The nonspacelike cut locus revisited

Pergamon Nonlinear Analysis, Theory, Methods&Applications. Vol. 30, No. 1. pp. 5894%. 1997

Proc. 2nd World Congress of Nonlinear Analysts Q 1997 Elsevier Science Ltd

PII: SO362-546X(97)00037-0

F’rintd in Great Britain. All ri ts reserved P 0362-546X/97 17.00+0.00

THE NONSPACELIKE CUT LOCUS REVISITED

PAUL EHRLICH

Department of Mathematics, University of Florida, Gainesville, FL 32611-8105, U.S.A.

Key words and phrases: Nonspacelike cut locus, first conjugate locus, timelike geodesic line, Lorentzian splitting theorem.

1. INTRODUCTION

We consider some aspects of the space-time cut locus which came to the fore in connection with the preparation of the Second Edition [l] of [2]. Since a first conjugate point along a Riemannian or nonspacelike geodesic is sometimes a cut point as well, the properties of the conjugate locus are lurking in the background when the cut locus is studied. In particular, in [2, Lemma 8.271 it had been noted that the locus of first future nonspacelike conjugate points in a globally hyperbolic space-time is closed. On the other hand, at the U.C.L.A. Summer Symposium of the American Mathematical Society, C. Magerin had presented an article [3] in which he noted that for Riemannian manifolds

“Although there is no specific reason of any kind for the conjugate locus to be closed, this seems to be taken for granted by many, see, for example Cheeger and Ebin’s textbook . . . when proving Prop. 5.4:‘the set of singular values of the exponential map is closed; an accumulation point of conjugate points is a conjugate point.“’

Contrary to this popular viewpoint, however, Magerin in [3] constructs several examples to show how dramatically an accumulation point of conjugate points can fail to be a conjugate point. He shows that even for a compact, positively curved Riemannian manifold, a sequence of conjugate points to a fixed point p may converge to a point which is not conjugate to p. Secondly, he gives an example of a compact Riemannian manifold with nonnegative sectional curvature such that the closure of the set of first conjugate points to a given point p may contain points which are not conjugate to p. These results suggested a careful re-examination of Lemma 8.27 of [2] to be in order.

A second issue which had arisen in connection with the splitting problem of S. T. Yau in space-time geometry, was the consequences for local space-time causality and geometry of the existence of a complete maximal timelike geodesic line, or even

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of the existence of a maximal geodesic segment. Yau had posed as Problem No. 155 of [4] the following:

“The splitting theorem of Cheeger and Gromoll says that if a Riemannian manifold M of nonnegative Ricci curvature contains a line y, (i.e., an absolutely minimizing geodesic), then M decomposes isometrically as a cross product Itgx N, the first factor being represented by y.

It would be of interest in studying the structure of space-time to prove that a geodesically complete Lorentzian 4-manifold of nonnegative Ricci curvature in the timelike direction which contains an absolutely maximizing timelike geodesic is isometrically the cross product of that geodesic and a spacelike hypersurface.”

Early studies of this problem assumed (cf. [5] f or instance) not timelike geodesic completeness as originally posed, but rather global hyperbolicity. For unlike timelike geodesic completeness, global hyperbolicity is a conformally invariant causality condition which is often placed on a space-time and which has the added bonus of insuring that any two chronologically related points may be connected by a maximal nonspacelike geodesic segment. But even with global hyperbolicity, which is a kind of substitute for Riemannian completeness, the Busemann function of a timelike geodesic line, unlike its cousin for a complete Riemannian geodesic, need not have the continuity and finiteness properties of the Riemannian Busemann function. In [5], it was found that the timelike co-ray condition, which was guaranteed for globally hyperbolic space-times of nonpositive timelike sectional curvature through the use of the space-time Toponogov theorem of S. Harris [6], insured the continuity of the space-time Busemann function on I(r) = I-(r) nl+(r). An important discovery of Eschenburg [7] was that if attention is restricted to a sufficiently small tubular neighborhood of a given timelike geodesic line in the globally hyperbolic space-time, then in that neighborhood, the Ricci curvature assumption Ric(v, w) 2 0 for all timelike V, would give control of the Busemann function in that neighborhood, despite the lack of the Toponogov theorem under the weaker Ricci curvature assumption.

In 1990, Newman [8] returned to the space-time splitting problem as originally for- mulated by Yau under the hypothesis of timelike geodesic completeness. Since general nonspacelike geodesic connectability was not available, Newman had to make a careful study of the implications of the existence of a complete timelike line for the local causality and timelike geodesic connectability in a tubular neighborhood (of sufficiently small radius) of a complete timelike line. In Galloway and Horta [9], these results were reconsidered from the viewpoint of the generalized timelike co-ray condition.

2. MAXIMAL GEODESIC SEGMENTS AND LOCAL CAUSALITY

The Lorentzian distance function d : A4 x M + [O, +oo] of an arbitrary space- time (M,g) may be defined as follows: set d(p, q) = 0 if there is no future causal

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nonspacelike curve from p to Q, and otherwise calculate d(p,q) as the supremum of lengths L(c) of all future causal curves c from p to q. Unlike the case for an arbitrary (not necessarily complete) Riemannian metric, the space-time distance function may be discontinuous and may take on the value +oo in general space- times. For example, if there is a closed timelike curve in the space-time and p lies on this curve, then d(p,p) = +oo. Thus d(p, q) = +oo if q lies on any future causal curve issuing from p. However, points can have infinite Lorentzian distance without such extreme breakdowns of causality as happens in certain Reissner-Nordstrom space-times, cf. Figure 4.2 of [l].

Implications for the Lorentzian distance function of causality conditions which are often placed on space-times have been studied extensively, cf. the texts [l], [2], [lo], [ll]. To recall just a few examples, first, if the space-time (M,g) is strongly causal, then for each p in it4 there exists a neighborhood U of p such that d](U x V) is finite valued. Second, if (M, g) is globally hyperbolic, then any metric gi for M conformal to g satisfies the finite distance condition d(gi)(p, q) < +oo for any p, q in it4 and also d(gi) is continuous on M x M.

On the other hand, one can study the implications for the local causality of the existence of a maximal nonspacelike geodesic segment, i.e., a nonspacelike geodesic c : [u,b] -+ (M,g) such that d(c(a),c(b)) = L(c). [Th e reverse triangle inequality then implies that d(c(s), c(t)) = L(cl[s, t]) f or all s < t.] As a preliminary example, suppose that a space-time contains a maximal nonspacelike geodesic segment c : [u,b] + (M, g). Then the space-time cannot be totally vicious, since such space-times satisfy d(g)(p, q) = +co for all p, q in M. Yet for any two points p = c(s) and q = c(t) on the given segment, we have that d(p, q) = L(c][s, t]), which is automatically finite.

Now suppose the space-time (M,g) contains a maximal null geodesic segment c : P, 11 - (Cd. Then if T E J+(c(s)) n J-(c(t)) for s < t, the reverse triangle inequality yields

0 = d(c(s), c(t)> 2 d(c(s), r> + 47.7 c(t)>

so that both d(c(s),r) = d(r,c(t)) = 0. H ence, chronology must hold at all points of J+(c(o)) f-w(c(l)). 0 ne cannot conclude more from the existence of a maximal null geodesic segment, for the space-time M = W x S1 with Lorentzian metric ds2 = d0 dt contains maximal closed null geodesics c(t) = (so, t) for any fixed so in W. Hence, the existence of maximal null geodesic segments does not preclude the failure of causality.

Suppose now that the space-time (M,g) contains a maximal timelike geodesic segment c : [0, l] + (M, g). Then causality must hold at all points of c([O, 11) by rounding the corner arguments. (A null segment p beginning and ending at c(to) can be postcomposed with cl[O, to] to produce a causal curve from c(0) to c(to) which cannot be maximal as the composition of a timelike followed by a null geodesic. Hence, cl[O, to] fails to be maximal, in contradiction). Once this elementary fact is established, then usual General Relativity type arguments (most easily carried out


using a lemma of Kronheimer and Penrose, cf. [ll,Lemma 4.161, on the failure of strong causality) imply that the following less elementary result is valid:

PROPOSITION 2.1. (Newman [8], Galloway and Horta [9])

Let c : [0, a] + (M,g) be a maximal timelike geodesic segment in an arbitrary space-time. Then for any to with 0 < to < a, strong causality holds at c(to).

We have mentioned above that in general the Lorentzian Busemann function is not as nicely behaved as the Riemannian Busemann function. Nonetheless, the existence of a complete maximal timelike geodesic line

has important implications for the behavior of the Busemann function in some tubular neighborhood of the given line. This realization was essential in the proof of the space-time splitting theorem, cf. [7], [8], and [9] for a partial list of articles in which this theme was successively developed.

For a future complete timelike ray 7, the associated Busemann function is formed by taking the limit

Hence, it is necessary to study s + d(q, r(s)). A first step in this procedure follows directly from the reverse triangle inequality: given any q in J+(r) n J-(-y), then 4$s), q) and 4q7 r(t)) are finite valued for any s, t. For given q as indicated, for all sufficiently large s,t we have y(s) << q < r(t). Hence, as L(r][s,t]) is finite and y is assumed to be maximal, we obtain

by the reverse triangle inequality. Thus both distances are forced to be finite as desired.

3. THE NONSPACELIKE CUT LOCUS

We have recalled above Magerin’s [3] surprising result that the conjugate locus of a complete Riemannian manifold is not as nicely behaved as is commonly assumed. Since a cut point in a complete Riemannian manifold is either a first conjugate point, or lies on at least two minimal geodesic segments, the generally incorrect closure of the first conjugate locus has been appealed to in the usual Riemannian proof of the closure of the cut locus. Thus let us present a more elementary proof of this closure which does NOT rely on this geometric dichotomy, but rather relies


directly on continuity properties of the exponential map and the continuity of the distance to the cut point as a function of the initial tangent to the unit speed geodesics ematiating from a given point.

THEOREM 3.1. Let (IV, go) be a complete Riemannian manifold. Then the cut locus C(p) of any point p in N is a closed subset of N.

Proof. Let s : SN = {v E TN : ge(v,v) = 1) ---+ [O,+oo] be defined as usual by s(v) =

sup{t 2 0 : 4~4% c,,(t)> = L(c,I[O,tl) = t), w h ere cv denotes the geodesic with initial condition c’,(O) = v. Thus s(v) is the distance to the cut point c,(s(v)) provided that s(v) < +oo. Now it is well known that if the Ftiemannian metric go is complete, then s : SN + [0, +CXI] is continuous. Take any sequence {q,,} c C(p) with q,, + q in N. Obtain a corresponding sequence {vn} c TpN with ]]z),,]] = 1 and qn = exp,(s(v,)v,). By compactness of the unit sphere bundle, extract a subsequence I.J~(~) + w in T,N. By continuity of the s-function,

in T,N. Hence, by continuity of the exponential map, we obtain

Q = /$&G(j) = jl;l& exP,(4%(j)bn(j)) = exp,M44

which implies that q E C(p).

Q.E.D.

Now we return to consideration of a space-time (n/i,g). Motivated by the Riemannian theory, we had treated timelike cut points “intrinsically” in [2, Section 8.11 by using the future unit observer bundle

TsIA4 = {II E TM; g(v, w) = -1 and v is future directed}

and studying the continuity properties of the s-function defined on T-IM as above. For a future inextendible timelike or null geodesic c : [O, u) -+ (M, g), the cut point to c(0) along c may defined by calculating

to := sup{t 2 0; d(c(O), c(t)) = L(c](O, t])}

and declaring ~($0) to be the future cut point provided 0 < te < a. Of course, for a null geodesic, this definition simplifies to

to := sup{t 2 0; d(c(O), c(t)) = 0).

Now difficulties arise in the treatment of the nonspacelike cut locus, since the set of unit timelike tangent vectors in T,M while closed is not compact, and also while a set of unit timelike directions can converge to a null direction, a set of unit timelike vectors cannot


converge to a null vector. Nonetheless, we had established in [2] that timelike geodesic completeness implies the upper-semicontinuity of s : T-1M + [0, +oo] (a more precise statement is provided in Proposition 9.5 of [l]), and that global hyperbolicity implies the lower semicontinuity of s : T-iM + [0, +CCJ]. On physical grounds, one usually assumes that space-times have nonpositive Ricci curvature on all timelike vectors, and hence one does not expect them to be timelike geodesically complete. Thus, one only has the lower semicontinuity of the s-function at one’s disposal when studying “physically realistic” space- times.

Nevertheless, employing

LEMMA 3.2. Let (M,g) be a globally hyperbolic space-time. Then the first future nonspacelike conjugate locus of any point is closed in M.

and the geometric characterization for globally hyperbolic space-times of a future causal cut point as either being a first future conjugate point or the endpoint of two maximal geodesic segments issuing from p, we had shown as in [12, Prop. 5.41 that the future causal cut locus of any point in a globally hyperbolic space-time is closed. Moreover, we remarked that this held despite the lack of upper-semicontinuity of the s-function on the unit observer bundle. (Of course, the examples in [3] invalidate the proof given in [12] for Prop. 5.4, but the result remains true in view of Theorem 3.1 above). The results in [3] suggested a careful re-examination of the proof offered in [2] of Lemma 3.2, but it remains correct; globally hyperbolic space-times are strongly causal and strongly causal space-times may not imprison inextendible nonspacelike curves in a compact set. Thus any nonspacelike geodesic must escape from any compact subset of a globally hyperbolic space-time in finite ai%ne parameter. However, Magerin’s examples show that the spacelike conjugate locus in a space-time may fail to be closed.

In working intrinsically, difficulties with unit timelike vectors converging to a null direction prevent the “intrinsic” s-function from being directly applied to give a proof of the closure of the causal cut locus for globally hyperbolic space-times along the lines of Theorem 3.1 above. This motivated us in collaboration with G. Galloway to define a modified but non- instrinsic s-function in Section 9.4 of [l] in order to give a simple proof of the closure of the nonspacelike cut locus along the lines of Theorem 3.1.

Let h be an auxilliary complete Riemannian metric for the space-time (M, g). Put

U(M) := {v E TM; h(v, TI) = 1 and v is future directed nonspacelike}.

Thus U(M) contains null as well as timelike vectors and also a sequence of timelike vectors in U(M) may converge to a null vector, unlike the case for T-1M. Let p : TM + M denote the projection map in the tangent bundle and define si : U(M) + [0, +m] by

31(v) := SUP@ 2 0; Q(v), G(t)> = J%I[O, w

where cv(t) = exp(tv) as before. Thus for v timelike, we have the relationship

1141~1(~> = 4vlll4l>~


Just as in [l, Section 9.11, the upper semi-continuity of si at v E U(M) with si(w) = +oo, or with si(~) finite and cV extending to [O,si(w)] may be established. Also, the global hyperbolicity of (M, g) implies the lower semicontinuity of sr . The following technical lemma shows how the assumptions of global hyperbolicity together with endpoint convergence give the control needed to show, using the si-function in an elementary manner similar to Theorem 3.1 above, that the nonspacelike cut locus of any point p is a closed subset of a globally hyperbolic space-time.

LEMMA 3.3. Let (M, g) be globally hyperbolic and let {cn : [0, b,] + (M, g)} be a sequence of maximal future nonspacelike geodesic segments with TJ~ = c;(O) in U(M) and ~~(0) = p for all 12. Put qn := cn(b,) and suppose that qn + q in M, b, + b > 0 in [0, +oo], and w, * w in U(M).

Then (i) b < a, so that b is finite and positive, (ii) c(a) = q, (iii) si(2r) > b.

In the proof of the space-time splitting theorem in whatever version, given a complete timelike ray c : [0, +w) + (M, g), it is necessary to have control over the past nonspacelike cut locus to c(t) as t ---+ +oo. In the globally hyperbolic case, the fact that the nonspacelike cut locus is closed aids in this effort, cf. [5, pp. 37 - 381. In the timelike geodesically complete case, however, the timelike cut locus may be less well behaved and no general structure theory is available. Nonetheless, Galloway and Horta in Proposition 3.9 of [9] make the interesting observation that the existence of a maximal timelike geodesic segment together with local finiteness of the distance functions d(-, c(t)) gives the required control. More precisely, given a maximal timelike segment

such that for each t in (0,l) the distance function f(q) := d(q, c(t)) is finite valued on a neighborhood of c(O), then there exists a neighborhood U of the segment c( [0, t]) which does not meet the past timelike cut locus of c(t).

REFERENCES

1. BEEM J., EHRLICH P. & EASLEY K., Global Lorentzian Geometry, Second Edition, Marcel Dekker, New York, 1996.

2. BEEM J. & EHRLICH P., Global Lorentzian Geometry, Marcel Dekker, New York, 1981. 3. MAGERIN C., General conjugate loci are not closed, in Proc. Symp. pure Math. (Edited by R.

GREENE & S.-T. YAU), pp. 465 - 478. Amer. math. Sot. 54, Part 3 (1993). 4. YAU S.-T., Problem Section, in Annals of math. Studies (Edited by S.-T. YAU), pp. 669 - 706,

Princeton University Press, Princeton, 102 (1982). 5. BEEM J., EHRLICH P., MARKVORSEN S. & GALLOWAY G., Decomposition theorems for

Lorentzian manifolds with nonpositive curvature, J. diff. Geom. 22, 29 - 42 (1985).

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6. HARRIS S., A triangle comparison theorem for Lorentz manifolds, Indiana math. J. 31, 289 -

303 (1982). 7. ESCHENBURG J.-H., The splitting theorem for space-times with strong energy condition, J. din

Geom. 22, 29 - 42 (1985).

8. NEWMAN R., A proof of the splitting conjecture of S.-T. Yau, J. din. Geom. 31, 163 - 184 (1990).

9. GALLOWAY G. & HORTA A., Regularity of Lorentzian Busemann functions, fins. Amer. math. sot., to appear (1996).

10. HAWKING S. & ELLIS G., Large Scale &kucture of Space-time, Cambridge University Press, Cambridge, 1973.

11. PENROSE R., Techniques of Digerential Topology in Relativity, S.I.A.M. Conference Series in Applied Mathematics, Sot. Indus. Applied. Math., Philadelphia, ‘7 (1972).

12. CHEEGER J. & EBIN D., Comparison Theorems in Riemannian Geometry, North Holland, New York, 1975.

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The nonspacelike cut locus revisited