (1980), and Ehrlich and Emch
(1992a), among others. Rom t h ~ natttlue of the isometry group, i t follows that
if the behavior of the exponential map at Po = (0.0,O. ug) is understood, then
as >sometries map geodesics to geodesics, the same behavior obtains at an
arb~trary point P with u(P) = uo. With choice of initial co~d~t ions y(uo) =
Z(UO) = v(u0) = 0. equation (13.15) simplifies to
In Emch and Ehrlich (1992a,b), a conjugacy index associated to this system
with the preceding initial condition was discussed. For u, E R, let
(13.17) L(75) = (solutions (y(a). z(u)) to (13.11) with y(uj) = z(u3) = 0).
Definition 13.2. (Conpgacy Index) For distinct real numbers u g < ul,
define the conjugucy indez I(u0, ul) of O.D.E. system (13.11) with respect to
uo and ul to be
which, given our dimension restriction, is either 0. 1, or 2. The pair {uo, ul) is
said to be conjugate provided i(uO. uI) > 0 and astagmatzc conjugate provided
that I(uo u l ) = I. Also, {uo, u l ) is said to be a fir& conjugate pazr provided
that l(uO. 2 ~ ~ ) > 0 and I(vo, T I ) = 0 for all u E (uo, ul). Also. !ntroducc thc
notation
(13.19) Conn(Po. u) = (Q E P(u) . there is a geodesic from Po to Q).
It is a consequence of Propos~t~ons 12.15 and 12.17 that every gravitational
plane wave space-time admits a first conjugate pax. This was esteblished
456 13 GRAVITATIONAL PLANE WAVE SPACE-TIMES
by direct considerat~ons in a symplectic context in Lemma 2.4 of Ehrlich and
Emch (1992a).
In Ehrlich and Emch (1992aj the following facts were established by study-
ing the exponential map and geodesic equation associated to a first astigmatic
conjugate pair uo < UI for an arb~trary plane wave: let PO E P(uo) b ~ . arii-
trary. Then
(1) if uo 5 u(Q) < ~ 1 , then there exists a unique geodesic (parametrize
as discussed above) from PO to Q; hence dim Conn(P0, u) = 3 for all
in [uo.ul).
(2) dimConn(P0. u l ) = 2 and Conn(P0~ul) is in fact a plane in P(ul
every point of which is conjugate to Po by a one-parameter family o
geodesics.
(3) If u(&) > u1, then Q E 14(Po).
(4) If uo 5 u(Q) < U I , then Q E I+(Po) if and only if Q lies on a uniqu
maximal timelike geodesic segment issuing from Po.
Also, the first future nonspacelike conjugate locus and the future non
spacelike cut locus of Po in Conn(&,ul) were seen to coincide. A no
ferentlable example was also given of a space-time of the above form
dimConn(Po,ul) = 1, so this possibility was not ruled out in the consider
ations of Ehrlich and Emch (1992a). Now, the proofs of these results calle
upon the estabhshment of a symplectic framework somewhat outside the view
point of the subject matter of this book Since it was seen in t h ~ s refercnc
that precisely the same behavior obtains in the polarized and nonpolarize
cases, we will restrict our discussion in the next section to the polarized c
for whlch the equations are somewhat less complicated and more easily de
with In order to illustrate the above points.
13.2 Astigmatic Gonjugacy and the Nonspacelike Cut Locus
In thls section we turn to the study of astigmatic or Index 1 conju
and the global geometric behavior of the geodesics issuing from Po As
mentioned, to avoid the complexities of dealing with the general case, which d
not in any way affect the qualitative outcome of the geodesic geometry, we wi
13 2 ASTIGMATIC CONJUGACY, NONSPACELIKE CUT LOCVS 41
restrict our attention to the somewhat simpler case of a polarized gravitation,
wave with g(u) = 0 and also assume f (u) 2 0 Hence the metric takes tk
form
and the geodesic differential equation for geodesics transverse to the null plane
P(u) uncouples as
y'' = f (u) y and
2'' = - f (u)t.
Under the above assumption on f , standard O.D.E. theory shows that equatio
(13.22) admits a nontrivial first conjugate pair solution z(u) with z(u0) - t (u l ) = O and z(u) > 0 for u with uo < u < UI. Fix such a number uo. A
f 2 0, any nontrivial solution y(u) to (13.21) with y(uo) = 0 satisfies y(u) # for all u > UO.
Let us denote by yl(u) [respectively, zl(u)] the solution to the 0.D.E
(13.21) [respectively, (13.22)] with initial conditions
(13.23) YI(UO) = zl(vo) = 0, and ?J:(uo) = -7: (uo) = 1.
Since the isometry group of any gravitational plane wave space-tirile a ~ t s :ran
sitivcly on the null planes P(uo) for any uo 6 W, it is sufficient to understanc
the behavior of geodesics issuing from Po = (0.0.0, uO) to understand expl
for any P E M with u(P) = uo. Evidently, any geodesic c(u) issuing from l?
which is not a straight line lying in P(uo) and which satisfies g(cl . c') = X ma:
be described as
for some choice of constants cl, c2 in R, in view of eqdation (13.16). For an!
u with uo < u < ul, we have arranged ;hat yl(u). zl(u) > 0 Hence, it 1s eas:
488 13 GRAVITATIONAL PLANE WAVE SPACE-TIMES
to check from (13.24) that not only does Conn(Po,u) = P(u) for all such u,
but also given any Q with u(Q) 6 [uo, ul), there is a unique geodes~c from Po
to 4. Now take any poiilk Q = (c, d. e, 2 ~ ~ ) in Conn(Po. 211). Note that
(13.25) C(UI) = 2
Hence, d = 0. Now cl is determined as cl = ~ / y ~ ( 2 ~ ~ ) , possible slnce yl ( u l ) > 0
Given this determination of cl, we then make v(ul) = e by choosing th
constant X to satisfy
(13.26) A = [2e + c2?4l'(W)lyl(ul)]
1'111 - ~ 0 1
and note that this setup is valid for any c2 E B. Hence. we have not on1
established that
(13.27) Conn(P0,ul) = P(u1) n { z = 01,
but also letting c2 vary over all possible values -m < cz < +co, we have show
that each point & in Conn(P0,ul) is conjugate to P by this oneparamete
family of geodesics as cl arld X = g(c', c') are uniquely determined by Q, but
c2 is undetermined. The "'unbounded'" nature of this one-parameter family
geodesics precludes the geodesic system from being pseudoconvex, and als
prevents this class of space-times from being globally hyperbolic [cf. Penros
(1965a)l.
To recapitulate. we have found that as u increases from u = 710. prlor t
reachlng P(ul) each point Q with u(Q) = a is joined to Po by a unique ge
desic, but when u = u1, the geodesic jo~n set Conn(Po, ul) drops from th
mensions to two dimensions, and as that happens, every point in Conn(P0, u
is conjugate to Po by a one-parameter family of geodes~cs as exhibited expl
itly above
The first future null conjugate locus of Po in the null hyperplane
may now be determined by taking X = 0 in the above discussion. Noting
yl'(ill)/yl(ul) > 0 under our hypotheses, and denoting they and v coordina
of ~ - ~ l = ~ l ~ ~ l ~ ~ l ~ ~ ~ l ~ l
, ul 2
13.2 ASTIGMATIC CONJUGACY, NONSPACELIKE CUT LOCUS 48
Conn(Po , u,) n ( z = 0
FIGURE 13.1. The conjugate locus of Po in Conn(Po. u l ) = P(uo)n
( z = 0) for a first conjugate palr (uO, 2 ~ ~ ) and Po = (0,0,0, uo) is
shown.
by Y and V respectively, we obtain the equation
(13.28) v = -[(~/~)YI'(~I)/YI(~I)]~~, 2 = 0,
for the first future null conjugate locus of Po in Conn(Po,ul). Hence. this
null conjugate locus, Nconj(Po), 1s a planar parabola in Conn(Po.ul). This
same phenomenon is shown to occur for an arbitrary gravitational plane wave
in Ehrlich and Emch (1992a, p. 214) The points in Conn(P3, u l ) whrch lie
on timeliice geodesics issuing from Po are precisely those which are In the
same component of Conn(Po,ul) - Nconj(Po) as the focus of the parabola
while those which lie on spacelike geodesics ~ s s u ~ n g from Po !ie in the other
component (cf. Figure 13.1).
490 13 GRAVITATIONAL PLANE WAVE SPACE-TIMES
More exactly, the Eollovii~lg result is established for the general plane wave
space-time in Ehrlich and Emch (1992a, p. 214).
Proposit ion 13.3. In the presence of astigmatic conjugacy from Po, the
future null (respective1,vi nonspacelike) cut locus of Po is prec~sely the first fu-
ture null (respect~vely, noespacelike) conjugate locus to Po as descr~bed abovc.
In particular, every point in the con~ponent of Zonn(P0, ul) -Nconj(Po)
lies in the connected component containing the focus of the parabola Nconj
is a future timclike cut point to Po.
Proof We first recall that these space-times have been known at least since
Penrose (1965a) to fail to be globally hyperbolic; thus we may not use the
existence of maximal nonspacelike geodesic segnleritv in this proof. Instead,
it is helpful to develop a version of the basic Proposition 3.4 which is more
adapted to the sublevel sets of the global coordinate function u : M -+ W rather than considering a small convex neighborhood centered at Po. While
is not a time function in the sense of Section 3.2, u does possess the following
two properties as considered in Ehrlich and Em& (1992a, pp. 200-204).
Definition 13.4. (Quasi-Time Fuactzon) A smooth function f : N
on an arbitrary space time (N, g) is said to be a quasi-tzme functzon provid
(I) The gradient V f is everywhere nonzero. past directed nonspaceli
and
(2) Every null geodes~c segment c such that f o c is constant is injectiv
From our description of the geodesics of gravitational plane wave spac
times and the fact that V f = dldv is past directed null, it is evident that t
global coordinate function u : M -+ P satisfies the properties of this definitio
Property (1) implies that a quwi-time function is also a semi-time func
as defined in Section 3.2, but it is the combination of both properties (1)
(2) in this definition which ensure that an arbitrary space-time which ad
a quasi-time function must he causal While a semi-time function pre
chronology from being violated it does not, in general, prevent the emst
of closed nu1 geodesics.
13 2 ASTIGEvfATIC CONJUGACY, NONSPACELIKE CUT LOCUS 491
Returning to the general setting of a plane wave, for any uo, ul with uo < ZLI,
introduce the notation
Arguing along the lines of (13 7 ) , it may be seen that any suchu-strip has the
convexity property that if m, n are contained in S(110, ul) . and y . [O,1] --+ M
is a future directed nonspacelike curve with y(0) = m and y(1) = n, then
Now we restrict our attention to a strip S(uD,ul) for whirh (t~o, Z L ~ ) is a first
conjugate pair. Then by our above considerations. expro is invertible on this
particular strip. Thus the Synge world function cP : S(u0,u1) + R given hy
is differentiable on S(uo, u1). Using the Sgnge world function. define the fol-
lowing partition of S(uo. Ill). Put
N = @-'(I)) = (points in S ( U ~ ! u l ) that are joined to Po
by a null geodesic),
T = {m E S(uo,ul) : +(m) < 0) = {points in S(UO, ul)
which are joined to Po by a "timelike geodesic),
and
8 = {rn E S(UO, %I) : +(m) > 0) = (points in S(uO, ul)
which are joined to Po by a spacelike geodesic)
Even though T may not be contained in a convex normal neignborhood of
Po, we still have VQ, existing and being future t~melike on all of T Now the
prom~sed technical lemma may be stated
13 GRAVITATIOKAL PLANE WAVE SPACE-TIMES
FIGURE 13.2. The null tail NT(P0) in P(uo) for a first astigmatic
conjugate pair uo < u1 is shown.
Points Q on the null tail NT(P0) then fall into one of two categories: first,
those points lying on Nconj(P0) and thus joined to Po by a one-paramet
family of null geodesics; and second, those points Q not lying on Nconjj
which are thus joined to Po by a one-parameter family of spacelike geod
Penrose (1965a) observed from the "noncompact" behavior of the family of n
geodesics of the type (13.24) from Po which refocus at Qo that PQ, happ
to be a limit curve of this family of null geodesics from Po to Qo, and e
more remarkably, this behavior prevents this class of space-times from b
globally hyperbolic. 111 Ehrlicli axid Emch (1992a, p. 218), we noted that
observation may be used as a step in the proof of the following result.
Lemma 13.9. Suppose uo < u1 is a first astigmatic conjugate pair
u(Q) > u1. Then Q E I+(Po).
Proof. First, choose c > 0 such that (al, .ul -t- c] centalm no u-value co
gate to u = 211. Then from the explicit form of the geodesics for gravi~atio
plane waves, it may be calculated that for any Q in the strip S(uI ,u l
there exists a future timeirke geodesic from a point R on the null ray PQ,
13.3 ASTIGMATIC CONJUGACY, ACHRONAL BOUNDARY 495
((0,O, s, ul) : s E R) to 4. But then since R E I+(Po) as Penrose observed,
we have Q E I+(Po) by future set theory, as I+(Po) = (rn E M : I+(m) 2 I"(P0)) (cf. Proposition 3.7 or corollary 3.5). Now glven an arbitrary Q with
u(Q) > u1, simply take any past directed timelike geodesic c(u) lssulng from
Q, and from knowledge of the quasi-time function u, we know for any u2 !n
(u1,ul + c), that S = c(u2) satisfies S << Q and S 6 S(uI ,u I + C) I+(PO).
Hence, Q E I+(Po) as well. U
Now F = I+(Po) is a particular example of a future sct. and the corre-
sponding achronal boundary is given by
The characterization [cf. Penr~se (1972, Section 5 ) ] of a general achronal
boundary in terms of what relatwists call the null geodeszc generators trans-
lates in our simpler context of (13.33) as follows
Theorem 13.10. Let (N,g) be an arbitra~y space-time, arid let B =
a(I+(Po)) for any Po E N. Suppose l? 6 B - (Po). Then there exists a
null geodesic /3 contained in B with future endpoint R which sat~sfies eilher
(1) p : [O, I] + N wjth P(0) = Po andfi(1) = R, or
(2) ,O : ( -a , 11 -+ B is past inextendible and 9(1) = R.
It is possible for a particular point on the achronal boundary B to possess
geodesies of both types (I) and (2) above. The null tail NT(Po) defined above
happens to be a subset of the achronal boundary B = d(I+(Po)). and po~nts R
of NT(Po) which lie on Nconj(Po) as well have null geodesics with endpoint R
of both type (1) and type (2) of Theorern 13.10. On the other hand, points R
of the null tail NT(Po) which do not lie on the null corijugate locus Nccnj(Po)
are future endpoints of a null geodesic 8 only of t jpe (2).
For this simple example of a future set. note that points R which satisfy
alternative (1) lie in the future horismos E f (Po) = J+(Po) - I'(Po) of Po.
Also, it should be noted that if a space-trme (N ,g) is causally simple, so that --
J f ( Q ) = J+(Q) = I+(Q) for any Q in N , then any future set B defined as in
(13 33) sat~sfies B = b(If (Q)) = E+(Qf Thus to get an example of such an
496 13 G W I T A T I O N A L PLAKE WAVE SPACE-TIMES
achfonal boundary B whlch contains polnts R satisfjing altei~iative (2) but
not alternative (I) of Theorem 13.10, it is necessary to consider space-times
which fail to be causally simple. Fortunately, ail gravitational plane waves
are known to fa11 to be causally simple, essentially because of the behamor
mentioned above that the null geodesic ray pq,, apart from its terminai point
Qo, is in J f (Po) but not in Jf (Po).
At present, we know that I+(Po) and B = B(I+(Po)) have the following
properties for the first astigmatic conjugate pair u0 < u l and Po = (0,0,0, uo).
(1) If R E I+(Po), then u(R) > UO.
(2) Tf uo 5 u(R) < u1, then R 6 I+(Po) if and only if R lies on a unique
maximal timelike geodesic segment c with c(uo) = Po and c(u(R)) = R.
(3) If u(R) > 211, then R E I+(Po) automatically.
(4) If u(R) = u1 and R E Conn(P0, ul), then R E I*(Po) if and only if R
lies on a uniquely determined 1-parameter family of maximal timellke
geodesic segments from Po to R.
What is not settled is the qaestion of what other points R in P(ul) which do
not lie on geodesics issuing from Po, i.e., z (R) # 0, are contained in I+(Po),
in simpler terms, what happens to the chronological future of Po as astigmatic
conjugacy occurs when u = u;?
The corresponding facts for the achronal boundary B = d(I+(Po)) are
fol~ows.
(5) If R E B. then u0 5 u(R) < u1. (6 ) Tf uo < u(R) < u1, then R E B if and only if R is the endpoint of
unique maximal null geodesic segment from Po to R.
(7) For R in Conn(Po,uI), it is known that R E B if and only ~f R
contained in the null tail NT(Po).
In the case of B, then, the issue 1s whether B n P(ul) = NT(Po), or equi
lently, whether P(ul) - Conn(Po, ul) 5 I+(Po).
I t may be seen for a subclms of polarized gravitational waves, which
termed "unimodal" in Ehrlich and Emch (1992a) and Ehrlich (1993).
these related ilssues may be settled affirmatively.
13.3 ASTIGMATIC CONJUGACY, ACHRON.4L BOUNDARY 497
Definition 13.11. (U~zmodal Gravztatzonal Plane Wave Metnc) A prr
larized gravitational plane wave metric g = i- f(u)ly2 - z2)du2 is said to
be mimodal ~f f (u) 2 0 and there exist A > 0 and T i 6 R with the following
properties:
(13.35) '?r
( ) < A < - ; 2 and
Requirement (13.35) serves to ensure that in a certain Priifer type trans-
formation for the solution of the 0,D.E. (13.22), a continuous determination
of arctan may be made [cf. Ehrlich and Emch (1992a, p. 188)]. What we need
for our purposes in this section, which is where the assumption of unlmodality
enters into the discussion below, is the following variant of a Sturm Separation
Theorem, which is implicit in the proof of Theorem 2.14 of Ehrlich and Emch
(1992a, p. 187).
Lern~na 13.12. Let uo < ul be a first astigmatic conjugate pair for a given
unimodal gravitational wave. Then there exist 5 > 0 and a strictly increasing
continuous function a : [O, 6) -+ [0, +x) with a(O) = 0 and such that
is a first astigmatic conjugate pair for all E in [O, 6).
Now we are ready for
Theorem 13.13. Let uo < u1 he a first astigmatic conjugate pair for
a unimodular polarized gavitational wat7e. T,et Po = (yo, 20, vo, uo) be an
arbitrary point of the null plane P(u0). If R E P(ul ) does not lie on any
geodesic issuing from Po, then R E It(Po). Hence, the null tail NT(Po) of Po
satisfies
498 13 GRAVITATIONAL PLANE WAVE SPACE-TIMES
Proof From the nature of the isometry group, ~t suffices to make explicit
calculations with Po = (0.0.0, u0) as above. With this particular cholce of Po,
for any gravitational plane wave metric with g(u) = 0 we have that P(s) =
(0,O. 0. s) is a future directed null geodesic issuing from Po Let P, denote the
point given by P, = P(uo + E) = (0,O. 0, u0 + r ) Further, let y,(u) and re(%)
denote respectively the unique solutions to the IVP's
(13.39)
Y" - f (2519 = 01 Y ( W + E ) = ~ , y ' (uo+e)= l and (13.40)
zl/ + f (U)Z = 0, z(u0-J-e)=O, zl(ugSc) = l
which arise in studying the exponential map from P,. In view of (13.24)
translated to P,, if a point Q = (y, z , v.u) with u > ug + E lies on a geodem
~ ( u ) issuing from P, with g(c'. cl) = A, then
(13.41) zv = _&.I9 y2 - i:(U) z2 + - uo - €), YE(u) zdzo
Now fix R = (a, b. c, ul) in P(ul ) which does not lie on any geodesic issuing
from Po, ensurlng b f 0. If we show that R lies on some future timelik
geodesic issuing from P, for somc E > 0. then R lies on a future null geod
from Po to P,, follouled by a future timelike geodesic hom P, to R, whe
PQ < R as desired.
But the point R will lie on a timelike geodesic from P, for somc c > 0
provided that the inequality
(13.42) Y~'(uI) z*'(uI) b2 2 c < - - yc(ul) ~ r (u1)
is satisfied for some E > 0. Since b2 > 0, this will ensue provided that
(13.43) %'(uI) yc(u1)
is uniformly bounded as E + O while
(13.44) z,/(u1) lim - = -oc
E+O ~ ~ ( 2 ~ 1 )
i3.3 ASTIGMATIC CONJUGACY. ACHRONAL BOUNDARY 499
Choose a constant E > 0 so that 0 5 f (u) < B2 for a with uo - ! 5 ?L < ul t 1.
Also as f (u ) 2 0, we have that y,(u) and g,/(u)/g, JU) are b o ~ h positive for
any ?I. > ?LO -I- E . Hence (13.43) follows by the usual b u c h comparison theory
techniques
The more technical part of the proof is to establish (13.44). Here there are
two cases. both of which occur in explicit examples: first, u i $ supp(f(u)),
and second, ul E supp(f (u)). In either case, let a ( € ) be as in (13.37).
Consider first the case that 7 ~ 1 @ ~11pp(f(u)), and so also UI + a ( € ) cf supp(f (u)) for all E > D and z"(u) = 0 for all ZL > ul . Hence there exist
constants A, so that
for all u > ul. Consequently,
so that limit (13.44) is immediate
Finally, suppose that 111 E supp(f(u)) L. Flaminio has remarked that the
analysis step (13.46) should remain valid because the Jacobi solution Z(U) near
~ t s zero at ul +a(€) is known to be asymptotically linear A detailed calculation
demonwrating thar, this intuition is valid may be given by drawing on the
variant of the Priifer transform technique used ai; in the proof of Theorem
2.14 of Ehrlich and Emch (199%). The IVP's (13.40) in the (16, z) plane are
transformed into the (s, 8) plane as in Theorem 2.14 in order to carry out the
details of the analysis. The crux of the matter is that the transform of zc (u )
to @,(s) enables a uniform bound on 6,'(s) to be given because of the presence
of the term sin[8,(8)] on the right hand side of equation (2.83) of Ehrlich and
Ernch (1932a).
CHAPTER 14
THE SPLITTING PROBLEM IN
GLOBAL EOPeENTZIAN GEOMETRY
An important aspect of global Riemannian geometry over the past 25 years
has been the investigation of rigidity in con~iection with curvature inequalities.
This concept was first widely disseminated in 1975 in the first two paragraphs
of the preface to the iduential monograph by J. Cheeger arid D. Ebin (1975),
Comparison Theorems in Riemannian Geometry:
In this book we study complete riemannian manifolds by devel-
oping techniques for comparing the geometry of a general manifold
M with that of a simply connected model space MH of constant
curvature W. A typicai conclusion is that M retains particular ge-
ornetacal properties of the model space under the assumption that
its sectional curvature KIM, is bounded between suitable constants.
Once this has been established, it is usually possible to conclude
that M retains topological praperties of MH as well.
The distinction between strict and weak bounds on KM is im-
portant, since this may reflect the difference between the geometry
of say the sphere and that of euclidean space. However, it is often
the case that a conclusion which becomes false when one relaxes the
condition of strict inequality to weak inequality can be shown to
fail only under very special circumstances. Results of this nature.
which are known as ngadzty theorems, generally reqmre a delicate
global argument.
A first example is provided by the topological sphere theorem established by
W. Raueh (1951), M. Berger (1960), and W. Klingenberg (1959, 1961, 1962),
among other authors.
502 14 THE SPLITTIKG PROBLEM IN LORENTZIAN GEOMETRY
Topological Sphere Theorern. Let (N. h) be a s~mply connected, com-
plete Riemannian manifold of dimension n 2 2. Suppose the sectional curva-
tures of (N, hj satisfy the curvature pinchag condition
(14.1) d . A < K ( a j < A
for all two-planes o in 6 2 ( M ) , where d satisfies
(14.2) 1
d > ;
and A > O is arbitrary. Then N is homeomorphic to the sphere Sn.
Notice that in this result we have a strict inequality in (14.2) and also in
the left inequality of (14.1) Now the rigidity in this context was first proved
by M. Berger under the weaker pinching hypothesis
(14.3) 1 - A 5 K(cr) 5 A. 4
Berger obtained that either the previous topological conclusion held and that
AT would still be homeomorphic to Sn. or if the previous conclusion faded to
hold, and hence N was no longer homeomorphic to Sn, then A: had not just to
be homeomorphic but had to be zsometnc to a compact rank one Rieinannian
symmetric space.
Here 1s a second example more germane to our context. It follows from the
work of Gromoll and Meyer (1969) that a complete Riemannian manifold
dimension n 2 2 wlth everywhere positive Ricci curvatures
(14 4) Ric(v, v) > 0 for all nonzero v in T M
is connected a t infinity. Rigidity now enters into this situatioil in the followln
manner. Suppose only that Ric(v, v) > O and that (N, h) fads to be connecte
at infinity. Then since (N, h) is assumed to be complete, there ex~sts w
is often termed a (geodesic) lzne, i.e., a complete geodesic c R - (N,
joining any two different ends of IV and minimizing distance between any tw
of its points. But then the Cheeger-Gromoll Splitting Theorem (1 971) m
applied to the line c to establish that (N, h) is zsometnc to a product mani
14 T H E SPLITTING PROBLEM IN LORENTZIriN GEOMETRY 503
Cheeger-Gromoll Spl i t t i~ ig Theorern. Lei (rV, h) be a complete Rie-
niannian manifold of dimension n 2 2 which satisfies the curvature condition
(14.5) R~c(v, z;) 2 O for ail 71 in T A . ~
and which contains a complete geodesic line. Then (iV. h) may he written
uniquely as an isometric product Nl x IWk wl~ere NI contains no lines and W k is given the standard Aat metric.
Thus in this example the curvature rigidity arises as condition (14.4) is
weakened to condition (14.5).
In 1932 Busemann (1932) introduced an analytic method of studying gener-
alizations of the classical horospheres of hyperbolic geometry to a very general
class of metric geometries. In the proof of the Cheeger-Gromoll Splitting
Theorem given in Cheeger and Gromoll (19711, this particular function was
independently rediscovered during the course of the proof as a key ingredient.
At about the same time. the Busemann function had also been seen to be
a useful tool in the study of complete Riemannian manifolds of nonpositive
sectional curvature (cf. Eberlein and O'Ne111 (1973), Eberlein (1973a), where
the function was named the "'Busemann function" in honor of iis discoverer)
This function has played a prominent role continuing up to the present time
in global Riemannian geometry.
In Chapter 12 singularity theorems for spacetimes which satisfy certain
global geometric conditions and certain curvature conditions have been dis-
cussed. A simple prototype may be stated as a reminder.
P ro to type Singularity Theorem. Let ( M : g) be a spacetime of dimen-
sion n > 2 which satisfies the following three conditions:
(1) ( M , g) con tans a compact Cauchy surface;
(2) ( M ; g ) satisfies R ic (~ , V) 2 0 for ail timelike (or ali nonspacelike) v;
and
(3) Every inextendible nonspacelike geodesic satisfies the generic cond~tion
(cf. Definition 12.7 and Theorem 12.18).
Then (M, g) contains an incomplete nonspacelike geodecsic
504 14 THE SPLITTING PROBLEM IN LORENTZIAN GEOMETRY 14 THE SPLITTING PROBLEM IN LORENTZIAN GEOMETRY 505
Now in this result the "strict curvature inequality" is condition (3), which In a more differential geometric vein, S. T. Yau an the early 1980's (unpub-
requires for each inextendible nonspacelike geodesic that a certain curvature lished) proposed the idea of a "rigid singularity theorem.' to use the language
quantlty be nonzero a t some point of the given geodesic. Thus, in this context, of Galloway (1993). This was later stated in Bartnik (1988b) as follows: curvature rigldity would suggest studying spxEr-dimes which satisfy all the Goeecture. Let ( M . g) be a space-time of dimension greater than two conditions of the Prototype Singularity Theorem but condition (3)
Earlier In an essay on singularities in general relativity, Geroch (1970b, (1) contains a compact Cauchy surface, and
pp. 264-265) had suggested for inextendible closed universes satisbing the Ein- (2) satisfies the timelike convergence condition Ric(c, v) > 0 for all timelike
stein equat~ons that, generically. such space-timm should be timelike geodesi- vectors v
cally Incomplete [cf. the m~ddle column in Figure 2 on p. 266 of Geroch Then either ( M , g) 1s timelike geodesically incomplete, or else ( M , g) splits iso-
(1970b)l. Geroch's conjectured v~ewpoint was stated as metrically as a product (R x V, -dt2@h), where (V, h) is a compact Rie~narlnian
Thus, we expect that the diagram for closed universes will be
almost entirely black. There are, however, at least a few white This philosophy apparently prov~ded Yau with the motivation to pose in points there exist closed. geodesically complete flat spacetimes the well-known problem sectlon of the Annals of Mathematics Studies, Vol- . . . Perhaps there are a few other nonsingular closed universes, but ume 102 [cf. Yau (1982)], the problem of obta~nlng the Lorentzian analogue of these may be expected to appear either as lsolated points or at least the Cheeger-Gromoll Ricci Curvature Splitting Theorem stated above. Speclf- regions of lower dimensionality in an otherwise black diagram. ically, Yau posed the question as follows: Show that a space-time (1M,g)
which is t~melike geodesically complete, obeys the t~mellke convergence con- Here Geroch is representing diagammaticauy the space of all solutions to
dition, and contains a complete timelike line, splits as an Isometric product Einstein's equations in the above discussion, with singular space-times repre-
(R x V, -dt2 $ h). Thls problem was then studled by Galloway (1984~) as well sented as a black dot and non-singular, inextendible space-times represented
as by Beem, Ehlich, and Markvorsen Galloway considered spatially closed as a white dot. Further, In Geroch (1970b, p. 288) the following problem was
space-times and employed maximal surface techniques stemming from results
in Gerhardt (1983) and Bartnik (1984). The approach of Beem, Ehrlich and
Prove or find a counterexample: Every inextendible space-time Markvorsen, which was directed toward globally hyperbolic rather than time-
containing a compact spacelike 3-surface and whose stress-energy like geodesically complete space--times, employed entirely different methods in
tensor satisfies a suitable energy condition is either 6-singular or which the use of the Busemann function for a future complete timelike geo-
flat. (The energy condition must be strong enough to exclude the desic ray ' i v s introduced into spacetime geometry. Under the less stringent
Einstein universe. Perhaps one should first prove that the 3-surface t~rnelike sectional curvature assumption K 5 0 a splitting theorem was ob-
cannot be, topologically, a 3-sphere.) tained; combining forces with Galloway yielded Beem, Ehrlich, Markvorscn,
and Galloway (1984, 1985)
Gdloway and Norta (1995) summarize Geroch's ideas here as "spatlall A Riemannian proof contained in Eschenburg and Heintze (1984) for the
closed spacetimes should fail to be slngular only under exceptional circu Cheeger-Gromoll hemannian Splitting Theorem, different than that originally stances." glven in Cheeger and Gromoll (1971). provided a helpful model for obtaining
506 14 THE SPLITTING PROBLEM IN LORENTZIAN GEOMETRY 14 1 THE BUSEMANN FUNCTION 50
results in the space-time context The d'Alembertian operator for a space- obtained by the use of "aimost maximizers" to construct to-rays rather t h a ~
t:me is hvperboiic rather than elliptic. but Cheeger and Gromoll (1971) used relyicg on maximal geodes~cs as in the previous papers and in the classica
analysis methods relying on ellipticity. Next Eschenburg (1988) made a key constructions done for complete Bemannian manifolds [cf. Newman (1990
new observation that if one only studied the geometry in a ne~ghhorhood of the Lemma 3.9)]. Since this method of proof fits rather well with the spirit o
given timelike line, then the hcc i curvature assumption would allow certain Proposition 8.2 of Section 8.1, ~t is this approach to the Lorentzian Splitting
arguments to be successfully modlfied from the K < 0 case to the Ricci curva- Theorem which will be treated here.
ture case, provlded that the space-time was globally hyperbolic and timelike An elementary complication in the proof of the space-tlme splitting theoren
geodesicaiy complete; then a "continuatlon type argument" would allow the occurs in the construction of what would be called in Riemannian geometry ar
splitting obtained in a tubular neighborhood of the given complete timelike ge. "asymptotic geodesic" to a given complete timelike geodeslc through a limltlng
odesic to be extended to the entire space-time. Galloway (1989a) removed the process In the space-time c s e , some care is needed to ensure that the llmiting
assunlption of tirnelike completeness from Eschenburg's work, and finally New- geodesic, which on a priori abstract grounds could be timelike or null, turns
man (1990), rounding out the circle of papers, gave a proof assuming timelike out to be timelike rather than null.
con~pleteness rather than global hyperbolicity. Here the new philosophy was 14.1 The Busemann Function of a Timelike Geodesic Ray that even though nonspacelike geodesic connectibility is not ensured without
the assumption of global hyperbolicity, within a tubular neighborhood of the We begin with a few remarks about the Riemannian Busemann function.
given complete timelike geodesic, the existence of the single complete geodesic Thus let (N, h) be an arbitrary Riemannian manifold with associated distance
enables certain limiting arguments to be made successfully in this tubular function do : N x N -+ R. Recall that even for incomplete Riemannian
neighborhood despite the possible general lack of geodesic connectibility in manifolds, the Riemannian distance function is always continuous and finite-
other regions of the space-time. It should also be recalled that certain el valued. Let c : [O, +m) 4 (N, h) be any future complete unit speed geodesic
mentary consequences of the existence of a maximal timelike geodesic segment my, i e , &(c(O),c(t)) = t for all t 2 0. Then we may consider a Busemann
have already been treated In Section 4.4. function b associated to the geodesic ray c by the formula
All of these results may be summarized in the following simple statement b(q) = t t ~ m [ t - do(c(t)? q)j of the Lorentzian Splitting Theorem = lim Ido(c(O), c( t ) ) - do(c(t), q)].
t-+m LorePltzian Splitting Theorern. Let (M. g) be a space-time of dimensio Since the Riemannian distance function a continuous and the triangle inequal-
76 > 2 which satisfies the following conditions: ity holds, elementary arguments reveal that b(q) exists, 1s finlte-~alued, and (1) (M, g) is either globally hyperbolic or timelike geodesically complete; is continuous. Let us review these arguments to have them firmlv fixed Put (2) ( M , g) sat~sfies the timelike convergence cond~t io~; and f (q, t) = t - do(c(t), q ) , so that b(qj is the iimi: of f (q. t ) a s t approaches +m.
(3) (M , g ) con t a n s a compfete t~melike line. First the triangle inequality yields
Then (M. g) splits isornetnca1ly ;is a product (R x V, -dt2 @ h), where (V, f (4, t ) l do(c(O), q: is a complete lijemmnim mmifold.
and second for any t > s that Many of the arguments given in the references cited above are rather corn
plicsted. Thus in Galloway and Borta (1995), considerable simplifications we f (41 t ) 2 f (9, s).
(14.7) and the finiteness of Riemannian distance, is bounded from above hence versions of the sphtting theoreem with the more desirable timelike convergence
has a limit. Further, one has the elementary estimate Ricci curvature condition were estabiished, an important new realization was
first found in Eschenburg (1988). This was the realization that with the weaker (14.9) If ( ~ , t ) - f (q, t)l 5 ~ o ( P , qf curvature condition, although regularity for the Busemann functions could
which shows the equicontinuity of this family in t and q and passes in the limit not initidly be obtained on large subsets of the given manifold, the existence
of the future complete timelike line ensured good behavior of the Busemann to the inequality
function and of asymptotic geocimics in a tubular neighborhood of the given
(14.10) I~(P) - b(q)l 5 do@> line. This control also sufficed to prove the Lorentz~an Splitting Theorem for
globally hyperbolic, timelike geodaically complete space-tlmes. For instance, which evidently establishes the continuity of the Eemannian Busemann func- in Section 3 of Eschenburg (1988) it was shown that the Busemann function tion. is Lipschitz continuous in a neighborhood of a given timelike geodesic ray.
Recall that for non-globally hyperbolic space-times, the Lorentzian dis- An important aspect of the differential geometry of a complete, noncompact tance function need not be continuous, only lower-semicontinuous (cf. Lemma Riemannian manifold (N,go) is the construction of an asymptotic geodesic ray. 4.4), and also that chronologically related pairs of points p << q may have starting at any p in N, to a given future complete geodesic ray c : 10, +m) -+ d ( ~ , q ) = +m (cf. Figure 4.2 and Lemma 4.2). Further, d(p,q) = 0 automati- (N,go). Let us review this construction to motivate some of the technicalities cally if q is not contained in Jf ( p ) , and the triangle inequality is replaced by which must be overcome in the space-time setting. Take it,) with t, -+ +oo. the reverse triangle inequality, assuming the points involved satisfy appropr By the Bopf-Rinow Theorem, there exists a minimal unit speed geodesic c, ate causality relations. Thus some care is needed in the preliminary analys with h ( 0 ) = p and e,(&(~,c(t,))) = c(t,). Letting v be any accumulation of the space-time Busemann function, especially as one case of the Lorentz point of the sequence of unit tangent vectors (cnl(0)) in T'n/l, put y(t) = splitting problem assumes timelilie geodesic completeness and not global exp,(tv), defined for all t > 0 by the geodesic completeness. As a limit of a perbolicity. Even assuming global hyperbolicity and hence guaranteeing a con- subsequence of the minimal geodesic segments {c, \ 0. d o ( ~ , ~(t ,))]) , c ~rlust be tinuous, finitevalued distance function, examples of space-times conformal globally minimal, recalling that Minkowsk'~ space show that it is possible for the space-time Busemann functio
to assume the value -co or to be discontinuous unless further assumptions (14.11) iim do(p, ~ ( t , ) ) = +oo
made. In Beern, Ehrlich, Markvorsen, and Gailoway (1985), which studied t since the triangle inequality gives the immediate estimate
Lorentzian splitting problem for globally hyperbolic space-times with time1
sectional curvatures K 5 0, it was shown that both Busemam functions do(^, c(tn)) L do(c(0). c(tn)) - &(P, ~ ( 0 ) ) = in - do(p, ~ ( 0 ) ) . and b- associated with the two different ends of a complete timelike geod
line c were continuous on the subset I(c) = 17(c) f7 I+(c), and it was Even though S. T. Uau formulated the Lorentzian splitting problem for
established that I(c) = M. Hence the sectional curvature assumption ens timelike geodesically complete space-times, as recalled in the introduction to
that the Busemann functions associated with a complete timelike line this chapter, the first attacks on this problenl were carried out instead for
continuous on all of M . globally hyperbolic spacetimes, for this is the class of space-times for which
510 14 THE SPLITTING PROELEILI IN LORENTZIAN GEOMETRY 14.1 THE BUSEhlANN FUNCTION 511
maximal geodesic segments exist connecting causally related pairs of points.
Hence, one may a t least start the construction as above by taking maximal
tirnelike unit speed segments c, : {O,d(p, c(t,))] -+ ( M . 9).
Now, however, a problem arises in considering {c,'(O)) in that even tl~ough
the space of futt~re nonspacelike directions at p is compact (despite the fact
that she set of unit future timelike tangent vectors is not compact). the limit
direction v obtained for this sequence might be a null vector rather than a
timelike vector. Dealing with this technicality motivated the introduction of
the t~melike co-ray condition in Beem, Ehrlich. Markvorsen, and Galloway
(1985) as well as the proof that this condition would be satisfied provided that
the timelike sectional curvature assumption K < 0 was imposed in order to
bring Harris's Toponogov niangle Theorem of Appendix -4 to bear.
Subsequent proofs of the Lorentzian Spl~tting Theorem in the globally hy-
perbolic case under the weaker Ricc~ curvature assumption dealt with showing
that the asymptotic geodaic construction would be well behaved in some
neighborhood of the given timelike ray. On the other hand. when Newman
(1990) returned to Yau's original formulation of the problem, he had to deal
with the further complication that the existence of maximal geodesic segments
connecting pairs of causally related points could not be assumed. Thls led to
the necessity of working with sequences of "almost maximal curves" as In
Section 8.1 and also in the treatment of Galloway and Horta (1995) to the
consideration of the "generalized co-ray co~~dition."
In the construction of asymptotic geodcslcs from almost maximal curves as
treated in Galloway and Horta (1995), a somewhat different formulation of
the basic nonspacelike limit curve apparatus is employed than that discussed
in Section 3.3 above. particularly Propositions 3.31 and 3.34. Their approach
ha- the advantage of not requiring the assumption of strong causality since
the topology of uniform convergence on compact subsets, rather than the C0
topology on curves, is used in Proposition 14.3 below.
Fix a complete Riemannian metric h for the space-time ( M : g) throughout
the rest of this section.
Convention 14.1. Given a future inextendible nonspacelike curve y in
(M,g) . y will always be assumed to be reparametrized to be an h-unit speed
curve unless otherwise stated.
Hence, with such a parametrization. the Hopf-Rinow Theorem guarantees
that
7 : [ao. +w) -+ (34. g)
(cf. Lemma 3.65). Similarly, a fiitltr~ causal curve c which is both past and
future inextendible may always be given an h-parametrization
c : (-00, +m) - ( M : g).
Starting with Galloway (1986a) and continuing In Eschenburg and Galloway
(1992) and Galloway and Horta (1995). the following formulation of the Limit
Curve Lemma has been found helpfill.
Lemma 14.2 (Limit Curve Lemma). Let y, : (-oe, +m) --+ ( M , g ) be
a sequence of causal curves parametrized with respect to h-arc length, and
suppose that p E M is an accumulation point of t h e sequence {y,(O)). Then
there exist an inextendible future causal curve y : (-00. +m) --+ ( M , g ) such
that ~ ( 0 ) = p and a subsequence (7,) which converges to y uniformly with
respect to h on compact subsets of B.
As noted in Eschenburg and Galloway (1092). dn advantage to the use of the
h-limit curve reparametrization is the following upper sernicor~tiriuity result.
obtained without the assumption of strong causality (cf. Remark 3.35).
Proposit ion 14.3. If a sequence 7,L : [a, b] -4 (hf, g ) of future causal
curves (parametrized with h-arc length) converges uniformly to the causal
curve y : !a, h ] --+ (M, g), then
L(y) L !im sup L(y,)
Proof, Partition [a, b ] as a = t o < t ; < . . . < t , = b such that each pub-
segment 7 / [t,, t,+l] is contained in a convex normal neighborhood A',. By the
512 14 THE SPLITTING PROBLEM IN LORENTZIAN GEOMETRY 14.1 THE BUSEMANN FUKCTION 513
assumption of uniform convergence, y, it,, is also contained in this neigh- A future directed nonspacelike geodesic ray in the sense of Definition 8.8 borhood N, for all sufficiently large n. Hence the known upper-semicontinmty may then be given an h-reparanletrization as an S-ray with S taken to be
of the Lorentzian arc length functional may be applied to the strongly causal S = <y(O)). Conversely, an S-ray y in the sense of Definition 14.4 may be
space-time (N,, g /N*) to obtain U? / [t,,t ,+~]) 2 l i m s u ~ , ~ - + ~ L(?, 1 it,, t , + ~ ] ) , reparametrlzed to be a nonspacetike geodesic ray in the sense of Definition 8.8 and now summing over z produces inequality (14.12). U
As was noted in Eschenburg and Galloway (1992, pp. 211-212). a second ad- d(y(01,dt j ) < d(S, -4)) = / [O, t j),
vantage of the use of the awdiary h-parameter is that the Busemann function which implies that y may be reparametrized as a Lorentman distance realizing may be defined for a h ture incomplete timelike geodesic ray 7 : 10, a) --+ ( M , g) geodesic and d(y(O), y ( t ) ) is finite for any t > 0. Of course, in the applications even though the finiteness of a does not permit t -+ +oo in the expression S will usually be taken to be a spacelike hypersurface unless S = { ? ( O ) ] . analogous to (14.6) above. But if we reparametrize y with h-arc lengtli, then The first steps toward establishing the regularity of the Busemann function
y : [0, +m) 4 (M. g), and we may put and the Lorentzian distance function in a neighborhood of the given timelike
(14.13) ray may now be taken [cf. Eschenburg and Galloway (!992), Galloway and
b(q) = lim b,(q) T-DI)
where Lemma 14.5. Let y : [O, +GO) 4 (M,g) be an S-ray pardmetrized by
(14.14) br(q) = d(?{O), ~ ( r ) ) - 4 9 % % T I ) h-arc length. Then
and where d denotes the Lorentzian distance function of the given space-time (1) d(p,q) < +GO for all p,q E I-(?) n It(S), and especlaily d(p,p) = O
(M,g) . Note that if q E n/i - I - ( y ) , then d(q, y fr ) ) = 0. Hence if y is future for all p E I-(?) n I+(S);
complete and thus a = -a, we have that b(q) = +co. Thus the space-time (2) For d q E I-(?) f7 I L ( S ) , d(S. q) > 0 is finice.
Busemann function b : M -, [-XI, +m] in general. (3) The Busemmn function b : I-(?) --+ [-GO, +oo] associated to y exists
As recalled above, for general noriglobally hyperbolic space-times, the Lor- and is upper semicontinuous;
entzian distance function can ediibit various pathologies. Nonetheless, as first (4) Thc Busemann function b associated to tile S-ray y is finite-valued and
noted in Eschenburg (1988), the existence of a timelike geodesic ray y give positive on I - ( y ) n I+(S);
finiteness of the distance function and Buseman11 function in I - ( y ) n I+(y(O (5) For p, q E I-(?) n I+(S) with p < q. the Busemmn function satisfies
It will be useful to develop these propesties in the more general context of b(q) 2 b(P) $. d(p, 4) ; and
rays, a s introduced in Eschenburg and Galloway (1992). Thus let S b (6) Suppose q,, = y(t,) and p, = y(s,) with s, 5 t,, and suppose t , - t subset of M, and recall that the distance from q to S is defined as d(S, q) and s, -+ s. Put q = y ( t ) and p = $8). Then d(p, q) = lim d(p,, q,).
sup{db, 9) . P S). Proof. (1) Unless p < q, d(p, q) = 0 automatically. Thus suppose p 9 q.
Definition 14.4. (S-rug zn a Space-tzme) The fkture inextendible caw Take any s E S n J-(p), and choose ro > 0 so that s G p $ q << ~ ( r ) for any
curve y with h-arc length parametrization y : [O, +m) + (M. g ) is said to b ch s and r 2 ro. The reverse triangle inequality then yields
an S-ray if $0) E 5' and 7 maximizes distance to S, l.e., d(s, P ) + d@, q) + d(q,y(r f 5 4 8 , y (r ) )
(14.15) L ( y j [O, a]) = d(S, y (a) ) for all a 2 0. 5 d(S, y(rj) = L ( y 1 [O. r ] ) < +oz
514 14 THE SPLITTING PROBLEM IN LORENTZIAN GEOMETRY
In particular. d(p.q) must be finite From Lemma 4.2-(1) we then have
d(p,p) = O for all p E I-(*) n d+(S)
Recalling that L(y 1 [O.r]) = d(y(Q), y ( r ) ) , the above chain of inequalitm
also yields d(s, p) + d(p, q) + d(q, y(r ) ) 5 d(y(O), y(r)) for any s in S n J-(p).
Since d(s,p) = O automatically if s E S - Jm(p) , we obtain from thls 1 s t
mequality
(14.16) d (S , P ) + d(P 9) + 4 % ~b-1) 5 d(y(O), ~ ( 4 )
for p < q and any r 2 1-0 chosen as above
(2) Take p = q in (14.16) to obtain
(14 17) 4% 9) + d(q,y(r)) 5 d(?(O), ?(TI)
for all suEciently large r. Since d(y(0). yjr)) is finite, d(S, q) must also be
finite. Since q E I+(S), d(s, q ) > 0 for some s in S ; hence d(S, q) > 0.
(3) Suppose ,T E I-(?). Thus there exists T > O with x < y(r) . Take any
s > r. Then
4x7 y(s)) L d(z .y ( r ) ) + d(-f(r),?(s)).
and also
d(y(Q), y(s)) = d(4(0), ?(TI) f d(y(r) , ?(§)) .
Hence
bs(z) = d(y(O),y(s)) - 4 2 , r j s ) )
5 d(y (O)*?(~) ) + d(y(r) , y(s) ) - Id(x.y(r)) + d(y(r) , y(s))I
= d(r(O), y(r)) - d(x, ? ( T I ) = b, (x) .
Thus for fixed x E I - (y) , r H b,(x) is decreasing monotonically in r . so t
b(x) = brn,,,, b,(xf exists, possibly with the value -a or +m. By the 1
semicontinuity of Lorentzian distance (cf. Lemma 4 4), each b,(x) is upp
semicontinuous. and hence the limit function b(x) is also upper semicontlnu
(4) Given any z E I-(?) n I+(S), choose s > O so that x < - / (s )
14.1 THE BUSEMANN FUNCTION 515
b,(x) = d(? ( O ) , y ( s ) ) - d(x, y ( s ) ) is finite by (1) applied to p = x , q = y ( s ) ,
since d(y(O), *(s)) = L(y 1 [0, s]) is automatically finite. Hence as r H b,(x) is
monotone decreasing, we have b(x) < +m It remains to rule out b ( z ) = -a.
To this end, take q = x in equation (14.17), and subtract d(z, y(r ) ) to obtain
b,(x) 2 d(S, x) > 0 for any r > s. Taking limits yields
for any x in I - ( y ) n I+(S) .
(5) Choose TO so that p < q 4 y(r) for all r 2 ro. Applying the reverse
triangle inequality to p < q y(r) yields
Let r -+ +m to obtain the desired inequality.
(6) Since y is maximal, we have d(pn, q,) = L(y 1 [s,, t,]) for all 11. Thus
limd(p,,q,) = lim L(y 1 [s,,tn]) = lim L(y 1 [s, t ] ) = d(p,q).
It has been noted above that in considering the constrilction of asymptotic
geodesics, somewhat different techniques must be employed to deal with the
timelike geodesically complete, but not necessarily globally hyperholir, case.
Hence we find certain conventions employed in Galloway and Horta (1995)
to be helpful and will use them below. First, we formalize certain aspects
of Section 8.1, recalling here that unlike Section 8.1, strong causality is not
assumed and also that curves are always given an h-arc length parametrization
in this section unless otherwise stated.
Definition 14.6. (Lirnzt Meximzzzng Sequence of Causal Curves) A se-
quence y, . [an, hn] 4 ( M , g ) of h-arc length parametrized future causal curves
is said to be limzt marcimizzng if
where en + 0 as n -+ +m.
516 14 THE SPLITTING PROBLEM IN LORENTZIAN GEOMETRY
Gwen any p, q in ( M . g) with 0 < d(p, q) < TCO, the definition of Lorentzian
distance implies that a sequence of l im~t maximal curves from p to q may
be constructed. In a manner similar to the proofs In Section 8.1, the lower
semicontinuity of dlstance and the upper semicontlnity of L established in
Proposition 14.3 combine to yield
Proposition 14.7. Suppose that -i, : [a,, b,] --. (M, g) is a limit maxri-
mizing sequence of future directed causal curves that converges uniformly to
a causal curve y : [a, b ] -+ M on some subinterval [a, b j E (),[an, bn]. Then
L(y) = d(y(a), y(b)), and hence y may be reparamecrized as a future directed
maximal geodesic from y (a ) to y(b) .
Proof. Putting y, = yn I [a, b], we have as usual
limsupL(7,) 5 L(y) 5 d(y(a),y(b)) n-+m
5 llm inf 47, (a), yn (b) ) n-+oo 5 lim inf (L(yn) + 6,)
,+too
= lim inf L(y,), n-+w
whence
(14 20) L(y) = hm L(-;/,) = lim d(y,(a),%(b)) = d(r(a),y(b)). n++m n-+m
The following lemma from Eschenburg and Galloway (1992) has been help
in constrricting asymptotic geodesic rays It should be emphasized that unl
(14.11) above for complete R~emannian manifolds, the condition d(z,, p,)
+oo in Lemma 14.8 dam not imply that (p,) divergs to infinity but only t
a, --+ +w.
Lernrna 14.8. Let (2,) be a sequence in (M.g) with z, 4 z. Let
IS(z,) with d(a,, p,) < +m. Let -/, . [0, a,] + (M, g) be a lirnit maxi
sequence of future causal curves with ~ ( 0 ) = z, and y,(a,) =
yn : 10. +m) -+ (M. 9) be any future lnextendible extension of 7,.
that d(p,, zn) + +DO. Then any limit curve y : [0, +co) + (M,g)
sequence {T,) is a nonspacelike geodesic ray starting a t z.
14.1 THE BUSEMANN FUNCTION 517
Proof. It is necessary to show chat a, 4 +a. Suppose not. By passing to
a subsequence, we may suppose that an -+ a < +x. Because the curves y,
are parametrized by h-arc length and h is a complete bemanman metric, it
follows that all y, are contained in a, compact subset S of M. Note also that
in this case we have Lh(yn) = a, -+ a < +w.
On the other hand, we will show that the assumption that d(z,,p,) -+ +DO
implies that Lh (7%) + +oo, yielding the required contradiction. To this end.
put a second auxiliary Riemannian metric ho on M more closely associated to
the given Lorentzian metric g by the following standard construction. Let T
be a unit timelike vector field on (M, g), and define an associated one-form by
r = g(T, .). Put
h 1 ~ = ~ + 2 r @ r = ~ ' + r @ ~ ,
and note that for this second Riemannian metric we have the given Lorentzian
arc length of any causal curve segment c and the ho-arc length related by
Since h and ho are Riemannian metrics and K is compact, there exists a
constant X > 0 such that h(v, c) 2 X2 hO(u, V ) for any v E TM / K, whence
for any curve segment c contained in K. Thus we have
Lh('Y731 2 ALg(%) > Xd(&,pn) - At, --+ +a by hypothesis. Cl
'4 more complex lemma is considered in Escheiiburg and Galloway (1992)
in order to allow for dealing with inextendible timelike rays which are not
necessarily future complete. But in view of the hypotheses involved in the
Lorentzian splitting problem, let us suppose now that y : :O, +oo) -i (M.g)
is a future complete t i d i k e S-ray, hence of infinite Lorentzian length. Fix
z E I-(y) n I+(S). Let z, -, z; and put p, = y(r,) where r, + -co. Then
d(z,,pn) < -kc for all sz suffciently large by Lemma 14.5-(1). On the other
hand, if we fix s > 0 with t << y(s), then for all sufficiently large a we have
518 14 T H E S P L I T T I N G P R O B L E M I N LORENTZIAN GEOMETRY
r , << ~ ( s ) as well. Hence the reverse triangle inequality yields d(r,.p,) >. d(zn,T(s)) + ~ ( Y ( s ) , " / ( T ~ ) ) --+ -too since d(r(s) .y(r ,)) = L(y / Is,r,]f
by hypothesis.
Thus wlth Lemma 14.8 in hand and the above verification, the coxtruction
of asymptotic geodesics to a future complete S-ray, y . 10, +m) + (M, g). may
now be accomplished Simply let {p,) be any sequence of limit maximizing
curves from z, to p,, and let p 10, +oo) --+ (M, g) be any limit curve guar-
anteed by Lemma 14.8 for the extended curves (5,). The use of the auxiliary
complete Riemannian metric h to pwametr~ze the curves by h-arc length is
what ensures the uniform convergence on compact subintervals necessary to
apply Proposition 14.7 and Lemma 14.8.
We will adopt the following terminology of Galloway and Horta (1935).
Definition 14.9. (Co-rays, Generalzted Co-yaps, and Asymptotes)
(i) Any ray p constructed in this fashion will be called a generalzzed co-ray
to :he given future complete S-ray y.
(2) The ray p constructed in this fashion will he called a co-ray to the
given future complete S-ray y if all y, are distance maximizers, I e , L(pn) = d(zn, p,) for all n.
(3) The ray p constructed in this fashion will be called an asymptote if all
p, are distance maxlmlzers and if t, = z for all n
In the space-time setting. the concept and term "co-ray" were first In-
troduced in Beem. Ehrlich, hlarkvorsen. and Galloway (1985), influenced by
terminology in Busemann (1955). Generalized co-rays for space-times were
first employed in Eschenburg and Gailoway (1992)
W-hiie considering limit maximizing sequences of timelike curves. Newm
(1990, Lemma 3.9) noticed an important ~mplication of timelike geodesic c
pieteness which is formulated In Galloway and Horta (1995) as follows.
Proposition 14.10. Let (M,g) be future timelike geodeslcally comple
Suppose p, q are points in (M, g ) mth p < q and d(p , q) < fm.
?;1 . [0, a,] -+ (M, g) be a Iimit rnaxirnizing sequence of future directed caus
curves from p to q. For each n, let Fn . [O, -%a) --t (M. g) be a future in
14.2 CO-RAYS A N D T H E B U S E M A N N F U N C T I O N 519
tendible extension of y,. Then if7 : [O. f a ) --t (.V, g) is a iimit curve of {?,I, either y maximizes from p to q or y is a null ray.
Proof. Let a = sup(a,). The exis~ence of convex neighborhoods centered
at p implies a > 0. By passing to a subsequence if necessary, we mav assume
that a, -. a. Then by Proposition 14.7. the iimit curve y [0, a) 1s a maxima!
nonspaceliko geodesic.
First suppose a < +w. Because we have diosen an auxiliary complete
Riemannian metric and parametrized the y, by l'L-arc length. we know that
y I [0, a ) extends to t = a. Also, we have that y / [O, a] is iriaxirnal, and ?(a) =
lim,,+,y,,(a,,) = q. Since d(p. q) > 0 and 7 is rnaxinial from p to q in this
case, y must be timelike.
Now suppose a = -too. If y is null. then y is the required null raj, and
we are finished. Thus suppose 7 is tirnclike. By the assumption of geodesic
completeness, y has infinite Lorentzian length. As d(p, q) is a finite positive
number. there exists T > 0 such that L(y ] [O, TI) > dfp. q) . Since a, -+ +GO,
equation (14.20) in the proof of Proposition 14.7 implies that for n sufficiently
large, we have T < a,, and hence
But this is impossible because y, is a timelike curve segment from p to q.
whence L(yn) < d (p . q) by definition of Lorentzian distance. Ci
14.2 Co-rays and the Busernann Function
It u7as first noted in Beem. Ehrlich, Markvorsen. and Galloway (1985)
that the acsumption for a globally hyperbolic space-time that all cc-rays con-
tructed to either end of a complete timellke geodesic line y : (-m. +m) +
M,g) turn out to be timelike rather than null lrnplies that the Busemann
functions bi and b- associsted to the two different ends of y are continuous
and also finite-valued on the beta I-(y) and I+(- / ) , respectlveiy. This assump-
tion was called the "timelike co-ray corid~tion.' It was also shown, employing
rris's Lorentzian version of the Toponogov Coii~parison Theorem for time-
geodesic triangles in globally hyperbolic space-times (cf. Appendix A).
520 14 THE SPLITTING PROBLEM IN LORENTZIAN GEOMETRY
that the assumption of everywhere nonpositive timelike sectional curLraturea
ensures that the given globaliy hyperbolic space-time satisfies this timellke
co-ray condition.
In the subsequent works, beginn~ng with Eschenburg (1988), more difficult
studies of the regularity properties of the Busemann function were made as-
suming only the timelike convergence condition on the Iticci curvature; hence
the space-time Toponogov Comparison Theorem could not be brought to bear
on this issue.
Another ingredient in the proof of the Lorentzian Splitting Theorems w
the fortuitous publication of Eschenburg and Weintze (1984) in which a ne
proof method for the Riemannian Cheeger-Gromoll Ricci Curvature Splitti
Theorem was employed. utilizing the technique of smooth upper and lower
support functions for the (a priori only continuous) Busemann functions of
a geodesic line. The origlnal proof method of Cheeger and Grornoll (1971
relied heavily on the ellipticity of the Laplacian and hence did not seem to be
adaptable to the case of the hyperbolic dlAlembertian.
We begin this section with the definition of these support functions as fir
treated in Eschenburg (1988) for the globally hyperbolic case and later treat
In Galloway and Horta (1495) in the current context of timelike geodesic co
pleteness. Let y : [0,+w) -+ ( M , g ) be a future complete S-ray in a spac
time ( M , g), and let b be the associated Busemann function given hy eqilatio
(14.13) and (14.14).
Definition 14.1 1. (Upper Support Fanctzon) Let a : [0, +m) -+ ( M ,
be a timeliie asymptote to 7 with a(O) = p E I - ( y ) f l I+(S) . For each 5 > let
b,,, : ,v -+ [-co, +m]
be defined by
(14.21) bp,s(2) = b(p) + d(p, 4 3 ) ) - d(x , 4 s ) ) .
Lemma 24.12. For any 9 > 0 the function given by (14.21) is
continuous upper support function for b at p, i.e , b,,,(x) 2 b(x) for all x n
p with equizlity holding when z = p.
14.2 CO-RAYS AND THE BUSEMAKN FUNCTION 521
Proof. By definition of an asymptote, there exists a sequence of maxi-
mal timelike geodesic segments a, : [O a,] --+ ( M , g) with a,(O) = p and
a,(a,) = y(r,). where r, -+ i-m as n --+ +m Also, since y is future com-
plete, a, -+ fix; slnce d(p.r(r,)) -+ i m . Let z be In the open neighborhood
U = I - ( a ( s ) ) f'l I+(S) of p for the fixed s > 0. Then for n sufficiently large.
x << a,(s) as well. By lower semicontinaity of distance, there exists 4, 4 0
such that d(x , cu,(s)) 2 d(x, a@))-S, for all large n. Then for all n sufficiently
large. the reverse triangle inequality and the maximality of y imply
By (14.20) also, d(p,a,(s)) -+ d(p ,a(s) ) as 71 -+ +w. Hence taking the
limit in (14.22) yields b(z) - b(p) 5 d(p, a ( s ) ) - d(x .a ( s ) ) , so that
as required. D
As noted by Eschenburg (1988) in the globally hyperbolic case, the expected
behavior of the Busemann function associated to the glven ray y on rays a
mymptotic to y , akin to that previously observed for complete Riemannian
manifolds, may now be obtained.
Corollary 14.1 3. Let y : 10, +m) -+ ( M , g) be a future comple~e S-ray
in the space-time (M,g ) , and let b be the Busemann function associated to
y. If a a 10. +m) -+ (M,g ) is a timelike asymptote to y starting at p in
I-(?) n I + ( S ) , then b(a( t ) ) = d(p, a ( t ) ) + b ( p ) for all t 2 0.
Proof. Taking x = a(t) and fixing any s > t in (14.23) gives
522 14 THE SPLITTING PIEOBLEhI IN LORENTZIAN GEOMETRY
since a is maximal. On the other hand, applying Lemma 14.5-(5) to p and
q = a ( t ) yields
b ( 4 t ) ) L b(p) + 4 ~ . 4 t ) ) . 0
Now we are ready to state the crucial new condition in Galloway and Norta
(1995) which leads to the local regularity of the Busemann function as well as
some control over maximal geodesic connectibility.
Definition 14.14. (Generalized Tzrnelzke Co-ray Condztzon at p) Let
( M , g) be a future tirnelike geodesically complete space-time, and let y
timel~ke S-ray Suppose that p E I - ( y ) n I + ( S ) . Then the generalzzed tzrnelz
co-ray condztzon holds a t p if every generalized co-ray to y starting at p i
tirnel~ke.
Arguments similar to thobe of Lemma 14.15 below show that this cond~tio
is an open condition, i.e , ~f the conditiori holds at a given p E I-(?) n I t (S
then the condition holds in some neighborhood of p.
Lemma 14.15. Suppose (hf, g) is future timelike geodesically comple
and Jet y be a timelike S-ray. Assume that the generalized tlmelike c
condition holds at p E I-(?) n I+(S). Then there exist a neighborhood
p and a constant R > 0 such that for all q E U and for dl r > R, there
a maximal timel~ke geodesic segment from q to y(r) .
Proof. Assuming that the desired conclusion is false, we will show that
generalized timelike co-ray condition cannot hold at the given point p
diagonalizing argument.
Supposing that the conclusion of the lemma is false, we may find { p ,
2-(y) n I+(S) with pn --t p, r, --t +m, and p, << y(r,) such that there
no maximal timelike geodesic segment from p, to y(r,) for ail n. In vie
Lemma 14 % ( I ) , we have 0 < d(p, , , ~ ( r , ) ) < +m for each n. Hence
n we may construct n limit maximizing sequence of future nonspacehke cu
ank [O, ank' + ( M . g)
from p, = ank(0) to y (r,) = ank(ank) with
L ( ~ n k ) 1 d(~nk (O) , ~ n k ( a n b ) ) - enk
14.2 CO-RAYS AND THE BtiSEMANh FUNCTION 523
where limk,,, e,k = 0. Because of the dshumed nonexistence of timelike
maximizers from p, to ?(m), the proof of Proposition 14 10 of Section 14.1
implies :hat l:mk,+, a,k = +m and also that the sequence {ii.,k) converges
to a null ray as k - +m, which we will call 0, . [O, +m) --t ( M , g ) , with in~tial
point .Dn(O) = p,. Hence equation (14.20) forces
for any s > 0.
Now do a diagonalizing procedure on the nonspacelike curves ( a n k ) to find
an increasing sequence k(n) --+ f oo with the property that for 7 , = a,&&(,)
and b, = u,k(,), the associated sequence jVn) satisfies
(1) b, + +o=:
( 2 ) L(7, / [O, 11) < 11% for all n; and
( 3 ) L(7n 1 [O,bn]) 1 d(~n(O).qn(bn)) - 1/n for a11 n.
By ( I ) , (3); arid Proposition 14.7, the sequence {q,) converges to a nonspace-
hke geodesic ray q with q(0) = p. By co~ldition (2) , the ray q must be null.
Thus negating the desired conclusion. we have produced a generalized co-
q to y with q(0) = p which is a null ray. But this contradicts the hypothesis
of the lemma that all co-rays to y at p are tlmelike. C
The next several results from Galloway and Horta (1995) treat the rela-
tionship between limit curve convergence and convergence by initial tangent
irection for timelike geodesic segments and in particular the property that
he timelike co-ray condition ensures that initial tangents of asymptotes con-
tructed in a neighborhood of p are bounded away horn the null cone. Out
these considerations. among others. the local continuity of the B~isemann
nction may then be established in Theorem 14.19,
It is helpful to adopt some mechanism to produce appropriate compact
sets of the noncompact set of unlt timelike tangent vectors. To be consistent
h Chapter 9, we will recall the notat~on T-liLII, from Definition 9.2 and
troduce a new notation following Galloway and Horta of
524 14 THE SPLITTING PROBLEM IN LORENTZIAN GEOMETRY 14.2 CO-RAYS AXD THE BUSEMANN FUNCTION 525
where C > 0 and h is the auxiliary Riemannian metric for ( M , g ) fixed above. But this is contrdictory since h(a'(O),a'(O)) is finite, yet by supposition,
These sets are compact and nonempty for C sufficiently large, and evidently limn,+, h(crnl(0). anf(0)) = +oo.
Corollary 14.17. Let (M, g) be future tlmelike geodesically complete, and
let y be a timeiike S-ray. Suppose the generdized tlmelike co-ray condition
Lemma 14.16. Let (M, g) be future timelike geodesically complete, and let holds at p E I-(7) n If (S). Then there exist a ne~&borhood U of p and a
r be a timelike S-ray. Assume that the generalized timelike co-ray condition constant C > O such that for ail q E U , if a : [0, +m) 4 (M, g) is a co-ray (or
holds at p E I-(?) n I'(S). Then there exist a neighborhood ti of p and asymptote) to y frorn q, parametrized with respect to Lorentzian arc length,
constants R > 0, C > 0 such that for alj q in U and any r > R, :f cx : [0, a) -, then a'(0) E Kc(q) .
(M, g) is any maximal timelike geodesic segment from q to y(r) parametrized The generalized timelike co-ray condition also has implications for the con-
tinuity of the Lorentzian distance function, as noted in Galloway and Norta
Proof Suppose that the lemma fails to hold. Then there exist sequenc
Proposit ion 14.18. Let (M, g) be future timel~ke g~odesically complete,
and let y be a timelike 5'-ray. Suppose the generdized timelike co-ray condition property that h(crn1(0),cun'(0)) --, +oo as n + +m. To apply the limit c holds at p E I-(7) f l I+(S). Then there exist a neighborhood U of p and a
number R > 0 such that for al! r > R the function 6 : U -i [O. fcc) defined
by 6(x) = d(x, yjr)) is continuous on U . produces a nonspacelike geodesic ray Z with Z(0) = p and with Z a co-r
Proof, Recall first that Lemma 14.5-(1) guarantees that 6 is finite-valued y, to which a subsequence of the 6,'s converge. which we may assume is
U. By Lemma 4.4, 6 1s lower semicontinuous, so it only remains to establish given sequence (6,) itself. Since the generalized timelike co-ray condition
Choose U. R, and C to satisfy the conclusions of Lemmas 14.15 and 14.16. speed timelike ray cu. : [O. +a) -i ( M , g). The domain is as indicated beca
any r > R. Suppose 6 is not upper semicontinuous at x in U. Then of the assumption of future timelike completeness.
Choose S > 0 so that 6([O, b ] ) is contained in a convex normal nei d(x, ?(T)). For each n, let a, be a maximal timelike geodesic segment whlch
borhood of p and also is contained in I - (7 ) . Because of this suppositi s parametrized by g-arc length, from x, to y(r) . In view of the conditions
tisfied by U , R, and 6, the initial tangents (anl(0)f- are contained in a d(E,(O), Gn(6)). By equation (14.20) we have E, 4 E. Now using the
ompact subset of T_lMI,. By this fact and the assumption of timelike future timelike completeness and choice of convex normal neighborhood, w
geodesic a frorn x to y(r). But then by equat~on (14.20), we ha%e €,an1(O) = exP2 (E(6)) -, exppl (~(6)) = E al(0).
Hence, a,'(O) -+ al(0) which implies that d(xn(k), Y(T)) = L(an(ic)) + L(a) = d(x, 7(r)).
ontradiction.
526 14 THE SPLITTING PROBLEM Ih' LORENTZIAN GEOMETRY
With these results now established. we are ready for the proof given in
Galloway and Horta (1995) of the continuity of the Busemann function in the
presence of the generalized tlmei~ke co-ray condition. The treatment is inspired
by that in Eschenburg (1988) for the globally hyperbolic case.
Theorem 14.19. Let (M, g) be future timelike geodesically complete, and
let y be a timelike S-ray. Assume that the generalized timelike co-ray condition
holds at p in I-(?) n If (S). Then the Buseiiiann function b associated to 7 is
Lipschitz continuous on a neighborhood ofp.
Proof. It is helpful to use a lemma given in the Appendix to Eschenbur
(1988).
Sub lemma 14.20. Let U be an open convex domain in Wn and f . U
a continuous function. Suppose that for each q in U there is a smooth 1
support function f, defined in a neighborhood of q such that l/d(fq)qll < Then f is Llpschltz continuous with Lipschitz constant L. i.e., for all x , y i
U we have
Proof of Sublemma. First treat the case that n = 1 and thus U is an op
interval and the estimate amounts to supposing that
in U such that by replacing f with - f if necessary, we may suppose t
y, and f satisfy f ( x ) < f(y) and also If(z) - j(y)I > Ljz - yl. G considerations evidently enable one to find an affine function l ( x ) with
slope I1(x) = Lo > L such that f (sf < E(s) and f ( y ) > l(y). Put qo = su
[z, y] : f ( t ) < l ( t ) ) . By continuity of f, evidently fm (qo) = f (qo) = l(qo)
f,, (t) 5 f (t) < l ( t ) for all t < qo sufficiently close to qo. Hence,
l1(q0) = Lo > L, in contradiction to inequality (14.26). Thus the
estimate (14 25) must be valid in the case that n = 1.
14.2 GO-RAYS AND THE BUSEMANN FUNCTION 527
Now consider the case that n > 1. Fixing arbitrary x , y in U , let c(t) =
(1 - t ) x + ty and I = W nc - l (U) . Put g = f o c : I -+ R. Note that In view of
the chain rule, the relevant constant in (14.26) applied to g is L1 = L / jx - y / .
Applying the n = 1 case to g ( t ) with t = 0 and t = 1 yields
required.
Proof of Theorem 14.23. Let y be parametrized by Lorentzian arc length
below. Choose U, R, and C according to Lemmas 14.15 and 14.16. Further.
take U sufficiently small that is compact and is contained in a convex normal
neighborhood ( N : 4) such that Q(U) is convex in Wn, n = dim M. Let q5 =
( x I 7 . . , x,), and let ho denote the associated Euclidean metric for T N given
by n
2=1
Since U has compact closure and h, ho are both R~emann~an metrics, there is
a constant K > 0 such that ho < K h on TU.
Put d,(x) = d(x, y ( r ) ) . It is sufficient to show that the functions b, = r-d,
are Lipschitz continuous on U with the same Lipschitz constant for all T > R. Since
br(x) - b,(y) = d,(y) - d , j ~ ) .
this is equiwalent to obtaining the Lipschitz continuity of the d,, r > R, on
U . From Proposition 14.13 we have the continuity of the d,'s; it thus remains
only to obtain the Lipschitz estimate using Sublemma 14.20.
Let D . N x N + [O,+m) denote the local Lorentzian distance function
for (N, g I N ) as described in Definition 4.25 and Lemma 4 26. Fix any point
q in U and r > R. Let a! be a maximal time!ike geodesic segment from q to
~ ( r ) parametrized by g-arc length, guaranteed by Lemma 14.15. Flx any q' in
U which lies on cu with q # q'. Having made these choices, a candida:e for a
local smooth support function for d, near q may then be defined by
528 14 THE SPLITTING PROBLEM IN LORENTZIAN GEOMETRY
For o in a neighborhood of q, the hnctions z I-+ D(x, q'), and hence f,,,(x), are
smooth by Lemma 4.26. Also the Iocd distance function D generally satisfies
D(x, ql) 5 d(x, q') for any x, q' in N. Thus by the reverse triangle inequality,
fq,r(x) 5 4 x > q') + 4 q 1 > ?(T))
5 d b , ?(TI) = dr(x),
and also
fp,T(q) = D ( ~ ' 4') + d(ql> Y(T)) = d ( ~ l ql) d ( ~ ' l ~ ( r ) ) = dr(q)
using the maximality of a.
I t remains to establish an estimate for the lower support functions fq,r o
the form needed to employ Sublemma 14.20. To this end, put
G=sup(lg,,(x)l:3:EU, l I % , ~ < n )
and l/vl/o = J w ) for w € TqM Since U has compact closure In N,
have G > 0. The basic differential geometry of the distance function d(. , q
from the point q1 prior to the past timelike cut locus of q' provides the gradient
calculation
v (fq,v) = 9(v. al(0))
for any v E T,M (cf. Lemma 14.26 above) Hence,
I d (f,,,) (v)l = Idv, I G lI~'(0)llo llvlio
I G K J ~ ( ~ ~ ' ( o ) , a'(0)) I I ~ I I O
5 G K C ~ Hallo.
Hence, L = G K C ~ has the property that lid ( fq,r), 11 ( L for all q E
Thus d, and hence 13, are Lipschitz continuous on U with Lipschitz const
L = G K C ~ by Sublemma 14.20 for any r > R.
Having presented four results on the pleasing consequences of th
alized timelike co-ray condition, let us now show as in Gdloway and H
(1995) that in the Important special case where S = (-/(0)), and hence
S-ray 1s an ordinary geodesic ray, the generalized timeiike co-ray co
holds in a neighborhood of the given ray, possibly excluding the initial p
y(0). This may be accomplished by &st proving the follo-wing resuit.
14.2 CO-RAYS AND THE BUSEMANN FUNCTIOK 529
Proposition 14.21. Let ( M , g) be future timelike geodesicdly complete,
and let y : [0, t w ) + ( M , g ) be a future directed timelike ray. Then any
generalized co-ray starting at p = y (a), a > 0, must coincide with y.
Proof. Here the limit curve techniques of Section 14.1 will be employed, so
all curves will be given a unit speed parametrization with respect to the fixed
auxiliary Etiemalnian metric h. Let a : [0, +m) -+ (M, g) be a generalized co-
ray to y at p = y ( ~ ) . Thus there exists a limit maximizing sequence of future
directed causal curves a, : 10, a,] 4 ( M , g ) from p, to y(r,) with p, --t p,
r, + +w, and a, -, +oo, which converges to a. Let U be any convex normal
neighborhood of p. Choose any q in U lying on y to the past of p; and take a
sequence {q,} of points on y between q and p such that q, --+ p. Since q, << p
and pk 4 p, by taking a subsequence of (p,) if necessary, we may suppose
that q, << p, for all n and that {p,) C U . Since pn, q, are contained in U and
limp, = lim q, = p, we may connect q, to p, by a timelike geodesic segment
whose length approaches 0 a s n + fw.
Now define a new sequence {on : 10, s,] ( M , g ) ) of causal curves by
following from q to q,, then going along the above timelike geodesic segment
in U from q, to p,. and finally following along a, &om p, to yjr,). Put
qn = an(an) and p, = cr,(b,). Fix r > a so that p, E Im(y(r ) ) and r, > r for
all sufficiently large a. We now need to establish the limit maximality of this
new sequence {o-,}.
Since y is maximal and the an's are limit maximizing, we have
L(mn I LO: snI) = L(cn I [Otan]) + L(mn I [an, bn]) + L(gn I [bn, s,])
L d(q, qn) + L(5n I Ian, bn]) + d ( ~ n , ~ ( r n ) ) - 6 ,
= d(q, P) - djqnl P) + L(G I [an. b,,]) + d(?n, ~ ( r n ) ) - 6,.
On the other hand, the lower semicontinuity of distance at (p, y ( r ) j implies
the foilowing inequality is valid:
= d(p,y(r,)) - 6, with 6, -+ O as n --+ +CG.
530 14 T H E SPLITTING PROBLEM IN LOREKTZIAN GEOMETRY
Combining these two inequalities yields
L(ocT, I [O. sn]) L d(q P ) - d(qn>?) + L(0n I [an, bn]) + d(p, ~ ( r n ) ) - 6n - en
= d(q. y(r,)) - d ( q n , ~ ) + L(an 1 [a,,b,]) - 6, - c,.
Bjr Lemma 14.5-(61, we have d(q,,p) - 0 as n -+ +w. Also by constru
tion, employing the normal neighborhood U , we have L(cr,, I [a,,,b,]) -+ O
n -+ +m Hence {on} is limit maximizing as desired
By Lemma 14.8, any limit curve a of this sequence is a maximal nonspace!~
consist of the portion of y from q to p, followed by CY But then by the u
which implies that a is contained in 3 as desired. 0
the following consequence is obtained from Proposition 14.21.
Corollary 14.22. Let ( M , g ) be timelike geodesically complete, an
y : [0, t o o ) --+ ( M , g) be a future d~rected timel~ke ray Issuing from p =
on an open set c~ntaining y - {pj .
port functions bp,, to the Busemann function given in equation (14.21),
necessary to have some control of the timelike cut locus. This was done in
assumption of timelike gcodeslc completeness rather than global hyperbo
so that the theory of Chapter 9 is not immediately appl~cable
Proposition 14.23. Let (11.1. g ) be future timelike geodesically corn
and let y be a timelike S-ray. Suppose that the general~zed timelike c
condition hdds at p E I-(?) n l+(S), and let a : \O,+m) -+ (M.g)
14.2 GO-RAYS AND THE BUSEMANN FUNCTION 531
Proof By taking unions. it is sufficient to show that for each s in [O,r],
there exi9ts a neighborhood C of a ( s ) which does not meet the past timelike
rut locus of a(r ) . There are three cases to consider s = r; O < s < r; and
s = O
Case (1) s = r . Since N is a timelike ray, strong causal~ty holds at a(r)
by Proposition 4.40. Hence by Theorem 4.27, there exists a convex normal
neighborhood U of a ( r ) such that the global Lorentzian distance function d(g)
agrees with the local distance function (D. U) Thus nonspnrelike geodesics
emanating from the center a ( r ) of this convex neighborhood contain no non-
spacelike cut points within Li, and U itself provides the required neighborhood
Case (2) 0 < s < T . This case, which essentially corresponds to Newman (1990.
Lemma 3.10), IS similar to the s = 0 case but simpler. Thus d e t a ~ h will be
omitted.
Case (3) s = 0 This is the case which makes use of the general~zed timelike
co-ray condition. It is helpful to establish two sublemmas.
Sublemma 14.24. Let the assumptions be as in Proposition 14.23. Then
for any T > 0, the function x H d.,.(z) = d(x. cr(r)) is continuous at p.
Proof of Sublemma 14.24. Let pn -+ p. Starting with Lemma 14.5-(5) and
the lower semicontinuity of distance, we have for the Busemann function b = b,
of y that
b(a(r) ) - b(pn) 2 d ( p n , c*(r))
2 d(p. 4 ~ ) ) - 6,
= b(a(r) ) - b(p) - 6,,
the last equality from Corollary 14.13 Smce the generalized timelike co-ray
condition is assumed to hold at p. the Busemann function h is continuolls a t p
from Theorem 14.19. Hence d(pn, rr(r)) - d ( p , tu(r)) as n --+ +GO. C1
Sublemma 14.25. Let the assumptions be as in Propos~tion 14.23. Then
for each r > 0 there exists a neighborhood U of p such that for each q in U ,
there exists a maximal timelike geodesic segment from q to cr(r).
ProoJ of Sublemma 14.25. Suppose the lemma is false. Then there exlst
T > 0, p, - - p, arid a limit maximizing sequence of past directed causal curves
532 14 THE SPLITTING PROBLEM IN LORENTZIAN GEOMETRY
an,& : [o. a , k j 4 ( M , gf such that for each k, a,,b(O) = a ( r ) and a,,k(a,,k) =
p,. Furthermore for each fixed n, by selecting a suitable subsequence. we
may suppose that as k - +ce we have u,,k -t a,, where a, is either a
past inextendible null ray or a past inextendible timelike ray. As was done
in Lemma 14.15, disgondize to obtain a limit maximizing sequence {qn :
[O, a,] ( M , g) ) with Vn(0) = ~ ( r ) , nn{an) = p,, a, + +m: and with {q,)
converging to a past inextendible ray 7 : [0, +oo) ( M . 9) where q ( 0 ) = a(?).
(Note here that the curves are parametrized with respect to the auxiliary
Riemannian metric h and also that 7 plays no explicit role in the rest of the
proof other than serving as a reference frame for the curves q,.)
Fix ro > r , and consider the past directed nonspacelike curves jj, : [0, c,] -+
( M , g ) obtained by concatenating each q, with tile segment along -a from ro
to T and reparametrizing appropriately (here c, = 1 f a, where I = TO - T ) .
Note that $,(O) = a ( r o ) , il,(l) = a ( r ) , and 17,(c,) = p,. Also put 6, =
d,(p) - d,(p,) -+ 0 as n + i c e by Sublemma 14.24.
Now recalling that the original sequence ( 7 % ) is hmit maximizing, the li
maximality of the new sequence {ij,) may be established. For
L(% I IO, %I) = d(a (T) , a:(T0)1 + L(k I I4GZl)
2 d ( a ( ~ ) , a ( r o ) ) + d ( ~ n , Q ( T ) ) - 6%
= d ( a ( ~ ) , a:(ra)) + d(p , a ( r ) ) - 6, - en
= d(p, a ( % ) ) - 6, - E n
where 6,, E , -+ 0 .
Since (ij,) is limit maximizing, this sequence converges to a past direc
nonspacelike geodeic ray ;j with i j(0) = a ( r o ) Since each 7j, containa
segment of ou from a ( r ) to &(TO), the limit ray must be timelike and fur
must coincide with a back to a ( 0 ) = p. Hence i j passes through p, an - -
constrllction, is contained in Jf(p) = I+(p). This implies, using Pr
3.7-(I), that p is in I f (p ) . whlch contradicts the finiteness of d(p ,p ) .
Proof of Case (3) of Proposztion 14.23: s = r. Let dl timelike ge
etrized with respect to Lorentzian arc length. S
this case. Then there exist r > 0 and p, 4 p suc
14.2 CO-RAYS AND THE BUSEMANN FUNCTION 533
p, is a past timelike cut point to a(r) for all n. Recall from Section 9.1 that
this means that there exists a past directed timelike geodesic
Cn : 10. d(pn: &(TI) + 6,) + (b.3 9)
with ~ ( 0 ) = a ( r ) and ~ ( d ( p , , a ( r ) ) ) = p, such that c , [ 10, d(p,, a ( r ) ) ] is
maximal but that d(&(t ) .a(+)) > L(c, I [ 0 , t ] ) for any t > d(p, ,a(r) ) . Be-
cause a is maximal, p and a ( r ) are not conjugate along a . Hence there is a
neighborhood V of -a1 ( r ) in T,(,)M which is diffeomorphic under expa(,) to
a neighborhood U of p. By re-indexing { p , } and shrinking U if necessary, we
may assume that (p,) 2 U and that C l satisfies the conclusions of Sublemma
14.25, so that there is a maximal timelike segment from a j r ) to each point of
U .
Let c, be a maximal segment from a ( r ) to p, for each n as described above
so that p, is the past timelike cut point, to a(r) along c,. If cnl(0) E V ,
by travelling along c, a little further than p,, we obtain a point q, in U with
q, << p, and such that c, from a ( r ) to q, is not maximal. BJ using Sublernma
14.25, we then obtain a maximal timelike segment y, from a(r) to q, which
must satisfy ynl(0) $ V. In this case, replace p, with q,.
Hence we may assume that we have maximal timelike geodesic segments
c, from a(? ) to p, with %'(Of $ V for any n. By passing to a subsequence
and reasoning as in Sublemma 14.25. ~e obtain a maximal geodesic segment
c from a ( r ) to p. Since a: is also a maxima! timelike segment from a ( r ) to p,
we must have that c is timelike. Also, since ct (0) is a limit point of { c n l ( 0 ) ) ,
we must have cl (0) $8 V. But this then forces CY and c to be two distinct
maximal timelike segments from p to a(?-). But then cu from p to a ( r f 6 ) is
not maximal; contradicting the fact that a was a timelike ray. 0
&examination of the arguments employed in the proof of Proposition 14.23
shows that the following alternative result may be proved about the timelike
cut locus and a maximal timelike geodesic segment [cf. Galloway and Horta
(1995, Proposition 3.9)j. Let y : [O, TO] --+ (M. g) be a maximal timelike ge-
odesic segment in an arbitrary space-time, and suppose that for each 7 with
< T < T O , the distance function
534 14 THE SPLITTING PROBLEM IN LORENTZIAN GEOMETRY
is finite-valued on a neighborhood of y(0). Then for each T with 0 < T < ro,
there exists a neighborhood U of the segment y(j0, r ] ) which does nos meet
the past tirnelike cut locus of yjr).
Now that control of the cut locus has been established, ~t is standard in the
above setting that the distance function a I-+ dr(s) = d(x,cr(r)) is not only
continuous near p as established in Sublemma 14.24 but also is smooth near
p. (This actually follows wizhout needing continuity first since the control of
the timelike cut locus ensures that Lorentzian distance may be expressed in
terms of the length in the appropriate tangent space of exponential Inverses of
points in M [cf. Lemma 4.263.)
We turn to an estirnate (14.29) for the d'alembertian U(d,) = troHdr,
which is an important ingredient in the proof of the splitting theorem. Similar
estimates have been important for a long time in global Riemannian geometry
[cf. Calabi (1957) for an early illustration]. In our current setting with U chosen
as in the proof of case (3) of Proposition 14.23, we first note the following basic
result.
Lemma 14.26. For q ~n U , let to = d(q, ~ ( r ) ) , and let c : 10. to] -- ( M , g )
be the past directed unit speed maximal tirnelike geodesic segmezt with c(0) =
~ ( r ) and c(to) = q. P u t f (x) = d,(x) = d(x, cr(r)). Then
(1) (pad fI(9) = -cl(to).
Hence,
(2) (gradf,gradf) = -1,
and the past directed maximal timelike geodesic segments from ~ ( r ) to poi
of U are integral curves of - grad f in U .
P~oof. Let v E T,M be given. and set c,(s) = expp(sv) For s suEcie
small, we may define y, : 10, d(c,(s), o(r))] -. ( M , g ) to be the unique maxirn
unit speed timelike geodesic segment from a(rl to c,,Is\. Then setting 7
( t) defines -
a variation ...
of the geodesic 1 ,
c through -, , eodesics with
14.2 CO-RkYS AND THE BUSEMANN FUNCTION
for all s. Hence by Corollary 12.24,
for any v. Thus. (grad f)(q) = -c1(to) is future directed tirnelike. and
(grad J, grad f ) = - 1
Lemma 14.26 then implies that we are now precisely in a well-known geo-
metric setup which is studled in Appendix B. We do our curvature calcula-
tions in this framework. Thus let Ul be an open subset of (M.g ) , and let
f : Ul --+ R be a smooth function on Ul which satisfies the eikonal differential
equation (grad f , grad f ) = -1. By Lemma B.1. the integral curves of grad f
are tlmelike geodesics. Moreover, we note in Section B.l that the (wrong way)
Gauchy-Schwarz inequal~ty for timelike tangent vectors Implies that any such
integral curve of grad f is maximal in the space-time (Ul, g /u,) We also note
that curvature calculations lead to the formulas (B.22) and (B.23) relatlng the
Ricci curvature of the integral curves off to the d'Alembertian O( f ) = tr o ~ j ,
Helpful in these calculations is the baslc identity
Iff (v, grad f ) = 0
for any v in T(Ul). since
In view of our desired application for which f = d,, we let c denote an integral
curve in Ul of X = - g a d f. (Thus there are certa~n slgn d~fferecces w ~ t h
(B.22) and (B.23) In Appendix B in which c is an integral curve of grad f )
Recall that V x X = 0 on U1 and also Nf (x, Y) = 0 for any Y in T(Ul ) .
536 14 THE SPLITTING PROBLEM IN LORENTZIAN GEOMETRY
Choose an orthonormal frame {El, E2, . . . , E, = X) along c and neighbor-
ing integral curves. Also, let V be any parallel field along c, also extended to
these nearby integrai curves as a vector field parallel along them as well Then
[V,X] = v v x - V,fV = V V X
since V is parallel along c Hence,
R(V,cl)c' = V v V x X - V,rVvX - VLv,xlX
= -v,~vvx - V(vvx)X.
Thus
(R(V, c')cl, V) = -(Vcl VvX, V ) - (V(o,x)X, V) n-l
= -cl ((VvX, V)) + (VvX, Vclv) - ~ ( ~ v x , E ~ ) ( v E , x , V) ,=I
using V v X = c,"~:(vvx,E,)E~, valid since N ~ ( v , B ~ ) = 0. Recalling that
V,lV = 0 and X = -grad f, we then obtain
d n-l
(R(v, c')cl, v ) = - { ~ f (v, V) 0 C) - C ( ~ f (v, E,)) . dt
j=1
If we put V = E, in this last equation and sum, we obtain
n-1
- RC(C', c') = - ( ~ ( f ) 0 c)' + C (HJ(E, E,:)~ %,?=I
n-l
r - ( W ) c)' 4- C (ilf(~,,~,))~. 2=1
The Cauchy-Schwarz inequality yields
n-1
Thus we are led to the Eccati inequality
14.2 60-RAYS AND TEE BUSEMANN FUNCTION 537
somewhat reminiscent of the Raychaudhuri equation (12.2). Of course, it
has long been known in the theory of conjugate 0 D E 's that such Riccati
inequalities lead to estimates on growth of solutions In this particular case.
we will verify that provzded Ric(cl, c') 2 0. integration techniques like those
already encountered in Proposition 12.9 may be used to obtain the desired
estimate for the d'Alernbertian of f = d,:
for any q in U. Hence, a key aspect of the hypothesis in the Lorentzian Splitting
Theorem that Ric(v, v) 2 0 for all timelike w is to ensure that estimate (14.29)
is valid.
To aid in the derivation of (14.29), introduce the notation +(t) = Dcrd,(c(t));
where c : [O,d(q, a(r))] + ( M , g ) denotes the past directed maximal timelike
geodesic segment from a(r) = c(0) to q = c(d(q, a(r)) j . Put t o = d(q, a(r ) ) .
Under the Ricci curvature assumption, inequality (14.28) impi~es
It may be checked that +(t) --+ -m as t -+ O+ by identifymg @(t) a s the
trace of the second fundamental form of the "distance sphere" of a(r) through
c(t) or by checking for Minkowski space [cf. Beem, Ehrlich, Markvorsen, and
Galloway (1985, p. 38)). Now
in view of (14.30). Integating from t = 0 to t = to yields
so that
Hence. recalling that to = d(q, a(?-)), inequality (14.29) is establislied.
538 14 THE SPLITTIXG PROBLEM IN LORENTZIAN GEOMETRY 14.3 THE LEVEL SETS OF THE BUSEMANN FUKCTION 539
14.3 The Level Sets of the Busemann hnction
In the differential geometry of simply connected, complete Riemannian man-
ifolds of nonpositi~e sectional curvature, the !eve! sets of the Busemann func-
tion are called "horosphcrcs" and have been much studied in differential geome-
try and dynamical systems Indeed, the already existing notion of a llorosphere
for the Poincar6 disk or upper half plane with the usual complete Eemannian
metric of constant curvature -1 seems to have motivated Busemann to define
what is now called the Busemann function [cf. Busemann (1932)) Much later,
in the proof of the Cheeger-Gromoll Riemannian Splitting Theorem. the letel
sets of the Busemann function provided the Riemannian factor in the isometric
splitting This aspect of the Busemann function suggested that perhaps the
Lorentzian Busenlann function should be studied as a means toward proving
a Lorentzian ,.,litting Theorem So far in spacetime geometry, the "hor
spheres" have primarily been employed as a tool In proving the space-ti
splitting theorem
Before returning to Galloway and Horta (1995), it is necessary to introduce
a standard concept from general relativity which has not been previously en-
countered in this book. Let A be an achronal subset of a space-time, i.e., no
two po~nts of A are chronologically related.
Definition 14.27. (Edge of an Achronal Se t ) The edge, edge(A). of th
achronal set A conslsts of all points p in such that every neighborhood i7 o
p contains a timelike curve from I - (p , U ) to I f (p, U ) which does 7 ~ 0 t meet A.
The set A is said to be edgeless if edge(A) = @.
Evidently, edge(A) C 2. O'Neill (1983, pp. 413-415) gives an excellent ex-
position of the basic facts concerning edge(Aj and topological hypersurfaces:
(1) The "achronal identity". - A - A & edge(A);
(2) An achronal set A is a topological hypersurface if and only if
edge(A) = 0, and
(3) An achronal set A is a closed topological hypersurface if end o
Note that being edgeless is somewhat like being "locally" a Cauchy surface
in that future timelike curves starting at points in I-(A) sufficiently close to
A must intersect 3. Galloway and Horta (1995) adopt the following definition of a topological
spacelike hypersurface. Recall that an acausal set fails to contain any causally
related points and is thus also achronal.
Definition 14.28. (Spacelike Hypersurface) -4 subset S of ( M , g) is said
to be a CO-spacelzke hypersurface if for eachp in S. there exists a neighborhood
U of p in M such that S n U is acausal and edgeless in (U,g 1 ~ ) .
The basic "edge theory' previously mentioned ensures that a spacelike hy-
persurface according to Definition 14.28 1s an embedded topological submani-
fold of M of codimension one. A smooth spacelike hypersurface, i.e., a smooth
codlmension one submanlfold with everywhere tlmellke normal, is a spacelike
hypersurface in the sense of Defin~tion 14 28. A somewhat stronger concept IS
that of a partial Cauchy surface.
Definition 14.29. (Partial Cauchy Surface) A subset S of (M, g) is said
to be a partzal Cauchy surface if S is acausal and edgeless in 1M.
In particular, a partial Cauchy surface is a spacelike hypersurface in the
above sense and 1s also closed.
Now we are ready to show that the level sets of the Busemann function of
a tirnelike 5'-ray are locally partial Cauchy surfaces, provlded the generalized
timelike co-ray condition holds [cf. Gallow-ay and Horta (1995, Proposit~on
4.2). Galloway (1989a, Lemma 2.3)].
Proposition 14.30. Let (M, g) be future timelike geodeslcdly complete
and let 7 be a timelike S-ray. Suppose that p E I - (? ) n I+ (S) IS a point of a
level set {b, = c ) where the generdlzed t~melike o-ray condit~on holds Then
there exists a neighborhood U of p such that the set C, = U n (b , = c j is
acausal in M and edgeless in U. In particular, C, is a partial Cauchy surface
in (U,G/U).
1 Proof. Choose an open neighborhood W of I, so that b = by is continuous
I on U by Theorem 14.19, and also such that Lernrnar 14.15 and 14.16 and
540 14 THE SPLITTING PROBLEM IN LORENTZIAN GEOMETRY 14.3 THE LEVEL SETS OF THE BUSEMANV FUNCTION 54
Corollvy 14.17 hold on U. Since b(q) 2 b(p) + d(p , q) by Lemma 14.5, we have the geodesic ray c, : [O. +m) -+ (N. go) by taking the union of distance balls
b strictly increasing along timelike curves, which impl~es that C, is achronal.
Suppose p is an edge point of C, in U. Then there exist s, y E U - C, and Bu= l J % ( c ~ ( t ) ) = U { q e (N,go) d o ( q , k ( t ) ) < t ) . t>o t>O
future timelike curves cl, c2 : [0, 11 -+ U joining x to y such that cl passes The horospheie Ii, associated to c, is then the boundary of the horobal
B,. If H, turns out to be sufficiently differentiable (C2), then there is
normal geodesic variation of c, along H, for which the stable Jacobi tenso
such that b o c2(t0) = 0. Thus cz(t0) lies on C, in contradiction. field D, is a variation tensor field [cf. Eschenburg and O'Sullivan (1980, p. 8
and Lemma 12.13 for the construction of the stable Jacobi tensor field for
maximal null geodesic segment 11 from x to y. By choice of U , there exist In the proofs of various versions of the Lorentzian Splitting Theorem, mostl;
sequence r, + +co such that there are maximal timelike segments a, from in the globally hyperbolic case (and also in separate investigations of hyper
surface families in Riemannian manifolds), various estimates have been rnadt
also the notation b,(x) = d(y(O), y ( r ) ) - d ( x , y ( r ) ) and the definition b( on the Hessian of the diitance function s ++ d(z, a ( t ) ) as t -+ +co where c
limp,+, b,(x). Now an aspect of ow choice of U is that the initial tange is a timelike asymptote to y , en route to obtaining various maximum prin
to timeiike geodesics remain bounded away from the null cone in making t ciples for the Busemann function and its sublevel sets [cf. Eschenburg (1987
asymptotic geodesic construction as a result of the neighborhood Kc( 1988, 1989), Galloway (1989a), Newman (1990)]. The discussion in the pre
Lemma 14.16 and Corollary 14.17. With this control, it follows that by cu vious paragraph shows that this step in the proof of the splitting theoren
the corner of the broken geodesic q u a , and comparing with the corner of qU corresponds to forming the horoball in Riemannian geometry. In particular
there exists E > 0 such that for all n sufliciently large, Riccati comparison techniques to estimate these Hessians have been well ex-
plored for Riemannian hypersurfaces by Eschenburg (1987, 1989), and it i~
~ T , ( Y ) - bTn(x) = d ( x ! ~ ( r n ) ) - d ( ~ > ~ ( h ) ) > E . noted that these techniques apply also to families of spacelike hypersurface
in space-times. Eschenburg, Xarcher (1989). and others take the viewpoint
Letting r , -+ +m; we obtain b(y) - b(x) 2 E. But since x , y E C,, we m
have b(z) = b(y) which furnishes the desired contradiction 0
rather than by using Jacobi equatio
542 14 THE SPLITTING PROBLEM IN LORENTZIAN GEOMETRY
We return to the case of interest in proving the Lorentzian Spllttlng The-
orem and thus implicitly to the "horobalK of a timelike geodesic asymptote
to y as given in Galloway and Horta (1995, Lemma 4 3 and Proposition 4.4)
under the asumption of future timelike geodesic completeness rather than
global hyperbolicity. In this result. all timelike geodesics are parametrized
Lorentzian unit speed curves, not as unit speed curves in the auxiliary R
mannian metric h.
Lemma 14.31. Let (M, g) be future timelikegeodesically complete, and I
y be a timelike S-ray Assume that the generalized timelike co-ray cond~tio
holds at p E I-(?) n I f ( S ) Then there exist a neighborhood U of p an
constants to > 0 and A > 0 such that for each q in U and each timeli
asymptote a to r from q we have
(14.32) f Jd t (v, v) 2 - A (u J. ,v i )
for all v E T,M and t 2 to, where dt = d ( . , a@)) and vi is the projectio
onto the normal space (a'(0))i.
Proof. Let U be a neighborhood of p with compact closure on which t
generalized timelike co-ray condition holds and also Lemmas 14.15 and 14
and Corollary 14 17 hold. By Corollary 14.17, the set of all initial tangents
asymptotes to y starting from points of U is contained in a compact sub
of the unit timelike tangent bundle. Then, by taking U and to sufficient
small, we may ensme that the set (al( t) : 0 5 t < to ) of all tangents to
initial segments of length to of all asymptotes a emanating from points of
contained in a compact subset of the unit timelike tangent bundle. Hence
set of all timelike two-planes Il containing these initial tangents is contdne
a compact subset of the set of all timelike two-planes in G2(M) . Thus, t
of all sectional cl~rvatures of such planes is bounded above by some co
k; K(rI) 5 k .
Now consider B = -Hd" along the segment a 1 10, to]. B = B, corresp
to the second fundamental form of the level sets
14.3 THE LEVEL SETS OF THE BUSEMANN FUNCTION 543
at a(to - s) for 0 < s < to. Moreover. u. ++ B,, u = to - s. obeys the Riccati
equation B,' -+- BU2 + R( . , @')a1 = 0 [cf Eschenburg f 1987, Equation f3))j
Hence given the sectional curvature bound K ( n ) 5 h-, the appropriate Ric-
cati comparison Theorem may be applied je.g., Proposition 2.4 in Eschenburg
(1987)] to deduce the existence of a constant A = A(k) such that for all v in
T,M
J I d i o (w, v) 2 -A (vi, 3').
The lemma now follows since t ++ Hdt(v , u ) is an increasing function of t
[cf. the proof of Lemma 12.131. 0
Before proving the next result and obtaining two corollaries, which are
employed in the proof of the Lorentzian Splitting Theorem, we recall from the
Preface of Protter and Weinberger (1984, p. v) a general philosophy important
in P.D.E.'s of whicii Propositio~~ 14.32 is an example.
One of the most useful and best known tools employed in the
study of partial differential equations is the maximum principle.
This principle is a generalization of the elementary fact of calculus
that any function f (z) which satisfies the inequality fl' > 0 on an
interval [a, b ] achieves its maximum value at one of the endpoints of
the interval We say that solutions of the inequality f" > 0 satisfy
a maximum principle. More generally, functions which satisfy a
differential inequality in a domain D and, because of it, achieve
their maxima on the boundary of D are s a ~ d to possess a maximum
principle.
We now turn to the proof of Proposition 14.32 following Galloway and Horta
(1995, Proposition 4.4). An earlier result for globallq hyperbolic space-times
was obtained in Galloway (1989a, Lemma 2.4), where the opposlte convention
for choice of normal was used for calcalat~on of the mean curvature [cf. Newman
544 14 THE SPLITTING PROBLEM IN LORENTZIAN GEOMETRY 14.3 THE LEVEL SETS O F THE BUSEMANPU' FUNCTION 545
Proposit ion 14.32. Let (M, g) be future timelike geodesically cornplece Let a : [O, 4- oo) -+ ( M , g) be a timelike asymptote to 7 with a(0) = p. Then
and suppose also that Ric(v, v) 2 0 for ail timelike v. Let y be a timelike for each t > 0, recalling Lemma 14.12 and Proposition 14.23.
S-rajr, and let W 5 I-(?) n l+(S) be an open set on which the generalized
timelike co-ray condit~on holds. Let C C W be a connected smooth spacelike b,,t(z) = b(p) + t - d(x. a@))
hypersurface with nonposrtive mean curvature, H z = divc(N) _< 0, where IV is the future pointing unit normal dong C. If the Busemann function b = b, is a smooth upper support function for b at p. Thus the function
attains a minimum dong C, then b, is constant dong C.
Proof. Suppose that b achieves a minimum b(q) = a at q in C. Let U be fc,t = bp,t + ~h
a neighborhood of q on which the generalized timelike co-ray condition hold is a smooth upper support function for f, at p, which implies that j,,t also has
and also Lemmas 14.15, 14.16, and 14.31 hold. Since bl C is continuous, a minimum at p. We will obtain a contradiction by calcula~ing Ax f,,,(p) and
suffices to show that b = a in a neighborhood of q in C. If this is not correc showing that it is negative for e sufficiently small and t sufficiently large.
then there exists a coordinate ball B with B C C n U centered at q such that
8(B) f DO(B), where First we digress to give a generd calculation for a smooth funct~on f :
M + W relating the dd'Alernbertian O(f) of f to the Laplaclan Ax( f 1 C) of f
D0(B) = {z E a(B) : b(x) = a). restricted to the spacelike hypersurface C. Fix p E C, and let (el, ez, . . . , en- )
be an orthonormal basis of spacelike vectors for T,C. Combining the sign Note that b > a on d(B) - DO(B). Also it follows from Lemma 14.31 that ther
conventions of Calloway and Horta (1995) with those in Chapter 3, Defin~t~on exists a constant C > 0 such that
3.43 for the second fundamental form, we ntay define the mean curvature Hc(p)
(14.33) ~ ~ * ( v , v ) 2 -C
for all asymptotes at 2, for all x in B, for all v 21 TT,C with (v, v) 5 1, and n-1 n-1
all t sufficiently large. H ~ P ) = - s ~ ( e , , e , ) = E(V,,N, e , ) t=1 %=I
By choosing B sufficiently small, we may construct a smooth function h
C having the following properties [cf. Eschenburg and Heintze (1984), Newm For the purposes of the calculations of Suble~nma 14.33, we denote the
(1990, p 177)]: adient in (M, g) by v( f ) and the gradient in (C, g lc) by V( f ). Also, let
(1) h(q) = 0; denote the Levi-Civita connection for (M, g), let V denote the Levi-Civita
(2) IjVchj( < 1 on B, where Vc denotes the gradient operator on C; onnection for (C, g j ~ ) , and let H; denote the hessian of f 1 C in (C, g (x).
(3) AGh < -D on B, where D is a positive constant and Ac is the indue Sublemma 14.33. Let C be a spacelike hypersurface of (,if. g), and let Laplacian on C; and
: M + R be a smooth function. Then for any p in C and with the mean
ure H z ( p ) given by (14 351, we have
-
546 14 THE SPLITTING PROBLEM IN LOREKTZIAN GEOMETRY 14.3 THE LEVEL SETS OF THE BUSEhIANN FUNCTION 547
Pmof o j Sublemma (1) We have Recalling inequality (14.29)) we have
n-I n - l
B(f)(P) = Z ( V ( f ) ( p ) . e , ) e , - @(f)(P)>N(P))N(P) W P , t ) ( p ) = -O(dt!(P) 5 7. t=l n-1 Because f,,t has a minimum on C at p, we have Vc( jE , t ) (P) = 0. Also,
= ei(f)ez - W f ) N ( P ) V(bt,,)(p) = -aJ(@) [cf. Lemma 14 261. Noiv using Sublemrna 14.33-(I), z=l
= V ( f 16) - lV( f )N(P) . V z ( f e , t ) ( ~ ) = Vc(bp,t)(p) S E V Z ~ ( P )
(2) Now = v(bp,t) ( P ) + (N(P) , V(bp , ) ( P I ) N ( P ) + E V C ~ ( P ) n-1
U(f )@) = Z g ( e z , e z W f ( e , , e,) + ~ ( N ( P ) . W P ) ) Hf ( N ( P ) , N ( P ) ) = -a'@) - (N(p) . crl(0)) N(p) + ~ V c h ( p )
z = l
n-I Thus, V ~ ( f , , , ) ( p ) = 0 implies that
= H f (ez , e,) - Iff (N , N ) I p . *=I N ( P ) = (N(P) , ~ ' ( 0 ) ) - ~ [ - ~ ' ( 0 ) + f(Vch)(p)j .
Hence we must calculate ~f (e,, e,) using (1). Now Since Hd$(vd t . .) = 0. we also have Hdc(al(0) . . ) = 0 Thus we obtain
Hf(e,.e,) = ('5,,Vf,et) = ( V e s V ( f I C).eZ) - ( a e , N ( f ) N . e z ) Hbppt (AT(p). N(P) ) = e2 (iv(p), ~ ' ( O ) ) - ~ H ~ ~ , ( V ~ h ( p ) , Vch(p) ) .
= (Ve ,V( f 16), e,) - ( e t ( N ( f ) ) N ( ~ ) + Np(f)Ve,N? et) By the (wrong way) Cauchy-Schvlarz inequality,
= Hi(e, ,et) - N p ( f ) (V,,N,e,)
Thus, I(N(p),a'(O))l 2 IlN(P)ll . l l~ ' i@)l l = 1
n-1 Since V z h ( p ) is tangent to C at p and also llVch(p)ll 5 1, u-e may apply U ( f ) ( p ) = A c ( f i WP) - N * ( f ) ):(Ve,N ez) - Hf t N ? N ) l F
r = l inequelity (14 33) to deduce H b p f t (Vch(p), Vch(p) ) < C and hence
= A c ( f I WP) - N,(f)Hc(p) - . f f f ( * i . N ) l p . a ~ ~ p ~ ( l v ( ~ ) , N ( ~ ) ) L ce2.
We now return to the proof of Proposition 14.32. By definition of f,,t Thus under the mean curvature assumption Nc(p) 5 0 and with the choice of
have N(p) as future directed tlmellke, we obtain
A~( f~ , t ) ( f - r ) = A z ( b p , t ) ( ~ ) + E A c ~ ~ P ) n - 1 nz(bt,,>(p) r + cE2
< A~(bp , t )@) - DE- -
By Sublemma 14.33 we have
548 14 THE SPLITTING PROBLEM IK LORENTZIAN GEOMETRY
Then taking t > 0 sufficiently large and E > 0 sufficiently small, we obtain
& 2 ( f t , t ) ( ~ ) < 0,
which is the desired contradiction 3
Proposition 14.32 has the following two corollaries [cf. Galloway and Norta
(1995)], which wlll be used in the proof of the splitting theorem in Section
14.4. Inequality (14.36) and these corollaries will indicate that the superlevel
sets
(4 E M : b,(q) > 4 are in some sense mean convex. This has been studied in the Riemannian case
in Eschenburg (1989).
Corollary 14.34. Let (M,g) be a future timeiike geodesically compl
space-time which obeys the timehke convergence condition Ric(v, v) 2 0
all timelike v, and let y be a timellke S-ray. Let W I - ( y ) f? IT(S) be
open set on which the generalized timelike co-ray condition holds. Let C be
connected xausal smooth spacel~ke hypersurface in W with nonpositive me
curvature. Suppose that C and C, = {b, = c ) n W have a point m com
and that C Jf (C,, W ) . Then
C c C,.
Corollary 14.35. Let ( M , g ) be a future timelike geodesically corn
space-time which sat~sfies the timelikc convergence condition Ric(v, v) 2 all timelike v, and let y be a timelike S-ray. Let W c I-(?) n I f (S) be
open set on which the generalized timelike co-ray condition holds. Let C b
smooth spacelike hypersurface with nonpositive mean curvature whose clo
is contained in IV -4ssume that C is acausal in W and E is compa
edge(C) C {b, 2 c ) , then
c r 1 el.
Section 14.5 of this ch
how the regularity for the Busemann
14.4 TEE PROOF OF THE LORENTZIAN SPLITTING THEOREM 549
a timelike S-ray on the set I - ( y ) n I + ( S ) may be extended to the larger region
1- ( y ) n [Jf (S) U D-(S)]: where D- ( S ) denotes the past Cauchy development
(or past domain of dependence) of S , i.e.,
D-(S) = ( q E M : every future inextendible nonspacelike
curve from q intersects S )
In particular, it is noted that if S is a compact acausal spacelike hypersur-
face in the future timelike geodesically complete space-time (M,g) , then the
generalized timelike co-ray condition holds on I-(-!) n [ J + ( S ) U D-(S)] . As a
consequence of these considerations, the following is obtained in Galloway and
Borta (1995, Corollary 5.6).
Corollary 74.36. Let (M,g ) be a globally hy.~erboIic, future timelike
geodesically cnrnprete space-time which contains a compact spacelike hyper-
surface S, and let y be any S-ray Then 6, : (M g ) -+ (0, +co] is continuous.
Moreover, the level sets C, = (q E M : h,(q) = c ) are partial Catlchy surfaces
in (M, g).
Galloway and Worta (1995) note that the de Sitter space-time provides an
example of ( M , g ) which is globally hyperbolic, geodesically complete, with
compact Cauchy surfaces S , yet contains S-rays y for which I - ( 7 ) # M .
Hence, the Busemann function b, of such a ray y is not finite-valued on all of
M ,
14.4 The Proof of the Lorentzian Splitting Theorem
The results have now been assembled which enable us to present a proof of
the Lorentzian Splitting Theorem in the timelike geodesically complete case.
as originally formulated by S. T. Yau (1982) in problem number 155. A proof
er this hypothesis, rather than that of global hyperbollcity, was first given
Fewman (1990). The proof given in this sectlon follows Galloway and Horta
, in which simplifications are obtained in the previous proofs cited in
oduction to this ckapter based on the systematic use of the generalized
550 14 THE SPLITTING PROBLEM IN LORENTZIAN GEOMETRY
Theorem 14.37. Let (M, g) be a t~melike geodesically complete space-
time which satisfies the t~mel~ke convergence condition Ric(v,v) >_ 0 for all
timelike vectors v. If ( M , g ) contains a timelike line, then ( M , g) is isometric
to (R x S, -dt2 h) where (S, h) is a complete Riemannian manifold.
Earlier work on the Lorentzian Splitting Theorem involving the Busemann
function approach assumed global hyperbolicity rather than timelike geodcsic
completeness, in part; because of the automatic existence of maximal timelike
segments connecting any pair of chronologically related points. Additionally,
the Lorentzian distance function of any globally hyperbolic space-time is al-
ways continuous as well a~ finite-valued. This version of the splitting theorem
is as follows:
Theorem 14.38. Let ( M , g ) be a globally hyperbolic space-time which
satisfies the timelike convergence condition Ric(v, v) > 0 for all timelike vectors
v. If ((M. g) contains a complete timeiike line, then ( M , g ) splits as in Theorem
In this second version, while completeness for all timelike geodesics is n
assumed, timelike complpteness of some maximal timelike geodesic is require
Proof of Theorem 14.37. Let y : I --+ (M.9) be any complete timelike 11
parametrized wlth (y1(t),-y'(t)) = -1. Let -y( t ) = y(-t) denote y wlth t
opposite orientation. As in all the proofs of the various versions of the splitti
theorems, define
and
bf = lim b) and b- = lim b;. r++m T-+w
Put I ( y ) = I+(?) r l I P ( y ) . Since y / [a. +m) and -y 1 la, +m) are tim
rays for all a E R, b+ and b- are defined and finite-valued on I ( y ) .
all the regularity results (and their time duals) hold on a neighborhood
point of y. It will be helpful for use in the sequel to note that Lemma 14 5-
translates into the following: given any p, q in I(?) with p < q, then
14.4 THE PROOF OF THE LORENTZIAN SPLITTING THEOREM 551
and
Now fix q in I ( y ) . Then there exist s. t > 0 so that y(-s) <C q << y ( t ) . Berice.
using the reverse triangle inequality,
whence
Thus B = b+ + b- 2 0 on I(?). Similar calculations show that B ( y ( t ) ) = 0
along the geodesic y(t) itself.
Let U be a convex normal neighborhood of :/((I) such that all the preceding
regularity properties hold for bf and b- on U . Consider the "horospheres'"
S+ = (b+(q) = 0 ) n U and S- = {b-(q) = 0 ) n U . By Proposition 1 4 30, both
Sf and S- are partial Cauchy surfaces in U. each containing y(0) Using the
topological manifold structure of S f . let W be a small coordinate hall in S+ centered at -(O) whose closure is also In S+. By applying the fundamental
existence result of Bartnik (1988a, Theorem 4 1) for spacelike hypersurfaces
with rough boundary data, we obtain a smooth maximal (i.e., Hz = 0) space
like hypersurface C U such that C is acausal in U with compact closure.
edge(C) = edge(W) C S f = (b+(q) = 0 ) n U, and C meets y. By deleting
points if necessary. we will also take C to be defined such that
First apply Corollary 14.35 to conclude that C C {hf ( q ) 2 0) n U But
also. since b-(q) 2 -b f (q) on I ( y ) , we have simultaneously that
Hence, by the time dual of Corollary 14 35 applied to --f, .rvc obtain
552 i 4 THE SPLITTING PROBLEM IN LORENTZIAN GEOMETRY
Thus
C G jb+(q) L 0) n {b-(4) L 0).
Now since C C (b+(q) 2 0) and points of edge(C) satisfy b+(q) = 0, we obtain
from Corollary 14 34 that b+ = 0 along C. Hence if we let x = y(to) denote
the point of intersection of C and y, we have b+(z) = 0. Thus as B = O on 7,
we obtain b-(x) = 0 as well. This fact and C {b-(9) 2 0) enable the time
dual of Corollary 14.34 to be applied to conclude that C C S-. whence
(14.38) b+ = b- = O along C.
Given any x in C, let a,f : [0, +m) -+ (M, g) denote any timelike asymptote
to yl [O,+CG) issuing from x, and let a; : [O,+oo) -+ (M.g) denote any
tlmellke asymptote to y ] [O. -m) = -7 I [0, +GO) issuing from x. With (14.3
in hand, it is now poss~bie to establish that at any x in C, the asympto
geodesics a; and a: fit together at x to form a smooth maximal geodesi
line, I.e., [a;i.]'(Q) = -[a,]/(O), and further, that through each such x in
there is exactly one tlrnelike asymptote a, to y passing through x. Towa
this end consider the (possibly) broken (at t = 0) geodesic
&(t) for t 2 0,
a;(-t) for t < 0.
Flrst, from Corollary 14.13 we obtain for t 2 0 that
(14.39) bf (a,(t)) = b+(x) + d(x. a,(t)) = O + t = t
and similarly b-(a,(t)) = -t for t < 0. Now take t < 0, and consider (14.3
w ~ t h p = a,(t) = a;(-t) and q = x. We obtain
0 = b+(x) 2 b+(a,(t)) + d(a, (-t), x)
= b+(@X(t)) + (4,
whence bf (a,(t)) 5 t for all t < 0. On the other hand, for m y t < O
B(Q,(~)) = h+(a,(t)) + b-(a,(t))
= hi (a, (t)) + b- (a; (-t))
= b'(a,(t)) - t
24.4 THE PROOF OF THE LORENTZIAN SPLITTING THEOREM 553
SO that as B 2 0 on I ( y ) , b+(a,(t)) 2 t for all t < 0. Combining these two
calculations yields b+(a,(t)) = t for any t < 0, hence for all t in R, recalling
(14.39). An analogous argument using inequality (14.37b) in place of (14.37a)
shows that b-(a,(t)) = -t for any t in R.
We are now ready to show that ax( t ) is globally maxlmal. For take any t l ,
t 2 with t l < 0 < tz. Applying (14.37a) with p = a,(tl) and q = a,(tz) gives
whence
d(ax(tl)ffs(t2)) 5 t 2 - t l .
Automatically,
Hence, d(a,(tl), a,(tz)) = tz - tl for any tl < 0 < t2, which implies that a,
is globally maximal. In addition, note that if a, were broken at t = 0, then
the usual "cutting the corner" argument applied at x = a,(O) would produce
a longer curve from a,(-to) to a , ( t ~ ) than a, / [-to, to] for a small to > 0,
contradicting the maximality of a, just established. Thus we have seen that
for x in C , any future directed asymptote crl to 7 1 10, +m) with ul(0) = x and
any past directed asymptote a2 to -y 1 [O,+m) with (TZ(O) = 3. fit together
at z to form a smooth, maximal timelike geodesic line This is of course the
geometric basis for the splitting theorem to be valid in a tubular neighborhood
of y. as we now establish.
First we establish that both a,f and a; are C-rays for any x in C. Re-
call that for any q in I-(? j [O, +m)) n f + (C), we have 0 < d(C, g) < bi(4)
[cf. inequality (14.18)j. Taking q = a, (t) with t > 0, we obtain
But dfX, cr,(t)) 2 d(c~,(O),a,(t)) = L(a, ' [0, t 1) since a, is maximal. Hence,
d(C,a,(t)) = t = L(a, / [ 0 7 t ] ) for all t > 0, and a$ I [O, +os) is a C-ray as
desired. Dud arguments show that a; is also a X-ray. It is then a consequence
554 14 THE SPLITTING PROBLEM IN LORENTZIAN GEOMETRY 14 4 THE PROOF O F THE LORENTZIAN SPLiTTING THEOREM 555
of basic theory in the calculus of variations that a: and a; are both focal po~nt hypothesis of timelike geodesic completeness, we may then define the normal
free and that a, meets C orthogonally (cf. Propositions 12.25 and 12.29). exponential map E : iR x C -+ (M, g) by E( t , q) = expq(tN(q)). i n view of
Let us ilow sl~ow that direct geometric arguments reveal that for any pan of the properties derived above for the asymptotic geodesics to y issuing from
distinct points p, q in C, yl = a," and 72 = a: do not intersect. For suppose points of C. we already have that E is nonsingular and ~njective ar.d hence a m = yl(s) = yz(t) for some s, t > 0. First, t = s = d(C,m) since both diffeomorphism onto its image.
geodesics are unit speed X-rays. Thus put to = t = s > 0, and note also that By our above calculations for b+(a,(t)) acd h - ( ~ , ( t ) ) , we have that
L(yl 1 10. t o ] ) = L(y2 I [0, to]) = d(X, m). Choose any 5 > 0, and let tl = to + S
and n = yl(tl). Starting with the fact that yl is a X-ray, we obtain h+(E(t.q)) = b+(exp(tN(q))) = t
d(C,n) = L(7l I 10,tlI)
= L(YI! [O, t o ] ) + L h l I [ to , tl]) b-(E(t, q)) = b - ( e ~ p ( t N ( ~ ) ) ) = -t.
= L(72 I [O,tol) + L(7l I [to, ill). Since E is a diffeomorphism onto its image, these equatioils reveal that both
b+ and b- are differentiable on the connected open set fi = E(W x C). Let c = yzlp,,ol * ~ l l ~ ~ ~ , ~ , ~ , a curve from q to n which, by the previous c
Fix x E C, and let a,,, denote the unit timelike geodesic from .L to ? ( r ) lation, has the property that L(c
for r sufficiently large. Then as b,f(x) = r - d(x, y ( r ) ) = r - d,(x), one has we have yll(to) # yzl(to). Hence,
Vbf (x) = -Vd,(x) = -a&(0) (cf. Lemma 14.26). Hence letting r, -+ +m. done using a convex normai neighborhood of m, show that we may construe
and remembering that U has been chosen so that the asymptotic geodesic future causal curve a from q to n with L(a) > L(c). But then L(a) > d(C,
construction is well behaved, we have a,'(O) = limn,+, cuL,,n(0) and thus which is impossible. Thus the future timelike geodesics a:, x E C, do
intersect. Dual arguments reveal Vb+(x) = lim Vb?=(x) = lim -c~j.~,(O) = -aZ1(0), n-TW n-+m
x E C, can intersect either.
Let us now turn to the non-i where a,(t) = E( t , x). More generally,
form yl = a: and 72 = a;, for any p, q E X except possibly at p = q Vb+ (az (t)) = -ayl(t) C. Suppose that there exist s, t > 0 such that yl(s) = yz(t) = m.
b+(m) = b+(yi(s)) = s > 0. On the other hand, taking any t l with 0 < t l r any t 6 W jcf. Eschenburg (1988)], This discussion shows that f = b+ 1s a
we have b+(-y2(tl)) = -tl from our previous work in the course of thih pr ifferentiable function on Ut which satisfies the eikond equation ('17 f. D f) =
Since b+ o yz 1s continuous, there exists t2 > 0 such that b'(yz(t2)) = 0.
this then contradicts the achronality of the set {b+ = 0) . Hence we h s that for any point q In Ul there exists a unique unit speed maximal
established the fact that no two normal geodesics to C intersect, except elike line asymptotic to y passing through q ) Put
the trivial possibility that both issue from the same point p of C and that Zt = E ( ( t ) x C).
intersect precisely at that point or coincide.
%call that C is a smooth spacelike hypersurface. Hence there exist
smooth future pointing unit ~imelike normal field N along C Recalling also
14.4 THE PROOF OF THE LORENTZIAN SPLITTING THEOREM 557
a certain Lagrange tensor field dong any unit timelike geodesic c(t) in U1 asymptotic to y calculates the mean curvature W ( t ) of the leaf of this foliation
at c(t). The Raychaudhuri equation associated to this Lagrange tensor fieid
then translates Into the following evolution inequality [cf. Galioway and Horta
Assuming that N fails to be parallel along the geodesic c and recalling the
curvature assumption Ric(N, N) 2 0, arguments similar to those in Proposi-
tion 12.9 of Chapter 12 show that the mean curvature H(tf to Ct a t c ( t ) must
blow up in finite time. But this contradicts the fact established above that all
such timelike asymptotes c to y are focal point free to Co at c(O). Hence, the
normal vector field N = -Vb+ must be parallel in U l , and the level surfaces
Ct are all totally geodesic (cf. Lemma B.5 of Appendix B). Moreover, the par-
allelism of N also implies that Ul splits locally isometrically by Wu's proof
(1964, p. 299) of the Lorentzian de Rham Theorem. More exactly, let h de-
note the Riemannian metric induced on the spacelike hypersurfxe C = Go by
inclusion in ( M , 9). Then E is an isometry of (R x C, -dt2 $ h) and (Ul. g lU, and hence provides the desired isometric splitting in the neighborhood Ul o
the given timelike line.
We summarize what has been accomplished so far as
Proposition 14.39. Suppose ( M , g) is timelike geodesicdiy complete an
satisfies fic(v. v) 2 0 for every timelike v , Let y : R -4 (M, g) be any complet
timelike line in (hif, g) . Then there exists an open neighborhood U of y (W)
M sucb that
( I ) C = (bt = b- = 0) n U is a smooth embedded acausal hypersurfa
(M, !?I, (2) (U. glu) is isometric to (W x C, -dt2 @ h), where h is the metric ind
on C by inclusion in ( M , g), and
(3) for each q in C, the curve R i- U given by cq(t ) = exp(iA
ma-imai timelike line ~assinp throud a which is ~ v m ~ t o t i c
given timelike line y. - - -
Further, any timelike asymptote to
-y / [a, foo), for any a in R, issuing from any point q in U , coincides
up to parametrization w ~ t h one of these flow !ines c,(t> of Vb+. (Here
N denotes the future directed unlt clmellke normal to C . )
It is now necessary to show that the local splitting given in Proposition
14.39 may be extended to a global splitting. In Eschenburg (1988) this wa;.
carried out by defining tmelike lines ci. ca to be equzvalent if they are joined
by a chain of timelike lines = c l , p2,. . . , /3h = c2 such that each successive
pair /3,, fl,+l forms the boundary of an isometrically immersed flat strip 4 : ( R x [al. aa], -dt2@ds2) -+ ( M , g ) satisfying 4 ( W x {al)) = P3, F3(Rx {a2)) =
pJ+l , and Fj(R x is)) is a timelike line for each s in [al.a2]. Later. following
the publication of Galloway (1989a), Galloway ind~cated to us that an alternate
proof of the global splitting could be given by maximally enlarging the local
hypersurface C of Proposition 14.39 originally obtained in a neighborhood of
a given timelike line. It is this second line of reasoning that will be pursued
next.
It will be useful for this purpose to recall the followirlg elementary rela-
tionship between parallel translation and parallel field&. Suppose that U is a
(path) connected open set and N is a srnootl~ vector field which is globally
parallel on U. i.e., for any q E U and any v 6 T q M , we have V,N = 0. Fix
any point ql in U , and let vl = N ( q l ) E Tq,M. Then for any qr: in U ,
If c : 10.11 + U is a smooth curve with c(0) = 41 and c(1) = qz. (14.44) and v2 E T,,M is obtained by parallel translating zl along c from
t = O to t = 1, then the resulting vector vz is independent of choice of path c from qi to qz.
This basic fact is valid because the parallel field P along c with P(0) = v l
rrtust be given by P(t) = A' o c(l) for all t i11 [O, I]. whence va = P(1) = N(q2)
which depends on N and q2 but not on the particular clloice of smooth ~ u i v e c
m ql to 92. Thus if the field 1%' on U is globally given. no holonorny problems
e encountered in parallel translation along different curves in U .
We turn now to the problem of compatibly enlarging a local splitting of
rm given in Proposition 14.39. Thus let 7 3 IR -+ (M, g) denote a given
i timelike line, and let W be a convex normal neighborhood of po = ~ ~ ( 0 ) such
55% 14 THE SPLITTING PROBLEhl IN LORENTZIAN GEOMETRY
that the regularity properties for timelike asymptote construction are valid on
V/ and the Busemann functions b$ and b, are continuous on W. Let Wo
be a second convex open neighborhood of po with T o 2 W, which is used
to obtain a local isometric splitting E : (B x Co, -dt2 @ hO) -+ (Uo,g) with
Co 2 Wo and E(t,q) = exp(tNo(q)), where No = -Vb$ is globally parallel
on U = E(R x C o ) Let pl E edge(C0) be arbitrary. Take (q,) C Co with
limn,+, E(0, q,) = pi. By continuity, b$(pl) = limn,+, bof(E(0, q,)) = 0,
and similarly bi(pl) = 0. Recalling that Co is totally geodesic and W is
convex, there exist unit speed ho-geodesics c, . 10, a,] + Co with c,(a,) = q,.
Let w in Tpo (CO) be an accumulation vector of {cnl(0)) and put c(t) = exp(tw),
c : 10, a) -+ (Ca, ho ) Since cz(t) = (2t, c(t)). 0 < t < a, is a unit timelike
geodesic in (M, g) and ( M , g) is timelike geodesically complete, it follows that
cz(t) extends to t = a. Hence, the geodesic c : [O,a) + (20, he) extends to a
geodesic ci : 10, a] -+ (&I, g) with cl (a) = pl.
Let P be the unique parallel field along cl : [O, a] -+ (M,g) with P(0) =
yol(0). Put v = P(1) in T,,M, and let yt : R -+ ( M , g ) be given by yl(t) =
expPl (tv). If we take {s,) (0, a ) with s, -+ a and put u,(t) = exp(tP(s,))
14.39, each u, is a maximal timelike line asymptotic to 70. Thus 71 is also
maximal timellke line.
Apply Proposition 14.39 to yl to get a local splitting E : (B x El, -rkt2 $
parallel on Ul, b$(pl) = b;(pl) = 0. We wish to show that Nl(q) = &(q)
all points q of Uo n U1, or equivalently, bf(q) = bof(q) (and thus also b l (q) =
b;(q)) at all polnts q of Uo n UI.
Choose b with 0 < b < a so that q = cl(b) 6 C1 n Co, where cl . [0, a]
(M,g) denotes the g-geodesic in W from go to pl constructed above. Rec
14.4 THE PROOF OF THE LORENTZIAN SPLITTING THEOREM 559
curve u : i0, I] -+ Cl P Co from q = a(0) to m = a(1) Since Pil and No are
both parallel at all points of C1 n Co and Nl(q) = No(q), ~t foilows agaln from
(14.44) that Nl(m) = No(m). Hence. f i = iv, at all points of Cl fl Co.
Now given any n E Ul Ti Uo, use either splitting to find a smooth curve
p(t) = E(t ,m) from a point m in El n Co to n. Since Nl(m) = No(m) and
Nl (n) and No(n) are obtained from Nl (m), No(rn) by parallel translation
dong p, we must have Nl (n) = No (n).
Thus we have obtained Vb$ = vb$ on Ul TiCro. Since bf (q ) = b$(q) for any
q i11 C1 n Co, it follows that bf = bz on Ul n lie. Put C = Co U C1 furnished
with the Riemannian metric h induced bj inclusion in (M. y). Fbrther, put
and define the extended splitting E : (R x C, -dt2 63 h) -+ (Lro U U, , g) by
E( t , q) = e w ( t N(q)).
Kow we ~ndicate how the or~ginal splitting may be extended from the given
local splittmg W x Co -+ Uo .'in all directions." Select sufficiently many points
(PI, pz, . . . , pk) in edge(&) and associated convex neighborhoods Mil , . . . , Wk
such that after determining local hypersurfaces El, C2,. . . , Ck and tubular
neighborhoods Ul, U2,. . . , Uk, that if any pair Up n Uq # 0 (or Cp n Cq # @),
then C, n C, contains a point of the starting hypersurface Co. With this
proviso, it follows as above using (14.44) that the Busemann functions (and
their gradients) obtained from the local splitting8 E, : R x C, -+ U, agree on
all overlaps, so that we may put
and
c = ~ o u ~ l U " ' u ~ k
and define a parallel field on U by N(q) = iVj (q) for q in U3 to obtain a well-
defined isometric splitting E : R x C + (U,g lo-) via E( t , qj = exp(t N(q)).
We now present the more general inductive szep necessary to conclude that
e given local splitting E : (B x Co, -dt2 @ h) -+ (Uo, g) may be globalized.
560 14 THE SPLITTING PROBLEM IN LORENTZIAN GEOMETRY g Thus we suppose that an open, smooth, path connected spacelike hypersurface
C with Riemannian metric h ~nduced by lncluslon In (hf,g) is given, wlth a
timelike unit normal field N parallel at all points of C so that the mapping
given by E(t ,q) = expg(t N(q)) is a diffeomorphism onto its image, ns well as
being an isometry, and that the timelike geodesics yq(t) = E(t , q) are timelike
lines for all q in C. Inductively. suppose further that if N is extended to
U by setting N(yq(t)) = yql(t), then N is globally parallel on (U. g). Finally.
although this is not necessary to the argument, suppose that C contains a given
local hypersurface Cg formed by starting with an originally given complete
timelike line yo and that h7 / U t = -Vb:o. as in Proposition 14.39.
Suppose that edge(C) is nonempty. Take any p E edge(E). To complete the
argument. we will show that C can be extended consistently past p to form
an even larger spacelike hypersurface C1 satisfying the above conditions. This
then shows that if we want to carry out "Zorn's Lemma" type arguments, then
Eb/ may be maximally extended to a smooth spacelike hypersurface ( S , h ) w ~ t h
edge(S) = 0, and we may thus obtain the desired global splitt~ng.
Let W & U be a geodeslcally convex open set with p E and W contamed
Inside a larger geodesicallv convex open set. By our prevlous discuss~on, ~f
we select po in W sufficiently close to p and let be a geodesically convex
neighborhood of po with p In d(Wo), then applying the local splitting Propo-
sition 14.39 produces a hypersurface Co contained in C (? Wo with po 6 Co
odesic el . \0,1) -+ ( M , g) with q ( 0 ) = po, q ( l ) = p, and c1([0,1)) &
14.4 THE PROOF OF THE LORENTZIAK SPLITTING THEORElvl 561
i e., using the Busemann function associated to c(t), the parallel field iV may
be consistently extended to C U El.
The crucial remaining step in this procedure is to show that this process just
described is well defined. That is. suppose a different point go E W is selected,
the geodesic r2 . [0, 11 --+ (M,g) with c2(0) = q ~ . cz(1) = p, and c2([0, 1)) S C
is constructed. and ?u2 6 T,M is obtained by parallel translating N(qo) along
c2 from t = 0 to t = 1. It is necessary to establish that wl = wz in Tp&f.
Thus suppose that url # w2 in T,M. Let P, denote the parallel field along
c, : 10, l] -, (M, g) with Pl(0) = N(po) [respectively, Pz(O) = N(qo) ] . Recall
that in fact P,(t) is equally well determined by the value of N(c3(t)) for any
t with O < t < 1. Thus in view of the proof of the local splitting Proposition
14.39, especially the use of Bartnik (1988a). we will replace the given initial
points po, go by points p& on c1 and qh on c2 stifficiently close to p such that
after the associated local splittings
are obtained, centered about ?,b(t) and %b ( t ) respectively, we have p E C;nCb
Fkom our previous studies of the compatib~lity of local spllttings, we also have
~ b ; ' = 0b2f at all points of C; n C;. and part~cularly. PI(p&) = -Vb;(& and I
P~(q6) = -Vb2 (¶A). Thus. in fact, wl = Pl(p) = -Vb:jp) = -'C7bz(p) = wa.
as desired.
Now that the existence of a global isometric splitting ( M , g) = (EX S. -dt2@
h) has been established, the only remaining detail in the proof is to show that
the Riemannian factor (S, h) is forced to be corriplete by the assumed t~mellke
completeness of (M, g). But this is the same type of argument that has already
been used above. Suppose that (S, h ) fails to be Riemannian complete. Then
there exkts an h-unit speed Riemannian geodesic c : (a , b) --t (S; h) where
either n or b is finite, which is inextendible either $0 t = a or to t = tr or
possibly both. Define 7 (a, b) -+ (M,g) by -{(t) = (2t,c(t)), which is a
geodesic in (iM,g) by formula (3.17) with f = 1. Further, g(-yJ(t), y l ( t ) ) =
2 + h(el( t ) ,c l ( t ) ) = -1. Thus y is a timdike geodesic in (A4.g) which is
either past incomplete or future incomplete, contradicting the hypothesis of
timelixe geodesic completeness of (M, g). O
562 14 THE SPLITTING PROBLEM IN LORENTZIAK GEOMETRY
Remark 14.40. It is perhaps amusing to note after the fact that the two
versions of the Lorentzian Splitting Theorem are not so far apart.
(1) Note that under the conclusion of Theorem 14.37. (M,g) must be
strongly causal by corollary 3.56. Alternatively, we know from the
proof of Theorem 14.37 that a complete timelike line passes through
every point of (M, g), hence strong causality also follows from Propo-
sition 4.40.
(2) Since the Riemannian factor (S, h) in Theorem 14.37 turns out to be
complete, we know that (M,g) had also to be globally hyperbolic (as
well as geodesically complete) by Theorem 3.67, (1) =+ (2), (3).
(3) Similarly in Theorem 14.38, the global hyperbolicity of (M, g) ensures
after the splitting is obtained that ( M , g) had to be geodesically com-
plete, also by Theorem 3.67, (3) * (2).
(4) The use of Theorem 4.1 of Bartnik (1988a) enables certain analytic
arguments involving smooth super support functions and subsupport
functions to show that the continuous function b+ o c is a%ne, hence
differentiable, to be dispensed with in the proof presented here. An as-
pect of this other proof method is to observe that by estimatw a k ~ n to
(14.36), these support functions have arbitrarily small second deriva-
tives [cf. Eschenburg and Heintze (1984), Beem, Ehrlich, Markvorsen.
and Galloway (1985, Lemma 5.1), Eschenburg (1988, Lemma 5.2 and
Proposition 6.3)j.
A different application of the Busemann function methods discussed in this
chapter has been given in Andersson and Howard (1994) to obtain results more
closely allied to Harris's Maximal Diameter Theorem A.5 of Appendix A. In
this setting. for example, estimate (14.29) translates into
O(d,) > -(n - 1) tan(a/2 - d,).
Here are two of the results obtained in Andersson and Howard (1994, The
14.5 RTGIDITY OF GEODESIC INCOMPLETENESS 563
Theorem 14.41. Let (M", g ) be a globally hyperbolic space-time which
satisfies the curvature condition %c(P~, v) 2 (n - 1) for any unit tlmelike vector
v. Suppose that (M, g) contains a mlwirnal timelike geodesic segment y :
( - ~ / 2 , ~ / 2 ) -+ ( M , g). Then there exists a comple~e Riemaqnian manifold
(S, h) such that (M, g) is isometric to the warped product
( ( - ~ / 2 , ~ / 2 ) x S, -dt2 + cos2(t) h)) .
As a consequence of Theorem 14.41, Andersson and Howard are able to
obtain a related result to Theorem A.5 of Harris (1982b). The definition of
"globally hyperbolic of order q = 1" is given in dppendix A, Definition -4.3.
Theorem 14.42. Let (M" , g) be a space-time which is globally hyperbolic
cf order 1 and also satisfies the curvature condition Ric(v, v) 2 (n - 1) for any
unit timelike vector v.
(1) If ( M , g) contains a maximal timelike geodesic segment y : r-;, $1 -+
(M, g) of length x joining x to y , then the causal lnterval D = If (z) ri
I - (9) is isometric to the warped product of the hyperbolic model space
(3, h) of constant curvature -1 and dimension (n - 1) and the interval
(-5, z ) , with warped product metric -dt2 - cos2(t) h.
(2) If (M,g) contains a timelike geodesic y : (-oo, +co) -+ ( M , g) such
that each segment y 1 [t, t + T] is maximal, then ( M , g) is isometric to
the universal cover of anti-de Sitter space.
14.5 Rigidity of Geodesic Incompleteness
Now that the space-time splitting theorem has been obtained. we are ready
to return to the question of rigidity of timehke geodes~e incompleteness as
discussed in the introduction to this chapter. The following formulation of
this ques~ion was given in Bartnik (1988b). Bartnik defines a space-tlme to
be cosmological if it is globallj hyperbolic with a compact Cauchy surface and
satisfies the timelike convergence condition Ric(v, v) 2 0 for all timelike v.
564 14 THE SPLITTIXG PROBLEM IN LORENTZIAX GEObIETAY
Co~ecture 14.43 (Bartnik). Let (M. g ) be a cosmoiogical space-time.
Then
(I) (M, g) contains a constant mean curvature Cauchy surface; and
(2) (M, g) is either timelike geodesirally incomplete or else ( M g) splits
isometrically as a product (R x V, -dt2 @ h), where (V h) is a complete
Riemannian manifold.
In Geroch (1966) the following condition was introduced
(14.45) For some p E &f. the set A4 - [If (p) n I-(p)] is compact.
Bartnik (1988b) proves that if condition (14.45) is satisfied, then there is a
constant mean curvature Cauchy surface in (M,g) which passes through p.
Also Bartnik (1988b) notes that if condition (14.45) holds, then either (M,g)
is a metric product or else is timelike geodesically incomplete, extending earlier
work of Galloway (1984~) which required that this condition hold at all p on a
compact Cauchy surface. Bowever. (1) is false in general [cf. Bartnik (1988b)l.
In Beem, Ehrlich, Markvorsen, and Galloway (1985), a rigidity result was
stated which may be obtained as a consequence of the Lorentzian Splitting
Thcorcm along fairly elementary lines, without any causality assumption like
(14.45), but under the weaker sectional curvature hypothesis. -4 complete
proof was published in Ehrlich and Galloway (1990)
Theorem 14.44. Let (M, g) be a space-time with a compact Cauchy sur-
face and everywhere nonpositive timelike sectional curvatures K 5 0. Then
either (M, g) is timelike geodes~cdy incomplete or else (-!. g) splits as a metric
product (B x V. -dt2 $ h), where (V, h) is cornpact
Prooj. Suppose that ( M . g) is timelike geodesically complete. Since (M, g)
is assumed to have a compact Cauchy surface S, it is not difficult to see that
(M, g ) is causally disconnected with K = S. Hence by Theorem 8.13, (M, g)
contains a nonspacelike geodes~c line. If the line is shown to be timeilke, then
the desired splitting follo~vs from the Lorentzian Splitting Theorem. Nence, it
only remains to rule out the alternative that the line is null.
This may be accompiishd by using a coroliary to Harris's Toponogov The-
orem of Appendix A [cf. Ehrlich and Galloway (1990)).
14.5 RIGIDITY OF GEODESIC INCOMPLETENESS 565
Theorem 14.45. Let (A4.g) be a space-time with compact Cauchy sur-
face and with everywhere nonpositive t~melike sectiond curvatures. If ( M , g)
is fl~tiire tjmehke geodesicajly complete, then each future inextendible null
geodesic contains a null cut point.
Pmoj. Assume that there is a future inextendible null geodeslc q : 10, a) -+
( M , g ) such that there is no cut point to p = q(0) along 7. Let (tk} be
an increasing sequence of nonnegative real numbers converging to a. and set
4.4 = rj(tk). Hence d(p, q k ) = 0 for all k since q contains no null cut point to p.
Fix a compact Cauchy surface S containing p, and for each k let A/k be a future
directed timelike geodesic segment from some point r k of S to q k which realizes
d(S, qk). (Her~ce yk is also a rrlaximal geodesic segment.) The sequence irk) has an accumulation point r on S, and thus some subsequence (y,) of (y,)
converges to a future inextendible norispacelike limit curve y with y(0) = r
which by the usual arguments is a maximal geodesic. Also, for each fixed t the
segment y 1 [O, t 1 realizes the distance from S to y ( t ) . klareover, for some t > 0
we have d(S, y ( t ) ) > 0 since y cannot be imprisoned in S. Hence, the limit
geodesic must be timelike rather than null. Note also that y is future complete
by the assumption of future timelike geodesic completeness. Furthermore,
(14.46) r n If (p) = 0.
For suppose y meets I+(p). Then y, meets I i ( p ) for large m. Hence q, E
I f (p ) , and thus d(p, q,) > 0, in contrad~ction.
Now we use the sectional curvature hypothesls to obtain a contradiction
to (14.46). Because y is a timelike limit curve of (y,}. we may choose m
sufficiently large so that q , E I+(r). Lsing I'(q,) I+(p). we then obta~n
I+( r ) n If (pj f 0.
Thus we may fix x in If (r) fl It(p) and let 6 : 10, L] -i ( M , g) be a maximal
timelike geodesic segment from r to z. S~nce K < 0 and 7 is future complete,
eorem 3.1 of Harris (1982b, p. 313), a consequence of Harris's Toponogov
e is some t o with x = b(L) <_ ?(to), which yields
?(to) in contradiction to (14.46).
566 14 THE SPLITTING PROBLEM IN LORENTZPAK GEOMETRY
At the end of Section 14.3, a summary was given of how Galloway and Norta
(1995, pp. 18-20) extend the regularity results for the Busemann function as
given in this chapter for I-(y) f lI+(S) to I-(?) n [J"(S) U D-(S)] . With this
accomplished, Galloway and Worta (1995, Theorem 5.1) are able to improve
upon earlier results in Eschenburg and Galloway (1992, Theorem B).
Theorem 14.46. Let ( M , g) be a space-time which contains a compact
acausal spacelike hmersurface S and satisfies the timelike convergence condi-
tion Ric(v, v) 2 O for all timelike v. If (M, g) is timelike geodesically complete
and contains a future S-ray y and a past S-ray 77 such that I-(y) n 1 + ( ~ ) # 0. then (M, g) splits as in the rigidity conjecture.
An entirely unrelated investigation of rigidity of geodesic incompleteness is
pursued in Garcia-Rio and Kupeli (1995). Suppose (I\.I, 9) is a stably causal
spacetime with smooth time function f : M -t R. i.e., g(V(f) ,V(f) ) < 0.
Rescale the given metric g to produce a new metric g, for M for which
g, (Vc(f),Vc(f)) = -1 by setting g, = -g(V(f), V(f) ) . g. Now the integral
curves of VC(j) are unit speed timelike geodesics (cf. Lemma B.1). Hence,
tirnelike geodesic conipleteness of (M, g,) implies that the flow of f in (M, g,)
is complete, and the flow may be used to produce a metric g(t) on the level
surface f -l(t) from that of (f -l(0), g lf-i(,)). Thus the following result is
obtained.
Theorem 24.47 (Garcia-Rio and Kupeli). Let ( M , g ) be a stably
causal space-time with smooth time function f : M -+ R. Let
Then either
(1) (M, g,) is timelike geodesically incomplete, or
(2) (Ad, g) is conformally difleomorphic to a type of pasametrized warped
product (R x N, -dt2 @ g(t)), where N = f-'(0) and g(t), t E R. is a
one-parameter family of Riemannian metrics for N.
APPENDIX A
JACOB1 FIELDS AND TOPONOGOV'S
TMEOBM FOR LORENTZIAN MANIFOLDS?
One of the important consequences of the Rauch Comparison Theorem
in Riemannian geometry is the Toponogov Triangle Comparison Theorem
[cf. Cheeger and Ebin (1975, pp. 42-49)]. Let M be a complete Riemannian
manifold (metric here and elsewhere in this appendix denoted by (. : . )) with
sectional curvature of all 2-planes a in M satisfying K(cr) 2 H for some num-
ber H. Let (yl,y2,y3) be geodesics in M forming a triangle: 31(0) = yz(0),
yl(L1) = "/3(0), and y2(Lz) = -y3(L3), where L, = L(y,). Suppobe that yz
and 7 3 are minimal geodesics and, in the case that H = q2 (q > 0), suppose
that L, _< r /q for i = 1,2,3. Assume the geodesics 75 are paranietrized by
arc length, and define a3 = (yli(0),y2'(0)) and a 2 = {-yI'(Ll),-~31(0)). For
a triangle ( T l , T 2 , ~ 3 ) , possibly in another manifold. 5 2 and Z3 are defined
similarly.
(1) In the simply connected two-dimensional Riemannian manifold
of constant curvature H , there is a geodesic triangle (7,. T2, y3) with
L(T*) = L,, i = 1,2,3, and 5 2 5 12, E3 5 02. This triangle is determined up to congruence if H < 0, or if H > 0 and all L, < r lq .
(2) In %, let 'J1 and T2 be geodesics with Tl(0) = T2(0), L(7,) = L1.
L(Yz) = L2, and Eag = aag. Let T3 be a minimal geodesic between the
endpoints of and 'J2. Then L(%) >_ L(y3).
It is possible to develop an analogous theorem for Lorentzian geometry: the
program will be sketched here, although the proofs wl!l be omitted Details appear in Harris (1979, pp. 3-41) and Harris (1982a).
fgy Stever, 6. Harris, Department of Mathematics, St Lours Umversity, St Louxs, Mlssourl
568 APPENDIX A JACOBI FIELDS AND TOPOKOGOV'S THEOREM
The first step is to modify the Timelike Rauch Comparison Theorem I
(ci. Theorem 11.11) so that it applies to tirnelike geodesics. not without con-
jugate points, but without focal points: If N is a submanifold of a manifold
M , then a point q f M wi!l be said to be a focal poznt o j N from p (p 6 N ) if
q is the image of a critical point of exp in the normal bundle of 1V at p. For
v a vector in the tangent bundle TM. -W(v) will denote the submanifold of -14
which is the image under exp of a small enough neighborhood of the origin in
the perpendicular space of v such that exp is an embedding on it. Thus N ( v )
is an (n - 1)-dimensional submanifold orthogonal to u.
In the following statement of the second Rauch Theorem, and elsewhere in
this appendix, A A B will denote the 2-plane spanned by the vectors A and B.
Theorem A.1 (Timelike &uch II). Let t/, be a Jacob] field along a unit
speed timelike geodesic y, in a space-time M,, z = 1.2, with y, : 10, L] -+ hf,;
let T, = 4%'. Suppose that (VI, VI)O = {VZ, %)o, (VI,TI)O = (V2, T2)0, and
(VT,V,)(0) = 0. Further suppose that for any vectors X, a t y,(t)
K(Xi A Ti) 2 K(X2 A 7'2)
and that 7 2 has no f0ca-I point of N(yzl(0)) from yz(0). Then for all t Jn [0, L],
An important corollary of this theorem shows how curvature can affect the
lengths of timelike curves.
Corollary A.2. Let y, : :O. L] --+ M1, i = 1,2 , be two timelike (or two
null or two spacelike) geodesics, and let E, be parallel unit timelike vector
fields along 7, with (E1,Tl) = (Ez,T2) (Te = %I). Let f . [O,L] -. R be any
smooth red-valued function. Suppose that for all t in [O, L], exp,,(,)(f (t)E,(t))
is defined; call this c,(t). Suppose further that for all t in [O, L], the geodesic
11 : s w exp,2(t)(~Ez(t)) has no focal point of N(vl(0)) from ~ ( 0 ) for s < f ( t ) ,
Finally, suppose that for all tirnelike 2-planes o, in M,,
APPENDIX A JACOBI FIELDS AND TOPONOGOV'S THEOREM 569
Then, for all t in [0, L],
(cll, cI')~ > (02/, c2')t
Thus if el is a nonspaceljke curve, then c2 is also nonsprtcclikc, and
Thls corollary makes possible a triangle comparison theorem for "thin"
triangles. The model spaces used for comparison are the two-dimensional
de Sitter and antl-de Sltter spaces of constant curvature [cf. Hawking and
Ellis (1973, pp. 124-134). Wolf (1961, pp 114-118)j. A triangle of timelike
geodesics (7lry2,?3) IS "th~n" In this context ~f the following holds: For a
given H. let yl and be limelike geodesics In the simply connected two-
dimensional Lorentzian man~fold of constant curvature H (denoted by h I H )
with ~ ~ ( 0 ) = ~ ~ ( 0 ) . L(Tl) = L1, L(y2) = Lz, and 0 3 = as. Flrst, suppose
there is a timelike geodesic T3 between the endpoints of 7, and T2. Let E be
the parallel translate of TI1(0) along Tiiz For each t m [O, Lz], there 1s a smallest
positive number f ( t ) such that exp(f(t)E(t)) lies on Y3. Second, suppose that
for ail such t , the geodesic s c-t exp(sE(t)) has no focal point of N ( E ( ~ ) ) from
T2(t) for s 5 f (t). If these two suppositions hold. and if y3 is maximal and ail
t~rnelike planes 0 in M satisfy K(u) < H. then L(y3) > L(7,).
The problem is to start with a more ger~eral tirr~elike geodesic triangle,
slice it up into "thin" triangles, apply the result jusi, mentioned to each sllce,
and then put them back together. In the Riemannian theorern. ~onipleteness
is used to ensare that In any triang!~ (yl. ~ 2 . 1 ~ ) . mlnirnal geodesics can be
extended from y3(L3) to yl. slicing up the original trisngle. In the Lorentman
context, global hjrperbolicity succeeds just as well as long as H 2 0. For
N = -q2, however. a problem arises: Not even the model spaces blH arc
globally hyperbolic; indeed, by Proposition 11.8, no tinlelike geodesic of length
greater than ~ / q tan be maximal in a Lorentzian manifold whose timelike
planes a satisfy K(o) 5 -q2. A sol~ltinn to this problem lies In a new concept.
a sort of global hyperbo1ic:ty in the small: For x and y in a l,orenizim manifold
M , let C(x,y) denote the space of nonspacelike curves in A4 from T to y ,
modulo reparametrizat~on. wlth the compact-open topology.
570 APPENDIX A JAGOBI FIELDS AND TOPONOGOV'S THEOREM
Definition A.3. A Lorentzian manifold M is globably hyperbolzc of order q
(q > Of if M is strongly causal and for ail points x and y in M with sup(L(y)
y E C(s, y ) ) < x/q, this space C(x , y) is compact.
The Lorentzian analogue of Toponogov's Theorem can now be stated (all
geodesics parametrized with un:t speed).
Theorem A.4 (Lorentzizen Triangle Comparison Theorem). Let M
be a space-time whose timelike planes a satisfy K(a ) 5 W for some constant
H; M is to be globally hyperbolic or, in case W = -q2, globally hyperbolic of
order q. Let yZ, -y3) be a triangle of timelike geodesics with the future
directed side becween the pastmost and futuremost of the three endpoints,
the other future directed side froxn yz(0), and y3 the remaining future direcsed
side. Suppose that 7 2 m d y3 are maximal and, if H = -qa, that L, < r / q ,
i = 1,2,3. Then
(1) There is in MH a timelike geodes~c triangle (TI, 72, T3) with L(7,) = L,,
i = 1,2,3, and wjthB2 2 a2 andB3 2 C Y ~
(2) For timel~ke geodesics 7, end constructed in MH with T1 (0) = 72(0),
L(T1) = L1, L(T2) = L2, and 3 3 = C Y ~ , if there is a tinlelike geodesic
T3 between the endpoints of Tl and %, then L(y3) 5 L(y3).
The Toponogov Triangle Comparison Theorem can be used to show ho
a bound on sectional curvature can rigidly determine a complete Rieman '
manifold lf the l~mits of the curvature-imposed constraints are attained.
such result is the maximal diameter theorem [cf. Cheeger and Ebin (19
p. 110)]: If a complete Riernannian manifold Mn satisfies K(u) >_ q2 > 0
all planes a , and if the diameter of M attains the maximum thus allowable
x / q , then M is isometric to the sphere Sn of curvature q2.
Theorem A.4 can be used similarly.
Theorem A.5. Let Mn be a space-time which is globally hyperboli
order q and satisfies K(IT) 5 -q2 for dl timelike planes a. Suppose t
possesses a. complete tirnelike geodesic y which is maximal on a2I inter
length nlq. Then M 1s isometnc to the s~mply connected geodesically co
n-dimensional Lorentz~an manifold of constant curvature -q2.
APPENDIX A JACOB1 FIELDS AND TOPOKOGOV'S THEOREM 571
In addition to tirnelike geodesics, it is possible to study the effects of curva-
ture on Jacobi fields along null geodesics. Sectional curvature cannot be used
for this since it is undefined for null (singular) planes, bu t a similar concept,
introduced here, will work.
Definition A.6. If o is a null plane and IV is any nonzero clement of the
one-dimensional space of null vectors in o, then the null aectzonal curvature of
u wzth respect to N, KN (IT), is defined by
where A is any nonnufl vector in a.
This expression for null sectional curvature is independent of the vector A and depends in a quadratic fashion on the vector N. Thls curvature quan-
tity has certain interesting relations with ordinary sectional curvature. For instance, see Proposition 2.3 in Harris (1982a).
Proposit ion A.7. If at d single point in a Lorenczian manlfold the null
sectjonal curvature5 are all positive (respectivel~ negative), then timelike sec-
tional curvature at that point is unbounded below (respectively. above): and
if the null sectional curvatures all vanish, thexi the tirnelike and spacelike sec-
tional curvatures are all equal. Thus, a Lorentzian rllanifold of clirner~sion a t
least three has constant curvature iff it has null sectional curvature everywhere
Null sectional curvature can be used much like timelike sectional curvature
ith regard to Jacobi fields.
Proposit ion A.8. Let ,8 : [0, L] --, iMn be a null (&neIy parametrized)
geodesic with T = 13' If for all nonnull vectors V perpendicular to T, KT(V A
< 0, then p has no conjzlgate points. If for some q, KT(V A T) 2 q2 for all
V or, more generallj: if Elic(T. Tj 2 (n - z ) ~ ~ , chen L 2 n/q impiies 0 oes have a conjugate point.
Nuli sectional curvature also lends itself to a Rauch-type theorem
572 .4PPENDIX A JACOBI FIELDS AND TOPONOGOV'S THEORZM
Theorem A.9 (Rauch Comparison Theorem for Nu11 Geodesics).
Let p, . [O. L] - M,, z = 1,2, be nuli geodesics; T, = Pa' Let V, z = 1.2 he
perpendicular Jacobi fields along P,, not everywhere parallel to T,, (and thus
nowhere parallel to T,), with V(0) = 0 and &'I' Vl')o = (Vz'. V2')o
Suppose hat for any t in 10, L] and for any nonnuli vectors X, at &(tj.
perpendicular to T,
K T ~ (Xi A Ti) < K ~ ~ ( x 2 A T2)
and that p2 has no points conjugate to &(O). Then for all t in (0, L],
{vi,pi)t L {v~,Vz)t.
The necessity of using perpendicular vector fields makes this theory less
tractable than that for timelike geodesics.
Details for Proposition A.8 and Theorem A.9 can be found in Harris (1982d).
Flnally in Harris (1985), consideration is gmen to the case where the rlull
curvature function of all null planes associated to a null congruence IS point
function. It is shoun that this condlt~on locally characterizes Robertson-
Walker metrics and that ~f further completeness and causality assuinptions
are added, then a global characterlzation may be obtained
APPENDIX B
FROM THE JACOBI, TO A mCGATI. TO THE
RAYCNAUDHURI EQUATION: JAGOBI TENSOR
FIELDS AND THE EXPONEKTIAL MAP REVISITED
The use of Jacobi field techniques in the so-called "comparison theory" in
Riemannian geometry stems from the close relationship between Jacobi fields
and the differential of the exponential map. One very b a i c example is provided
by Proposition 10.16 in Section 10.1, in which the Jacobi field J along a unit
timelike geodesic c with J(0) = 0 and J1(0) = ui is gi\ien by
In this appendix we give a second illustration of the relationship between the
differential of the experiential map and Jacobi fields which has come to promi-
nence in the 1980's version of cornparison theory in Riemannian geometry. In
this comparison theory, the use of the Jacobi equation has been partly s u p
planted by the use of a Riccati equation for the ae~ond fundamental tensor of
a local foliation of a Riemannian manifold by hypersurfaceb arising a, the level
sets of a differentiable function with gradient of constant length [cf. Karcher
(1989)) Eschenburg (1975, 1987, 1989)j. A recent applicat~on of the Rccati
inequality method to submanifolds of semi-Riemannian manifolds is given in
Andersson and Howard (1994).
B.1 Generalities on Semi-Riemannian Manifolds
Let (M, g) be a semi-Eemannian manifold with semi-Remannlan metrlc g
of arbitrary signature ( 5 , ~ ) but which is nondegenerate. Let U be an open
subset of &I. This section is cor,cerned with the basic differential geometry of
the foliation induced on U by the level sets of a siriooth fidx~ction f : U -, R wlth
574 APPENDIX B JACOB1 TENSCR FIELDS
.e constant nowhere vanishing gradient vector field V f = grad f of everywhe*
length. Upon resealing, it may be supposed that
(B.1) gfgrad f , grad f ) = X
for some choice of X in (-1. 0, +I). Such functions have been termed solutions
to the eikonal equasion.
This formalism has been useful in a number of different geometric con-
texts. The first and perhaps most well-known example occurs in the case
where (M, g) is Riemannian, U is a deleted neighborhood of p which does not
intersect the cut locus of p, and f = d(p, .), where d denotes the Riemann-
ian distance function of (M,g). In this case: X = 1 in (B.l). The analogous
situation may be considered on a strongly causal space-time where U could
be taken to be a subset of It(p) which does not meet the future timelike cut
locus of p, and f = d(p, .), where d this time denotes the Lorentzian distance
function of (M,g). In this case, X = -1. The case X = -1 also arises (a pos-
teriori) in the course of the proof of the Lorentzian splitting theorem, as in
Beem, Ehrlich, hlarkvorsen, and Galloway (1985, p. 391, in which U could
be taken to be a tubular neighborhood of a complete timelike geodesic line
c : R -+ (M.9) in the globally hyperbolic space-time ( M , g ) , and f = b+
(or f = b-) 1s the Busemann function associated to the future timelike ray
(respectively, the past timelike ray) determined by the given timelike line.
The third case, X = 0, arises in the context of semi-time functions for space-
times, i s . , g(grad f , grad f ) < 0 IS required for a semi-time function The case
that grad f is identically null 1s encountered for the spare-time M = R x S1
with Lorentzian metric ds2 = d @ d t and f ( t , @ ) = -t. A second larger class
of examples is furnished by the gravitational plane wave space-times, Here
the global coordinate function f ( 2 . y,u. v) = u hm gradient grad f = d/Dv,
which is a global null parallel field. The geometric properties of this so-called
"quasi-time fi~nction" have been systematically studied for these space-times
in Ehrlich and Emch (1992a,b, 1993) [cf. Chapter 131.
The following basic remit explains the significance of constant gradient
length in (B.1).
B.l GENEIRALITIES ON SEMI-RIEMANNIAN MANIFOLDS 575
Lemma B.1. Let X = -1 (respectively, -1, 0). Suppose f : U - R is a
smooth function with g(V f. Vf) = A. Let c : [a, b ] -t U be an integral curve
of grad f . Then c is a tirnelike (respectively. spacelike, riullj geodesic.
Proof. Fix any p in U and 2. in T,M. Denoting the Hessian D ( D j ) by ~ f ,
we have
Since v was arbitrary and the semi-Riemannian metric ~vas assumed to be
nondegenerate, we have
on U . Thus the integral curves of grad f are geodesics. O
In the case that (M. g) is Lorentzian and X = -1, or in the case that (,2.f. g)
is Riemannian and X = +1, it is standard that (B.l) has the even stronger
implication that any such integral curve of grad f is a maximal geodesic in
the space-time (U, i , ] ~ ) (respectively, a minimal geodesic in the Riemannian
manifold (U,glu)) by virtue of the reverse (or wrong way) Cauchy Schwarz
inequality for timelike vectors [cf. Sachs and Wu, (1977a)J (respectively, the
Cauchy-Schxvarz inequality for tangent vectors).
We present here the argument for the space-time case, under the assumption
that the vector field grad f is future directed. Thus let y . fa b] + U be any
future directed nonspacelike curve in U with y(n) = r( f 1 ) and y (6) = c(tz) ,
where c is a maximal integral curve of grad f lying in U . Then we first have
that 4.f 07) , g(grad f , 7') = -$---
since both vectors are future directed and grad f is timelike. Secondly. by the
wrong-way Cauchy-Schwarz inequality in the case that yi ( t ) is timelikc, and
trivially if y l ( t ) is null, we have
576
so that
APPENDIX B JACOB1 TENSOR FIELDS
as required. The argument for the Lorentzian case with grad f null is deferred
until after Lemma B.3.
Let us now record an elementary identity in the context of a differentiable
map f : U -+ B for U an open subset of the (nondegenerate) semi-ftiernannian
manifold (M, g) that will be useful in the sequel:
for any p in U and v in TpM. This identity follows immediately from the
The identity (B.3) then has the following consequence.
Lemma B.2. For X in (-1,0, fl), let f : U -+ IR have constant gradie
length X as in (B.1). Then im(f,p) = Tj(@ for each p in U. Hence, f
a submersion and dim( f -l ( T ) ) = dim(M) - 1 for each r in f (U) in a11 t
cases simultaneously.
Proof. This is an immediate consequence of the assumed nondegeneracy
the given semi-Riemannian metric and (B.3). For tl
- he only way that f,, (
B.l GENERALITIES ON SEMI-RIEMANNIAN MANIFOLDS 577
fail to be surjective is that Imf fa,) = (01, since d i ~ n ( T ~ ( ~ ) R ) = 1. But In this
case, we then have from (B.3) that g(v, grad f (p)) = 0 for all v in T,M. Bus
this then implies that grad f(p) = 0, in contradiction to the assumption that
grad f is nowhere vanishing. O
The following is the translation to the semi-Riemannian context of a basic
result in elementary differential topology.
Lemma B.3. Let c : [0, a) - - U be an integral curve of grad f with c(0) in
f - ' ( r ) Then
(1) I f X = 0, then c(t) E f -l(r) for all t ;
(2) I f X = +l, then e(t) E f - l ( r + t ) for all t , and
(3) I f X = -1, then c(t) E f -l(r - t ) for all t .
Proof. Since c(t) is an integral curve of grad f , wc have from identity (B.3)
that
( f 0 c)'(t) = f*c'(t) = g(c1(t), grad f (c ( t ) ) )
= !?( Vf (clt)) , Vf (c(t)j 1 = A.
Hence if X = 0, then (f 0 c)'(t) = 0 for ali t and (1) follows. The other cases
are similar. (if X = -1, then (f 0 c)'(t) = -1 so that (f o c](t) = T - t.)
Thus in the case of a null gradient, instead of mapplng a level surface to a
different level surface, the gradient flow o f f maps each level surface into itself.
This phenomenon is encountered for the gravitational plane wave space-times
where f (3. Z,U,V) = 2 is the "natural" qu~ i - t ime function for this class of
space-times and has everywhere nonvanishing but null gradient (cf Chaptcr
13).
Also as mentioned above, we may derlve the following consequence from
Lemma B.3. Let (M,g) be Lorentzian, and let c . ( a , b) --, U be an :ntegral
curve of f : U -+ IW with gradf = 0. Then c is globally maximal with respect
to arc length among future directed nonspacelike curves lying in U For by
considering -grad f if necessary, we may assume that c is future directed. If c
fails to be maxima1 in the above sense. then there exist s, t with a < s < t < b
518 APPENDIX B JACOB1 TENSOR FIELDS
and a future dirccted timellke curve 5 : [O, 11 -4 U with ~ ( 0 ) = C ( B ) and
a(1) = c(t ) (cf. Section 9.2). But then ( f o a)'(u) = (V f ( o (u ) ) , o l (u ) ) > 0 ,
so that f (c(s)) = f (n(0)) > f (a(1)) = f (c(t f) , in contradiction to part (1) of
Lemma B.3.
Related to Lemmas B 2 and B.3, and also a consequence of equat~on (3.31,
is the following description of the tangent spaces and normal vertor field to
the local foliation ( f -"u) : :L E W) induced on U by the level submanifolds of
f .
Lemma B.4. For any X in (-1: 0, i-11, we have simultaneousiy
(1) The tangent space at any p in f -' (r) is given by
(B.4) T ~ ( f ( r ) ) = ker( f e p ) = (2. E TpM : g(v, grad f (P)) = 0) ;
(2) N = grad f is a normal field of constant length for each level subman-
ifold f - ' (r); and
(3) For any v in T p ( , f - l ( r ) ) we have D, grad f E T p ( f ( T ) ) .
Proof. (3) In view of (B.3) it suffices to note that (as before)
In the case that (M,g) is Lorentzian and grad f is timelike (respectively.
spacelike), the induced metric on the level submanifolds 1s nondegenerate and
the submanifold is thus spacelike (respectively, timelike) in the sense of Definl-
tion 3.47. In the case that grad f is null, the induced metric on the level sub-
manifold 1s degenerate in view of (B.4) slnce grad f (p) is both in Tp( f - ' ( f ( p ) ) )
and orthogonal to all vectors in Tp( f - ' ( f Hence, in the case that grad f
is null, the level submanifolds are not "semi-Riemannian submanifolds" i11 the
sense of OWeill (1983), so that some care needs to be taken in considering the
concepts of the second fundamental form and of a totally geodesic submanifold.
To be consistent with Section 3.5 in notation, given grad f on U we define
an operator L 011 Tp( f - ' ( ~ ) ) by
B.l GENERALITIES ON SEMI-RIEMANNIAN MANIFOLDS 579
In the case that A = +1 or X = -1 and ( M , g ) is Lorentzian, since the level
submanifolds inherit nondegenerate induced metrics, i t is well known that
is the shape operator whose %an~shing determ~nes whether the level submani-
fold is totally geodesic. In the case that A = 0 (or ( M , g) is not Lorentzian),
it is a consequence of part (3) of Lemma B.4 rather than familiar abstract
nonsense that (B.6) is still \slid, so we may contlnue to regard L as the shape
operator. This ib further onf firmed by the following lemma which is standard
when the induced metric is nondegenerace.
Lemma B.5. Let (M. g) be an arbitrary (nondegenerate) semi-R~eman-
njan manifold, and let X E (-1.0, $1). Suppose f : U -+ R satisfies the
further condition
03.7) Ct, grad f = 0
for all v in T,M and for all p IR U. Then every Jet-el subrnanifold f - ' (r)
in U is totally geodesic in the mowing sense: Let v E T,( f - ' ( r ) ) , and let
c . (a. b) -, U be (the restriction of) the geodesic In ( M , g ) with cl(0) = V.
Then c((u, b)) is contained in ~+-'((r).
Proof. Consider the function F : (a, 6) --+ R given h>r ~ ( t ) = f o c ( t ) .
Then we have F(0) = r , and we need to show that F1(t ) = O for all t to
obtain the desired result. Now F1( t ) = ( f o c ) ' ( t ) = g(V f (c j t ) ) , cJ( t ) ) and thus
F1(Q) = g ( V f ( p ) , v ) = O by (B.4). Further,
since c is a geodesic and (3.7) is assumed. Thus F1( t ) = FJ(0) = 0 m r e
quired.
580 APPENDIX B JACOBI TEh'SOR FIELDS
B.2 Jacobi Tensors and t h e Normal Exponential Map in the
Timelike Cease
In this section. we will assume that ( M , g ) is Lorentzian and that U is equipped with a smooth function f : U -+ R with everywhere unit t~melike
gradient. i.e., g(grad f , grad f ) = -1. Since grad f is normal to the level
surfaces of f, we will denote N = g a d f . Also, since we are only interested
in local calculations in this seetlon, we will not bother to carefully specify
domains of geodesics or of maps In the t or s-parameters.
Fix r in R with r In f ( U ) . With the above proviso in mind, recall that it
is traditional to study the geometry of the submanifold f-'(r) by considering
the s ~ c a l l e d normal exponential map E ( . , t ) on f -l(.r), which may be defined
Since the integral curves of N = grad f are geodesics, it follows that in this
particular geometric situation (B.8) is equivalent to followrng the gradient flow
of f starting at points of f -'(T), and thus Lemma B.3 implies that E( . $ t ) is
a map
(B.9) E ( - . t ) : f -"r) -+ ff-'(r - t).
The normal exponential map may be placed in a variational context with
notation ;I? in Chapter 10 for the purpose of studying Jacobi fields along
timelike geodesics initially perpendicular to f -l(r) as in Karcher (1989) for
Riemannian manifolds.
Fix p in fe l ( r ) , and let c : [O,a) -+ M denote the integral curve of N with
c(O) = p; by the above comments, c(t) = exp(tN(p)). Now let v E T,(f-'(r))
be glven, and choose any smooth curve b : (-E, E ) -+ f -'(r) with b(0) = p and
bt(0) = v for purposes of computation. Then define a variation a(t , s ) of c by
(B. 10) 3) = E(b(s), t ) = ex~b(s)(tN(b(s)))
so that
~ ( t , . ) : f -l(r) 4 f -'(r - t )
B.2 THE NORMAL EXPONENTIAL MAP 58 1
and
a, 8/& = grad f o a.
Since s H a ( . , s ) is a variation of c through geodesics, the variation vector
field
is a Jacobi field. By using the chain rule or by introducing some fancy notation,
one may remite the third term of equation (B.11) in a form closer to that of
Proposition 10.16 as recalled in the introduction to this Appendix. Explicitly.
if we let
i : ~ x I R i f-'(r) xR
be the map given by d(s,t) = (b(s),t), then a = E o i and i, =
(b' (0); 0) = (v, 0). Hence, we have
for all t in [O, a). Hence, the set of Jacobi fields J ( t ) as v varies over all vectors
tangent to f-'(r) at p is identified with the differential E, of the normal
exponential map E by means of equation (B.12).
Lemma B.6. The variation vector field J ( t ) of the variation a(t . s) given
by (B. 10) satisfies the differential equation
(B. 13) J I M = -L t (J ( t ) )
for all t in [O,a) with initid condi~ions J ( 0 ) = 1: and J'(0) = -L(v), where L
is the second fundarr~ental tensor for the level surfaces { f - ' (T - t ) ) defined as
in (B.5) by Lt(w) = -D,N.
Proof. Using (B. 11). one calculates
J'it) = D~iat (a* g) I(,,,, = (a*&) / ( t , O )
= Dalaa grad f o a
= Dn.(~/ds)(t,o) grad f = Dq9N = -L,(J(t)) . n
582 APPENDIX B JACOB1 TENSOR FIELDS
Thus in our particular geometric situation, the usual submanifold initial
condition for Jacobi fields, Ji(0) = -L(v) = -L(J(O)) , is propagated for all t
along the level submanifolds f - l ( r - t ) and the transversal timelike geodesic c.
Further, we may reinterpret equation (B.13) in the language of Jacobi tensor
fields as utilized in Section 10.3 and Sections 12.2-12.3. Let
be the Jacobi tensor along c : [0, a) + M satisfying the initial conditions
(B.15) A ( O ) = E and A1(0 )=-L .
Then as me remarked In Lemma 12.21, A is a Lagrange tensor field which
intuitively represents the vector space of Jacobi fields J along c which satisfy
the initial condition J1(0) = -L(J(O)) . In view of equation (B.13), we have
in the present context the further identification for the associated tensor field
B = AIA-I occurring in the Raychaudhuri equation that
for all t in [O, a) for which A( t ) is non-singular, i.e., which are not focal points
top. Also, one may interpret equation (B.12) as identifying the differential E,
of the normal exponential map with the Jacobi tensor A. Equation (B 16) then
has the implication that the Rqchaudhuri equation for A (or equivalently, E,)
is related to the trace of L and its derivative L', hence to the mean curvature
of the ievel surfaces ( f - ' ( r - t ) ) (cf. equation (12.2) in Section 12.1).
B.3 The Rieeati Equdion for L
There is a venerable technique in the disconjugacy theory of ordinary dif-
ferential equations in which a Jacobi equation such as u" + q(t)u = 0 is linked
to an associated Riccatr equation r' + r2 + q( t ) = 0 by the change of variables
r = ul/u [cf. Bartman (1964)]. The n x n matrlx version of this theory involves
forming B = AIA-' exactly as is done In general relativity in passlng from
the Jacobi equation A" + RA = 0 to the associated hychaudh>~ri equation
(cf. Sectioris 10.3 and 12.2-12.3).
B.3 THE RICCATi EQUATION FOR L 583
In this section, we shall follow the route suggested in Karcher (1989) and
obtain an intermediate Rtccatr equatlon satisfied by the second fundamental
forrn operator in the geometric context under consideration. Thus. let (M, g)
be a Lorentzian manifold and f : U -+ R be a smooth function with everywhere
timelih gradient of constant length -1, as in Sectron B.2. Let p E f - ' ( r ) be
fixed, and let c : [O, a ) --, U denote the integral curve o f f starting at p. Choose
any v E Tp(f - ' ( T ) ) , and let J = J, denote the Jacobi field along c with initial
conditions J(O) = v and J1(0) = -L(v) . We established in Section B.2 the
identity
(B.17) J1 ( t ) = - L o J ( t )
for all t in [0, a). Recalling here that X1(S) = DaIatX for vector fields and
tensors along the geodesic c, we have
or in more compact notation,
Thus differentiating (B.17) covariantly with respect to t yields
J" = - (LO J)' = -L1 (J ) - L(J1 )
= -L'(J) - L( -L(J ) )
again using (B.17). Combining this last result with the Jacobi equation J" + R(J, c1)c' = 0 then yields the desired Riccati type equation for J,:
(£3.13) L1(J,) - ( L o L)(J,) = R(J,. d)cl for a11 t in [O. a )
if we denote the composition L o L by L? then prior to the first focal point
to t = 0 along c to the submanifold f - ' ( r ) , we h ~ v e from (B 18) the operator
equatlon
(B. 19) L' - L~ = R( - , c')cl.
584 APPENDIX B JACOB1 TEXSOR FIELDS
Recall from equation (B.16) that B = A'A-"may be identified with -L prior
to the &st focal point to p along c, and recall also the earlier notation of
Definition 12.2 of Section 12.1 of 6 = tr(B). We rnay then rewrite equation
(B.19) as
(B. 20) B' = -B o B - R(,, ct)c'
in exact accord with equation (12.1) of Section 12.1. Recalling further that
tr(I3') = [tr(B)I1, we may obtain either from (B.19) or (B.20) the equation
(B.21) 8' = -tr(B o B) - Ric(cl~ c') = -tr(L o L) - Ric(cl, c')
which corresponds (with change in sign conventions) to equation (1.5.5) in
Karcher (1989): d/ds[trace (S ) ] = - t race(R~) - trace(S2), which is as far
as Karcher proceeds in decomposing this particular quation. In general rel-
ativity, the Raychaudhuri equation for 6' is derived from (B.21) by further
decomposing B in terms of the expansion, shear, and vorticity tensors and
calculating B2 in terms of this decomposition (cf. the derivation of equation
(12.2) in Section 12.1).
Note also that equation (B.21) occurs in another related form in connection
with the "Boehner trick" type identities derived by manipulations of the curva-
ture tensor and by utilizing the fact that the normal field used to calculate the
second fundamental tensor is a gradient, i.e., N = grad f. In this formalism,
taking a parallel orthonormal basis {El, E2,. . . , En) along c with En = c', one
is then led to the following version of (B.21) (U denotes the d2Alembertian of
f 1:
and hence the inequality
1 (B.23) -Ric(cl. c') b (m(f) 0 c)' + n_i(I(f) 0 el2,
which plays a role in the proof of the so-called splitting theorems in both
Riemannian and Lorentzian geometry. In Beern, Ehrlich, Markvorsen, and
B.3 THE XCCAT! EQUATION FOR L 585
Gdloway (1985, Lemma 4.3 and Corollary 4 4 , for instance. the untraced
version of equation (B.23) corresponding to equation (B.20) is employed.
Finally, let us conclude this section by deriv~ng as in Karcher (1989) a
we!l-known second link between E, and the mean curv~ture tr(L(t)) in the
present context by using equations (£3.12) and (B.16). Put a( t ) = det(E,),
or more formally, a(t) = det(A(t)). Tnen using the identity tr(AIA-') =
(detA)-"det A)' of Section 12.1, Definition 12.2, we obtain
Thus the expansion tensor B(A(t)) is, up to sign, precisely the mean curvature
at c ( t ) of the hypersurface in the family of f-level surfaces passing through the
point c(t), Karcher's viewpoint is that the Jacobi equation should be regarded
as giving geometric control by considering the Jacobi equation as equivalent
to the so-called Riccati inequality (B.23) and to equation (B.24). Of course,
equation (B.24) also demonstrates rather explicitly the well-known fact that
in the hychaudhuri framework, the expansion tensor @(A) is the quantity
that measures the infinitesimal change in hypersurface volume; indeed, in Es-
chenburg (1975, p. 32) the tensor 6 is even called the "volumen-expansion''
tensor.
In the context of equation (B.241, the following elementary calculation has
been noted. Suppose that a( t ) and b(t) are differentiable functions which
satisfy the ordinary differential equations
and
respectively, Then
Hence if a(t) and b(t) are positive valued and f 2 g , then a/b is monotonic
nondecreasing. This remark provides, after integrating along radial geodesics
586 APPEXDIX B JACOB1 TENSOR FIELDS
emanating from p, a proof of the Bjshop-Gromov Volume comparison The-
orem for Riemannian manifolds, choosing $ = d(p, . ) and taking g to be the
correspo~lding distance function on the appropriately chosen model space of
constant curvature [cf. Karcher (2989, pp. 186-I87)j.
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manifold, Archiv. der Math. 32, 92-96. Woodhouse, N. M. J. (1973), The diferentzable and cawaE stmctures of space-
time, J. Math. Phys. 14, 495-501. Woodhouse, N. M. J. (1976), An application of Morse theory to space-time
geometry, Commun. Math. Phys. 46, 135-152. Wu, H. (1964), On the de Rham decompositzon theorem, Illinois J. Math. 8,
291-311. Yau, S. T. (19821, Problem Sectzon, in Ann. of Math. Studies (S. T. Yau, ed.),
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490-493. Zecman, E. C. (1967), The topology of Minkowskz space, Topology 6, 161-170.
LIST OF SYMBOLS
BALLS
B+(P, €1, B- (PI 6) inner balls: future and past. 142, 143
O+(PIE), O-(P,E) outer balls: future and past, 144
BOUNDARIES
aa -44 dbM BCM d i f , i- io
abstract boundary, 198, 232-238 Schmidt boundary, 214 causal boundary, 215-218 topological boundary, 495 timelike infinity: future and past, 178 spacelike infinity, 178
BUSEMANN FUNCTIONS
b Busemann function, 512 b+, b- Busemann functions: future and past, 550 b ~ , bp,, various Busernann type functions, 512, 520
CAUSALITY
D+(s), D - ( s ) Cauchy development: future and past, 287 E + ( s ) , E- (S) horismos: future and past, 287
H + ( s ) , f f - (S) Cauchy horizon: future and past, 287
I+ ( P I , I - (P) Chronological future and past, 5. 6, 54
J+ { P I , J - (P) Causal future and past, 5, 6, 54
P K q p in chronological past of q, 54 P Q ~ p in causal past of q. 54
CUT LOCUS
rz", Iff
AS V
[X, Yl grad f = Vf
DIAMETER
GROUPS
INDEX
LIST O F SYMBOLS
nonspacelike cut locus, 311-322
nompacelike cut locus: future and past. 311-322 null cut locus: future and past. 296-318 timelike cut locus: future and past. 296-305 cut locus in T,M, 302
Christoffel symbols, 18 Hessian of f, 24 Laplacian of f , 24, 120
(Levi-Civita) connection, 15, 22 Lie bracket. 18 gradient of f , 24. 307
timelike diameter, 399. 401, 402
Riemannian distance, 2-4 Lorentzian distance, 8, 137
fundamental group, 399, 419 isometry group, 188 isotropy group at p, 186 left and right translation, 190
index of vector fields X, Y, 325 quotient index, 326 index for hypersurface, 458 extended index, 326, 342, 395 index, 326, 342, 395
LIST OF SYMBOLS
INNER PRODUCTS
g(V, W ) = (V, W ) inner product on tangent space, 327
(w , Xr) = u(X) on one forms and vector fields. 20 << , >> inner product on T,(T,:W). 338
J Jacobi field, 31
Jt (4 Jacobi fields vanishing at a and t , 326
Jt (P) quotient Jacobi tensors vanishing at a and t . 326
A @ ) , a t ) Jacobi tensors, 426, 431 B (= A'A-I), fS used with Jacobi tensors, 428, 431
LENGTH
L(Y length of curve y. 8, 137
MANIFOLDS AND SPACE-TIMES
MAPS
ring of smooth, real-valued functions on M, 16 class of srnooth vector fields on hl, 16 Euler characteristic, 50 warped product, 95
semi-Euclidean space. 181 (universal) anti-de Sltter. 183, 199, 306 hlinkowski, 25, 116. 177 de Sitter, 183, 286 real projective 3-space, 189
exponential map, 53 canonical isomorphism, 337
METRICS
c ( M . 9 ) set of all metrics conformal t o g. 6, 142 Con(iM) conformal equivalence classes, 239
i sww) quotient metric. 370-398
g1 < g2
partial order on metncs, 64
91 < 92 partial order on metrics, 64 4 Lor(M) i
'iet of all Lorentzian rnetrics on M , 50, 63, 89
620 LIST OF SYMBOLS LIST OF SYMBOLS
SECOND FUNDAMENTAL FORMS
second fundamental form operator, 93, 165 second fundamental form, 92 in direction of n. 93
TANGENT
T ~ N normal space to N, 444
TM tangent bundle of .nA, 16
T,M tangent space to M a t point p, 16
T-lM unit observer bundle, 298-304
TENSORS AND CURVATURES
sectional curvature, 29-32 curvature, 19, 20 Riemann-Christoffel, 22 scalar curvature, 23 Ricci curvature, 23, 30 energy-momentum, 44-48 torsion, 13 Wronskian, 385 expansion, 428,431 shear, 429, 431 vorticity, 428, 431
TOPOLOGIES
C0 topology on curves, 49. 72, 79
CO topology on metrics, 63, 89, 239, 247
C' topology on metrics, 63, 247
VECTOR FIELDS, CURVES, AND BUNDLES
C ( ~ + 7 ) piecewise smooth timelike curves from Conn(P0, u) in P(u) and connected to Po by
a geodesic, 485 p to g, 354-365
G(P) quotient bundle, 326, 368 space orthogonal to c'(t), 327
orthogonal to /?'(t). 368 null conjugate locus, 489 null tail. 433 null hyperplane in R4 481
piecewise smooth and orthogonal to c, 327, 369 piecewise smooth, orthogonal to c, vanishing at endpoints, 327 piecewise smooth and orthogonal to a, 366 piecewise smooth, orthogonal to P, vanishing at endpoints, 367 change in X'. 329 path space from p to q, 326
a-boundary (abstract boundary 3,). 198, 232. 233-238
Achronal. 57, 142, ~dentity. 538
Adapted coordmates. 251 normal neighborhoods, 251
Adm~ssl ble chain, 252, 253-255 deformation, 389, 390 measure. 67 variation, 376. 388
Affine f~mctlon, 332. 333 parameter, 12, 17. 112 parameter, generalized, 208
Alexandroff topology. 7, 8, 39-60 Almost maxrrnal curve. 146,272-275 Angular momentum for Kerr space-
time, 181 Anti-de Sitter space-tlrne, 183, 199.
306, 569 Arc length
functional, 356 Lorentzim, 135 upper semicontinuity, 83.273, 278,
51 1 Arzela's theorem, 75, 76 Astigmatic conjugate. 485, 486-4'39
~ s y m p t o t e. 518
b. a. complete (bounded aceelerati 197, 207
b-bcundary (bnndle boundary at), b-complete (bundle c~rnplet~e), 15
208 Basis
orthonormal. 21, 30, 404 natural, 1 G pseudo-orthanormal, 36, 460
Big bang cosmological model, 10 174, 185, 190, 477
see also Friedmann cosmologic, models
see also Robertson-Walker spac t!me
Bi-invariant Lorentziari merrlr, 19( 191-195
Birkhoff's Theorem, 132 Black hole. 480
Schwarzschild. 173, 179-182 Kerr, 174, 179-181
Bonnet-Myers Theorem. 399-400,4 Bore1 Measure, 67 Boundary pomnts
a-boundary (d,), 198, 233-238 approachable, 234 b-boundary ( d b ) . 214 causal boundary (8,). 21 5-218
425, 474 C"-rep~llar 219
624 INDEX
from enveloped manifold: 230 quafi-regular, 225 re,@%, 225 smooth, 472478
Bounded wceleration, 197, 207 Bounded parameter property, 233 Boyer-Lindquist coordinates, 181 Busemann functions, 503-566
Lorentzian, 512, 550 Riemannian, 507 see also horoball see also splitting theorems
C(M,S), 6, 142 c-topology
on c ~ w e s , 49,72, 79,81-83,272, 273, 278,288, 297
on Lor(M), 50, 63, 89, 239 Q-stable, 242 Cr-topology, 63, 247 Canonical
isomorphism (r,) , 337 variation, 331, 388
Cauchy complete, 4. 209, 425 development (D+, D-), 287,466 horizon (Nf, H-) , 287 partial surface, 539 reverse Cauchy-Schwarz, 575 surface, 65, 362 time function. 65 timelike complete, 211 see also horismos
Causal boundary (a,), 215-218,425,472-
478 future ( J f ) , 5, 6 past (J-), 5, 6 relation (I), 54 space-time, 7 stably, 63, 89 vector (= nompaeelike), 24 see also globally hyperbolic
Causally continuous, 59-60, 158-160,479 convex, 59, disconnected, 272,228-293 related, 54 simple, 479
Chistoffel symbols (r,k,), 18, 248 Chronologicd
future ( I f ) , 5, 6, 54, 55 past (I-), 5, 6, 54, 55 relation (a), 54 space-time, 7
Closed geodesics, 145-146 trapped surface, 148-151, 463
Cluster curve, 73 Compact
space-time, 58, 418 Complete
b, 197 b. a., 197 geodesic, 12, 17, 202 geodesicdly, 4, 425 nonspacelike, 12 m111, 12 pregeodesic. 1 7 Riemannian manifold, 4 spacelike, 12 timelike, 12 timelike Cauchy, 21 1 see also Hopf-Know Theorem
Con(M), 239 Cone
null, 15, 25-29 Conformal
changes and ndl cnt point,s, 306 changes and null geodesics. 308 factor, 91 global transformations, 28, 307 metrics, 6. 28
Conformally stable property of Lor(M), 249
Conjugacy index, 485
INDEX 625
Conjugate points, 295, 298, 302-322, null sect~ond cllrvature. 571 413 Riemann curvature (R or RiJkm),
definition of, 314, 328, 374 19, 20, 30 locus of first need not be closed Kernam-CIzristoEel (RJk,), 22,
in Riemannian, 315 30 Conn(P0, u), 485, 489-496
Kcci (Ric or K J ) , 23, 30, 45 Connection (V), 16 coeficients (I':), 18, 22 R(X,Y)Z, 19
scalar (R or r ) , 23 curvature of ( R ( X , Y)Z), 19, 22 mduced on submanifold (VO), 92 sectional(K(E,p)), 29-32, 399.
Levi-Civita, 15, 22 403
symmetric, 19 singularity, 225 torsion free, 19. 22 tensor, 20
torsion of ( T ( X , Y)), 19 Curve Conservation laws, 45 dmost maximal, 146, 272-275 Convex 6'-topology on space of, 79, 81-
hull, 268, 270 83, 272, 273, 278, 288, 297 neighborhood, 53-54 endless, 61 normal neighborhood. 53-54,292 endpoint of, 61, 233-234
Co-ray, 518 future d~rected, 50 asymptote, 518 geodesic, 17, 18 generalized, 510, 518 impr~soned, 61 , 221, 264 (generalized) timelike co-ray con- inextendible (= endless), 61
dition, 519, 522, 531 lirmt. 72, 81, 297 Cosmological constant (A), 44, 189 limit point, 233-234
see Einstein Equations maximal. 11, 66, 146, 166-171, Cosmologicd models
272, 275. 278, 296. 298, 356 see F'riedmann cosmologicd mod- null (= lightlike), 25 els
partially imprisoned, 62, 264.426 see Robertson-Walker space-tlmw past directed, 50 Cosmologieaf space-time, 563
Covariant pregeodesle, 86, 295, 286, 307-
curvature tensor, 22 309
derivative (;), 18, 370 spacehke, 65 Covering boundary sets, 231 timelike, 25
Covering space, 52, 86, 148, 419, 473 Gut points Critical point, 355, 356, 361 comparison to conjugate po~nts, Curl, 120 302-322 Curvature
bounded, 31 components ( R 3 k m ) , 20 constant, 31, 181-185 covariant tensor, 22 identities. 22-23
nonspacellke, 311-323 null, 296-298, 305-318. 395 Riemannian, 295. 31 7 time!ike. 296, 302-305 see also conjugate point
CW-complex, 355. 363. 354
626 INDEX
d'Alembertian (a), 114.534,535,545 Degenerate plane section. 29 de Sitter space-time, 163, 184, 286,
549, 569 Diameter, 399
timelike (diam(M, g)), 399, 401, 402
Disprisorment, 401, 415, 416, 418, 420, 422
Distance function, Lorentzim, 8, 137 continuity of for globally hyper-
bolic, 140 distinpshing space-tlmes, 158 finite, 142, 162 finite compactness. 211 globally hyperbolic spacetimes,
11, 65. 140, 142 inner (metric) balls (B', B-), 142,
143 local distance function. 160 lower semicontinuity of, 140. 277,
290-291, 508 nonsymmetric, 138 outer (metric) balls (Of, 0-), 144 preserving maps, 151 reverse triangle inequality, 140.
274. 279, 508 totally vicious space-t~mes, 68.
137 Distance function, Riemanman, 2-4
complete, 4 finite compactness, 4 see also Hopf-Rinow Theorem
Distance homothetic map, 151 Distingilishng
space-time, 58. 70, 158 Distribution, 121, 123 Diverge to infinity, 271, 272. 283.
288 Domain of dependence, 287
see Cauchy developrneilt
Edge, 538 Eikonal equation. 541, 574 Eimtein
equations, 44-48 manifold, 133 space-time, 117 static universe. 189,286, 290, 306,
307: 310. 313, 318, 323,477 silmmation convention, 23
Einstein-de Sitter universe, 115 Embeddings in high-dimens~onal Min-
kowski spacetime, 354 Empty spacetimes. 180
see also Ricci flat spacetimes Ends, of manifolds, 272, 282 Endpoint, 61, 234 Energy
conditions, 434 density, 47, 189 function, 375
Energy-momentum tensor (T, ) ,44- 48, 180
see also Einstein equations Enveloped mmifold, 2.30 Equivalent boundary sets. 231 Euler characteristic ( x ) , 50 Expansion tensor (8). 428, 585
quotient expansion (8), 431 Exponential map (exp,), 53. 314-
322, 399, 410 normal exponential map, 580 see also conjugate points see abo normal coordinates
Extended index for null geodesics; 395 for timelike geodesics, 342, 407-
408 Extemion, 198, 21 9. 219-225
C1-extemion, 232 local, 198, 220, 221-225
INDEX
given a pseudo-orthonormal sis, 37
Finite related to Ricci curvature, 39 compactness, 197, 21 1 related to sectional curvature distance condition, 142, 162 nonnull, 34
First variation formula, 360, 450 77.
Geodesic r lat f i n e parameter, 17, 202
metrics, 45, 141 closed, 149, 295 Ricci flat, 45, 124-127, 180 complete, Lorentzian, 91, 20, see also semi-Euchdean spaces
n, . I complete, Riemamm, 4 r m a
see perfect fluid Focal point, 444, 445, 446-449, 462
to a spacelike hypersurface, 447 Free, action of a group, 419 Freely falling, 31, 43, 425 Ekiedmann cosmological models, 286,
306, 311. 402
comwtedness, 400,417,418, equations of, 17, I8 incomplete, 108. 202, 309 instability of completeness, 2 instability of incompleteness, line, 272, 519 line, nonspacelike, 273, 28.3: 2
29.1 -- - see also Robertson-Walker space- line. null. 286
times Fundamental group (T I ) , 399,419 Future
Cauchy development (D'), 287 causal ( J + ) , 5 chronological ( I+) , 5 directed vector field, 5 horismos (E'), 175-177,287,288-
293, 317, 322, 468 horizon (H'), 287 ~mprisoned, 221 one-connected, 351,352,353,400,
414. 415 set, 55 trapped set, 287. 288, 465, 468 unit observer bundle (T- lM), 298-
304, 523
1 - - - line, timelike, 273, 290, 519 loop, 295. 312 maximal. 66, 146, 166-171. 2
275, 278, 296, 298, 356 minimal, 3, 4, 272 pregeodesic, 86, 295, 307-309 ray, 271, 279, 281,289
Geometric realization for quotient dle, 368, 369, 372, 373
Geroch splitting theorem, 65, 102 Global
conformal transformations, 28 time function, 64, 665
Globally hyperbolic, 11, 65 embedding in high-dimensions
kowski space-time. 354 of order q, 570
Gradient, (grad f ) , 24,307, 363,4
N Gallss' Lemma, 338, 379 General position, 40 Hadamard-Cartan theorems, 399- Generalized affine parameter, 208 411-423 Generic condition, 33-44, 309-3 11, Hausdorff
433 closed limit, 74 given an orthonormal basis, 33 distance, 258
lower limit (Lirn i d {A,)). 74 upper limit (Lim sup (A,)), 74
Hessian (Hf), 24 see also Laplacian see also d'Alembertian
Homogeneous completeness of (pos. def. case).
185 implied by isotropic, 186, 187 spatially, 155-188,261 two point. 185-187
Nomothetic, 97, 99, 151, 155, 188 Womotopy, 149, 400
groups, 399. 419 type, 325,326 see also fundamental group
Wopf-Rinow Theorem, 4 , 75, 197, 205, 271, 276, 399, 417
see also Cauchy complete Horismos (E+, E-1, 175-177, 287,
288-293, 317, 322, 468 Horizon (H', H-), 257 Noroball, 540, 541
see also Busemann function Norosphere, 538, 541 Hypersurface
aehronal, 57 spacelike, 64, 539
Ideal boundaries, 198,214-218,232- 238
Imprison&, 221,264 Incomplete
geodesic, 108, 202, 309 nonspac&ke, 425 null, 108 space-time, 108 timelike, 108
Inequality reverse Cauchy-Sehwarz, 575 reverse triangle, 10,140,274,279
Indecomposible future (past) set, 215- 218
Index, 323-398 comparison theorem, Lorentzian,
400. 406-410 ramparison theorem. Riemannian,
400. 406 extended index (Indo), 342 forms, 325, 328, 375, 377, 406,
458 geodesic, 324 timelike (Ind), 342, 407-408 to a spacelike hypersurface, 458
Inextendible space-time, 219, 220
Infinity diverge to, 271. 272, 283, 288 future timelike (if) , 178 spacelike (io), 178 past timelike (i-), 178
Inner continuo^^, 59 Inner products
metric (g and < , >), 21 on T,, (T,(M)) (< . >>) . 338 on vectors and co-vectors ( w ( X f ) ,
20 signature ( ( 8 , r ) ) , 15, 21
Inner (metric) ball (B+, B-) , 142. 163
Interval topology, 241 Irrotational, 120 Isometry, 97, 99, 354
local, 181 group ( I ( M ) ) . 188
Isotropic, 185, 186, 261 see also two point homogeneous
Isotropy group (I ,(M)), 186
Jacobi classes, 372, 373-398 classes, mslxirndity of, 988 equation, 31, 41, 328 fie&, 51 , 328, 332-398 fields, maximality of, 342 tensor, 383, 426
INDEX
see also second variation see also tidal accelerations
Kernel of a (1,1) tensor field along a null geodesic, 382
Kerr space-time, 179-181 Killing field, 120, 121, 482 Koszul formnla, 109 Kruskal diagram. 182
Lagrange tensor field. 385,427,428, 432
Laplacian (A), 24, 120, 545 see also Hessian see also dlAlembertian
Length Lorentzian, 8, 136 see also energy function
Levi-Civita connection (V), 15, 22 metric compatible and torsion free,
22 Lie bracket ([X, Y)), 18 Lie group, 190-195
translations of, 190 see also bi-invariant metric
Light cone. 15, 25-29 used to partial order metrics, 64 see also null cone
Lightlike (= null), 24 Lirni t
curve lemma, 5.7 2 curve of a sequence, 49, 12, 297 in Co topology, 79, 81-83, 272.
273, 278, 288. 297 Hamdorff, 74 maximizing sequence of causal curves,
515 point of a curve, 233-234
Line, maximal geodesic, 273-293 Lipschitz, 57, 75, 136 Local extension, 198. 220, 221-225 Loop space, 363, 364
Lor(M), 50, 63,64,89; 101. 23s Lorentzian
arc length. 135 distance function, 8, 137 manifold, 5, 25 metric, 5 product, metric, 95, 176, 31 quotient metric (g) , 370 signature (-, +, . . . , +), 15
51, 400 triangle comparison theorem warped product, 49-50, 95,
133 180-189
Madfold Lorentzlan. 25 Riemannian (= pos. def.), 1 seml-Riemannian, 15, 264 tangent bundle of ( T M ) , 16
RIaxlmal curve, 146, 296, 298 geodesic, 66. 166171, 356 space-time, 198
Maxlmal~ty of Jacob: classes. 388 of Jaeobi fields. 342
Mean clirvature. 213, 544, 585 Measure zero, 355 Metric
compatible with conneetion, , complete Lorentzian, 2/33 complete %emaman, 4 inner ball, 142. 163 Levi-Civita connection of, 22 Lorentzian. 25, 51 nondegenerate. 21, 92 outer ball. 144-145 quotient, 370 Rimannian (= pos. def.), I
25 Schwarzschild. 45, 131. 132, 17
179-182 semi-Rlernannian, 20 21 264
630 INDEX
semi-Eudidean, 181, 245 signatufe of, 21 tensor (g or g,), 21
Minimal geodesic segment, 4. 272 hilinkowski space-time (Ln = Ry),
25, 116, 159, 177, 173-184, 270, 286, 311. 354, 410: 480
Morse function, homotopic, 355,356, 363
Morse Index Theorem null, 365, 398 Riemannian, 324 timelike, 346, 405
Natural basis, 16 Nconj (Po), 489-497 Nondegenerate
metric tensor, 21 plane section, 29, 403 submanifold, 92
Nongeneric, 33 Noiiimprisoment, 263 Nonspacelike (= causal)
curve, 54 cut locas, 311-323 geodesic line, 273, 283, 285-293 geodesic ray, 279, 281 geodesically complete, 202 geodesically incomplete, 202,425 i i d t curve, 73, 297 vector, 24
Normal coordinate neighborhood, 53- 54
see also convex normal neighbor- hood
Null boundary point, 4 74 cone, 15, 25-29 conjugate point, 306, 395 convergence condition, 434 curve, 25 cut locus, 296-298,306309,395 index forms, 325,375,377,395
geodesic completeness, 91, 202 geodesic incompletenms, 202,306.
309 geodesic ray, 289 geodesics and conformal changes,
308 line, 286, 290 Morse Index Theorem, 324, 365,
398, 405 pregeodesics, 110, 307-309 tail (NT), 493, 494497 tangent vector, 24
Orientation by timelike vector field, 5, 25, 50
Orthogonal vectors, 21 Orthonormal basis, 21, 30, 404 Outer continuous, 59 Outer (metric) ball, 144
P
P(UO)I 481 Parallel translation, 1 7 Partial
imprisonment, 264, 265, 416 order on metrics, 64
Past Cauchy development (D- ) , 287 causal ( J - ) , 5 chronological ( I - ) , 5 directed vector field, 5 set, 55
Path space from p to q, 326 timelike (C(,,q)), 146, 354-365, 400
Penrose diagram, 179, 284 Perfect fluid, 47, 189, 311 Piecewise smooth
variation, 329, 389-391, 450 vector field, 327
Plane section degenerate, 29 nondegenerate: 29: 92, 403
sectional curvature of, 29-32,403 spaceiike. 29 timelike. 29, 403
Plane wave solution, 479-499 definition of gravitational plane
wave, 483 definition of plane fronted waves,
480 polarized, 483 sandwich wave, 483 unimodal gravitational plane wave,
497 Positive definite, 15 Pregeodesic, 17, 86, 110, 111. 295,
307-309 Pressure, 47, 180. 189 Proper variation, ,329. 389-391 Properly discoi~ti~mously. 419 Pseudoeonvex, 265, 268, 269, 270.
401, 415, 416. 418, 420, 422, 488
nonspacelike, 269 Pseudo-orthonormal bais, 36, 460
INDEX 6
Quasi-regular singularity, 225 Qliotient
bundle (G(P)), 326, ,368 bundle index form, 377 covariant derivative. 370, 371 c~irvatme tensor, 371 expansion tensor (8 ), 431 metric, 370, 371-398 shear tensor ( 8 ), 431 vorticity tensor ( G ), 431
Quotient topology on Con(hf). 241
Railch Comparison Theorems. 400, 406. 408, 572
Ray. 271 nonsparelike geodesic, 279 null geodesic, 289, 290
Flaycha~idh~lr~ eqnatiun, 402
along mdl geodesics, 432 along timelike geodesics. 430 v~rt~icity free, 430, 432
Real projective space, 189 Reflect,lng, 69 Regular boundary points. 225 Reissner-Nordstrcm space-time, 15
179 Residual set, 355 Retraction. 357 Reverse triangle inequality, 10, 14t
274, 279 Riccati equation. 573. 582 Rieci clirvatlire (&c or a3), 23, 30
45, 180 Riemann-ChristofFel tensor ( a 3 k s n )
22 Riemannian
distance function, 2-4 manifold, 25, 30 nletric (= pos. def.), 15, 24 t,wo point homogeneous, 185-188
Robe1 tso~i-Walker space-tlme, 103, 174. 185-190. 250-263, 311, 477
having closed trapped surfaces, 472
Root warping ftinction (S = a), 119
see also warping function Rotation group (SO(3)), 180
Scalar c~~rvature ( R or T ) , 29 Schmidt &boundary ( d b ) , 197. 214 Schwarzsch~ld space-time. 45. 131,
132, 173, 179-182 Second fundamental forms (S. S, L,)
165, 445 1x1 general (S), 92 in the direction n (5,). 93 opelator (L,), 93, 456, 461 see also closed trapped surface see abo totally geodesic
632 INDEX
Second variation formula, 324, 329 Sectio~lal curvature (K(E.p)) , 29-32,
399. 403 bounded, 31,403 null, 5.iI timelike, 29-32, 403
Self-abjoint, 430 Semicontinuity. 140, 273, 277, 278.
290-291.299-301, 511 Semi-E~iclidean (R:), 181, 24 5
area, 29 classification of planes, 29
Semi-Riemannian metric, 15, 264 Semi-time function, 68, 490 Shear tensor (a), 428
quotient ( 5 ), 431 Signature
degenerate, 29 Lorentzian, metric, 21, 25, 51.
400 Riemannian (pos. def.), 15, 25 semi-Eudidean. 29, 181, 245 semi-Riemannian, 21 type (ti, r), 21, 181, 245
Simply connected. 352 Singalar
C'-singular point. 237 spacetime. 12,203,225-238.426
Singtilarity theorems, 468, 469, 472, 478
Smooth boundary point, 474, 472- 478
Spacelike botindary point, 474 curve, 25 hypersurface, 539 infinity (to), 178-179 stibmanifold, 92
Space-time, 25, 51 a-boimndary, 198, 232-238 anti-de Sitter, 183, 199, 306, 569 big bang models, 103-104 b-boundary, 214 b complete, 208
b. a. complete, 207 causal, 7. 58, 73 causal bo~mdary, 215-218, 425.
472-478 causally continuous, 59, 71, 73,
158-160,479 causally simple, 65, 73, 479 chronological, 7, 58, 73 cosmological, 563 c~trvat,u~re sin~ilarity, 215 definition of, 5. 51 de Sitter, 183, 184, 286, 549, 569 di~t~i~lguishing, 58, 79, 73, 158 Einstein static, 189, 286,289, 306,
307. 310, 313, 318, 323, 477 flat, 141 Riedmann cosmological models,
286 306, 311, 402 f~iture one-connected, 351. 352.
353, 400, 414, 415 geodesically complete, 91. 203 globally hyperbolic, 11, 65, 71,
73, 354 incomplete, 203, 309, 425 Kerr, 174. 179-181 locally inextendible, 198,220-238 Minkowski (Ln = R;), 25, 116.
159, 177, 173-184, 270, 286. 311, 354, 410. 480
nonspacelike geodesirally complete, 203
null geodesically complete, 253 plane wave solution, 479-499 qi~nsi-reglllar singularity, 225 refiecting, 69 Reissner-Nordstrom, 139, 179 Ricci flat, 45, 124, 180 Robertson-Walker, 103, 174, 185-
190, 250-263 311. 477 Schwarzxhild, 45, 131, 132, 173,
179-182 singular, 203, 425, 468 stably catlsd, 63, 73, 89, 242 static, 121, 180
INDEX 6:
strongiy causal, 7, 59, 73 timeiike geodesically complete, 2&? totally vlcious, 68. 137, 168
Sphere Theorem, topological. 502 Sph~tmg theorems
Cheeger-Gromoll. 503 Geroch, 49, 65, 102 Lorentzian. 506, 550, 562
causality, 63. 89, 242 conformally, 242 in the C' t?opology. 239, 242 properties in Lor(M). 239, 247
Static. 121, 180 Strong callsalits 7, 59 St,rong curvature singularity, 225 Strong energy condit~on, 434
In the first edztzon of thzs book, we used strong energy con- dition to mean what zs now termed the timelike conver- germ condition.
Submanifold indticed connection (VO), 92 nondegenerate, 92 second ftindamental form. 92-94 spacelike, 92 timelike. 92. 164 tottally geodesic, 99 94. 98, 165
summation con.i.entio11. 23 Synge world filnct~on. 491
Tangent b~lndle ( T M ) . 16 Tangent space (T,M), 16. 25. 336
null (= lightlike). 24 spacelike, 24
cnrvatiire (R(.Y. Y ) Z ) , 19 energy-momentum (T or T,,), 4
48 expansion (8) . 428. 585 expansion, quotient ( i? ), 4 31 Jacobi. 383. 426 Lagrange. 385. 427, 428. 432 metric (g or g,J). 21 Rcci (Ric or a?), 23, 30, 180 Bemann-Christoffel (R,J,,), 2; scalar ctirvatiire (R or T ) , 23, 21 shear (c) 428 shear. qt~ot~erit ( 8 ), 431 torsion ( T ( X , Y ) or T , J k ) . 19 vortit~ty (w) , 428 vorticity, quotient (Z j ), 431 Wrorlskian (W(A. B)). 385
Term~nsl indecomposible futtire set (TIF), 215-218 past set (TIP), 215-218
Theoreins. selected Arzela's Theorem, 75, 76 Birkhoff's Theorem, 132 Bonnet-Myers Theorem, 339-40(
405 comparison theorems. 400, 408 Geroch Sphtt~ng Theorem, 65,
102 Hadamard-Cartan 'Theorems. 39'
401, 411-423 Hopf-Rinow Theorem, 4.75.197
205. 271 276, 399, 417 Maximal Diameter Theorem. 570 blorse Index Theorems. 324. 346,
365, 398, 405 Ranch Comparison Theorems, 40(
406, 408. 572 singi~larity theorems, 468, 469,
472, 478 splitting theorems 49. 506, 550,
562 Timelike Index Comparison The-
orem. 406 Topolog~cal Sphere Theorem, 502
634 INDEX
Toponogov's Dlameter Theorem, 406
Toponogov Triangle Comparison Theorem, 567
Tkiangle Comparison Thecrems, 567, 570
Tidal acceleration, 31-32. 42 ~inboilndedness of, 32, 42-44 see also Jacobi fields
TIF's and TIP'S, 215-218 Time
function, 64, 70 generalized time function, 70 oriented. 5, 25. 50, 52 q11asl-time fi~nction, 490, 577 semi-time function, 68, 490
Tlrnelike convergence condition, 46. 112
see comment afier strong energy conditton
Timelike boundary point, 4 74 Caiichy complete, 21 1 chain, 356, 357 conjugate point, 328. 413 convergence condition. 46, 1 12 curve, 25 curve space jiWb,,)). 357 cut loc~~s , 298-305 diameter, 399, 401, 402 geodesic completeness. 209 geodesic incompleteness, 203 geodesic line, 273, 290 index comparison theorem, 400,
4 08 index form, 325. 328, 407-408 infinity (zi, z-), 178 Morse Index Theorem, 324, 346.
405 path space (C( , , ) ) , 146, 354-365 sectional curvature ( K ( E , p ) ) , 29-
32, 403 spares, 209 siibma~fold, 92, 164
two plane, 29. 403 vector field, 50
Topology Alexandroff, 7, 8, 59-60 Co on cilrves, 79, 81-63. 272,
273, 278, 288. 297 C' on Con( M), 239 C' on Eor(M), 239, 247 interval on Con(A4), 241 quotient on Con(M), 241
Toponogov Triangle Comparison The- orem, 567
Torston tensor ( T ( X , Y) or T , J k ) , 19 Totally
geodesic, 93, 94, 98. 165 viciotis. 68, 137. 165
?l-apped set, 287, 465, 468 s~irface, closed, 463
Triangle cornparison theorems, 567, 570
Two point homogeneo~is, 185-155
u Unit
observer biindle (T-I hf), 298-304 vector. 21
Upper slipport fiinction, 520. 530
Vactium space-times. 482 see also Ricri flat and empty
Variation, 329 admissible. 376, 388-391 canonical. 331, 388 first, 360, 450 plecewise srnooth. 329, 389-391 proper. 329, 389-391 second. 324 vector field. 323. 331, 389
Vector field along a cirrve. 17 Killing, 120. 121 nonsparehke, 24
INDEX
nilli, 22 , 24 spacelike. 24 timellke. 24 ~ m i t , 21
Volnme form, 67 Vorticity tensor (w). 428
quotient (IJ ). 431
Warped producr (M x f H ) . 95, 94- 133. 180 163. 250-263
fibers. 119 horizontal vectors, 119 leaves. 119 local warped products, 117 133 vert~cal vectors, 119
Tarping fl~nctlon, 112. 189 see also root warptng fiinct~on
Weak energy condition, 434 Wheeler universes, 399, 401 Wlder light cones. 64 !Vroiiskidn (IV'(A. B)), 385