Gravitational-Wave Physics and Astronomy

Jolien D. E. Creighton, Warren G. Anderson

An Introduction to Theory, Experiment and Data Analysis

WILEY SERIES IN COSMOLOGY

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Jolien D. E. Creighton andWarren G. AndersonGravitational-Wave Physicsand Astronomy

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Jolien D. E. Creighton and Warren G. Anderson

Gravitational-Wave Physicsand Astronomy

An Introduction to Theory, Experimentand Data Analysis

WILEY-VCH Verlag GmbH & Co. KGaA

The Authors

Dr. Jolien D. E. CreightonUniversity of Wisconsin–MilwaukeeDepartment of PhysicsP.O. Box 413Milwaukee, WI 53201USAjolien@uwm.edu

Dr. Warren G. AndersonUniversity of Wisconsin–MilwaukeeDepartment of PhysicsP.O. Box 413Milwaukee, WI 53201USAwarren@gravity.phys.uwm.edu

CoverPost-Newtonian apples created by TevietCreighton. Hubble ultra-deep field image(NASA, ESA, S. Beckwith STScl and the HUDFTeam).

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V

JDEC: To my grandmother.

WGA: To my parents, who never asked me to stop asking why, although they did stopanswering after a while, and to family, Lynda, Ethan and Jacob, who give me the space Ineed to continue asking.

VII

Contents

Preface XI

List of Examples XIII

Introduction 1References 2

1 Prologue 31.1 Tides in Newton’s Gravity 31.2 Relativity 8

2 A Brief Review of General Relativity 112.1 Differential Geometry 122.1.1 Coordinates and Distances 122.1.2 Vectors 142.1.3 Connections 162.1.4 Geodesics 242.1.5 Curvature 252.1.6 Geodesic Deviation 312.1.7 Ricci and Einstein Tensors 322.2 Slow Motion in Weak Gravitational Fields 322.3 Stress-Energy Tensor 342.3.1 Perfect Fluid 362.3.2 Electromagnetism 382.4 Einstein’s Field Equations 382.5 Newtonian Limit of General Relativity 402.5.1 Linearized Gravity 402.5.2 Newtonian Limit 432.5.3 Fast Motion 442.6 Problems 45

References 47

3 Gravitational Waves 493.1 Description of Gravitational Waves 493.1.1 Propagation of Gravitational Waves 55

VIII Contents

3.2 Physical Properties of Gravitational Waves 583.2.1 Effects of Gravitational Waves 583.2.2 Energy Carried by a Gravitational Wave 663.3 Production of Gravitational Radiation 693.3.1 Far- and Near-Zone Solutions 693.3.2 Gravitational Radiation Luminosity 743.3.3 Radiation Reaction 783.3.4 Angular Momentum Carried by Gravitational Radiation 803.4 Demonstration: Rotating Triaxial Ellipsoid 803.5 Demonstration: Orbiting Binary System 843.6 Problems 91

References 95

4 Beyond the Newtonian Limit 974.1 Post-Newtonian 974.1.1 System of Point Particles 1044.1.2 Two-Body Post-Newtonian Motion 1094.1.3 Higher-Order Post-Newtonian Waveforms for Binary Inspiral 1144.2 Perturbation about Curved Backgrounds 1144.2.1 Gravitational Waves in Cosmological Spacetimes 1194.2.2 Black Hole Perturbation 1234.3 Numerical Relativity 1304.3.1 The Arnowitt–Deser–Misner (ADM) Formalism 1304.3.2 Coordinate Choice 1394.3.3 Initial Data 1414.3.4 Gravitational-Wave Extraction 1434.3.5 Matter 1434.3.6 Numerical Methods 1444.4 Problems 145

References 147

5 Sources of Gravitational Radiation 1495.1 Sources of Continuous Gravitational Waves 1515.2 Sources of Gravitational-Wave Bursts 1575.2.1 Coalescing Binaries 1575.2.2 Gravitational Collapse 1655.2.3 Bursts from Cosmic String Cusps 1695.2.4 Other Burst Sources 1705.3 Sources of a Stochastic Gravitational-Wave Background 1715.3.1 Cosmological Backgrounds 1725.3.2 Astrophysical Backgrounds 1915.4 Problems 194

References 196

6 Gravitational-Wave Detectors 1976.1 Ground-Based Laser Interferometer Detectors 198

Contents IX

6.1.1 Notes on Optics 2036.1.2 Fabry–Pérot Cavity 2076.1.3 Michelson Interferometer 2116.1.4 Power Recycling 2146.1.5 Readout 2166.1.6 Frequency Response of the Initial LIGO Detector 2216.1.7 Sensor Noise 2266.1.8 Environmental Sources of Noise 2306.1.9 Control System 2396.1.10 Gravitational-Wave Response of an Interferometric Detector 2416.1.11 Second Generation Ground-Based Interferometers (and Beyond) 2446.2 Space-Based Detectors 2516.2.1 Spacecraft Tracking 2516.2.2 LISA 2526.2.3 Decihertz Experiments 2566.3 Pulsar Timing Experiments 2566.4 Resonant Mass Detectors 2606.5 Problems 265

References 267

7 Gravitational-Wave Data Analysis 2697.1 Random Processes 2697.1.1 Power Spectrum 2707.1.2 Gaussian Noise 2737.2 Optimal Detection Statistic 2757.2.1 Bayes’s Theorem 2757.2.2 Matched Filter 2767.2.3 Unknown Matched Filter Parameters 2777.2.4 Statistical Properties of the Matched Filter 2797.2.5 Matched Filter with Unknown Arrival Time 2817.2.6 Template Banks of Matched Filters 2827.3 Parameter Estimation 2867.3.1 Measurement Accuracy 2867.3.2 Systematic Errors in Parameter Estimation 2897.3.3 Confidence Intervals 2917.4 Detection Statistics for Poorly Modelled Signals 2937.4.1 Excess-Power Method 2937.5 Detection in Non-Gaussian Noise 2957.6 Networks of Gravitational-Wave Detectors 2987.6.1 Co-located and Co-aligned Detectors 2987.6.2 General Detector Networks 3007.6.3 Time-Frequency Excess-Power Method for a Network of Detectors 3037.6.4 Sky Position Localization for Gravitational-Wave Bursts 3057.7 Data Analysis Methods for Continuous-Wave Sources 3077.7.1 Search for Gravitational Waves from a Known, Isolated Pulsar 309

X Contents

7.7.2 All-Sky Searches for Gravitational Waves from Unknown Pulsars 3167.8 Data Analysis Methods for Gravitational-Wave Bursts 3177.8.1 Searches for Coalescing Compact Binary Sources 3187.8.2 Searches for Poorly Modelled Burst Sources 3327.9 Data Analysis Methods for Stochastic Sources 3337.9.1 Stochastic Gravitational-Wave Point Sources 3447.10 Problems 345

References 347

8 Epilogue: Gravitational-Wave Astronomy and Astrophysics 3498.1 Fundamental Physics 3498.2 Astrophysics 351

References 353

Appendix A Gravitational-Wave Detector Data 355A.1 Gravitational-Wave Detector Site Data 355A.2 Idealized Initial LIGO Model 359

References 361

Appendix B Post-Newtonian Binary Inspiral Waveform 363B.1 TaylorT1 Orbital Evolution 366B.2 TaylorT2 Orbital Evolution 366B.3 TaylorT3 Orbital Evolution 367B.4 TaylorT4 Orbital Evolution 368B.5 TaylorF2 Stationary Phase 369

References 370

Index 371

XI

Preface

During the writing of this book we often had to escape the office for week-longmini-sabbaticals. We would like to thank the Max-Planck-Institut für Gravitations-physik (Albert-Einstein-Institute) in Hannover, Germany, for hosting us for one ofthese sabbaticals, the Warren G. Anderson Office of Gravitational Wave Research inCalgary, Alberta for hosting a second one, the University of Minnesota for hostinga third and the University of Cardiff for our final retreat.

We thank, in no particular order (other than alphabetic), Bruce Allen, PatrickBrady, Teviet Creighton, Stephen Fairhurst, John Friedman, Judy Giannakopoulou,Brennan Hughey, Lucía Santamaría Lara, Vuk Mandic, Chris Messenger, EvanOchsner, Larry Price, Jocelyn Read, Richard O’Shaughnessy, Bangalore Sathyapra-kash, Peter Saulson, Xavier Siemens, Amber Stuver, Patrick Sutton, Ruslan Vaulin,Alan Weinstein, Madeline White and Alan Wiseman for a great deal of assistance.

This work was supported by the National Science Foundation grants PHY-0701817, PHY-0600953 and PHY-0970074.

Calgary, June 2011 J.D.E.C.

XIII

List of Examples

Example 1.1 Coordinate acceleration in non-inertial frames of reference 4Example 1.2 Tidal acceleration 6Example 2.1 Transformation to polar coordinates 13Example 2.2 Volume element 14Example 2.3 How are directional derivatives like vectors? 15Example 2.4 Flat-space connection in polar coordinates 18Example 2.5 Flat-space connection in polar coordinates (again) 20Example 2.6 Equation of continuity 21Example 2.7 Vector commutation 23Example 2.8 Lie derivative 24Example 2.9 Curvature 27Example 2.10 Riemann tensor in a locally inertial frame 28Example 2.11 Geodesic deviation in the weak-field slow-motion limit 34Example 2.12 The Euler equations 37Example 2.13 Equations of motion for a point particle 39Example 2.14 Harmonic coordinates 42Example 3.1 Transformation from TT coordinates to a locally inertial frame 53Example 3.2 Wave equation for the Riemann tensor 55Example 3.3 Attenuation of gravitational waves 57Example 3.4 Degrees of freedom of a plane gravitational wave 60Example 3.5 Plus- and cross-polarization tensors 62Example 3.6 A resonant mass detector 65Example 3.7 Order of magnitude estimates of gravitational-wave amplitude 72Example 3.8 Fourier solution for the gravitational wave 73Example 3.9 Order of magnitude estimates of gravitational-wave luminosity 75Example 3.10 Gravitational-wave spectrum 76Example 3.11 Cross-section of a resonant mass detector 77Example 3.12 Point particle in rotating reference frame 83Example 3.13 The Crab pulsar 84Example 3.14 Newtonian chirp 89Example 3.15 The Hulse–Taylor binary pulsar 90Example 4.1 Effective stress-energy tensor 100Example 4.2 Amplification of gravitational waves by inflation 122

XIV List of Examples

Example 4.3 Black hole ringdown radiation 129Example 4.4 Analogy with electromagnetism 135Example 4.5 The BSSN formulation 136Example 5.1 Blandford’s argument 156Example 5.2 Rate of binary neutron star coalescences in the Galaxy 159Example 5.3 Chandrasekhar mass 166Example 6.1 Stokes relations 204Example 6.2 Dielectric mirror 205Example 6.3 Anti-resonant Fabry–Pérot cavity 209Example 6.4 Michelson interferometer gravitational-wave detector 212Example 6.5 Radio-frequency readout 219Example 6.6 Standard quantum limit 228Example 6.7 Derivation of the fluctuation–dissipation theorem 232Example 6.8 Coupled oscillators 261Example 7.1 Shot noise 272Example 7.2 Unknown amplitude 278Example 7.3 Sensitivity of a matched filter gravitational-wave search 280Example 7.4 Unknown phase 284Example 7.5 Measurement accuracy of signal amplitude and phase 288Example 7.6 Systematic error in estimate of signal amplitude 289Example 7.7 Frequentist upper limits 292Example 7.8 Time-frequency excess-power statistic 295Example 7.9 Nullspace of two co-aligned, co-located detectors 302Example 7.10 Nullspace of three non-aligned detectors 302Example 7.11 Sensitivity of the known-pulsar search 315Example 7.12 Horizon distance and range 322Example 7.13 Overlap reduction function in the long-wavelength limit 336Example 7.14 Hellings–Downs curve 337Example 7.15 Sensitivity of a stochastic background search 343Example A.1 Antenna response beam patterns for interferometer detectors 358

1

Introduction

This work is intended both as a textbook for an introductory course on gravitational-wave astronomy and as a basic reference on most aspects in this field of research.

As part of the syllabus of a course on gravitational waves, this book could be usedto follow a course on General Relativity (in which case, the first chapter could begreatly abbreviated), or as an introductory graduate course (in which case the firstchapter is required reading for what follows). Not all material would be covered ina single semester.

Within the text we include examples that elucidate a particular point describedin the main text or give additional detail beyond that covered in the body. At theend of each chapter we provide a short reference section that contains suggestedfurther reading. We have not attempted to provide a complete list of work in thefield, as one might have in a review article; rather we provide references to seminalpapers, to works of particular pedagogic value, and to review articles that will pro-vide the necessary background for researchers. Each chapter also has a selection ofproblems.

Please see http://www.lsc-group.phys.uwm.edu/~jolien for an errata for thisbook. If you find errors that are not currently noted in the errata, please notifyjolien@uwm.edu.

Conventions

We use bold sans-serif letters such as T and u to represent generic tensors andspacetime vectors, and italic bold letters such as v to represent purely spatial vec-tors. When writing the components of such objects, we use Greek letters for theindices for tensors on spacetime, Tα� and uα , while we use Latin letters for theindices for spatial vectors or matrices, for example v i and Mi j . Spacetime in-dices normally run over four values, so α 2 f0, 1, 2, 3g, while spatial indices nor-mally run over three values, i 2 f1, 2, 3g, unless otherwise specified. We em-ploy the Einstein summation convention where there is an implied sum over re-peated indices (known as dummy indices), so that Tαμ uμ D P3μD0 Tαμ uμ andMi j v j D

P3j D1 Mi j v j . In these examples, the indices α and i are not contracted

Gravitational-Wave Physics and Astronomy, First Edition. Jolien D. E. Creighton, Warren G. Anderson.© 2011 WILEY-VCH Verlag GmbH & Co. KGaA. Published 2011 by WILEY-VCH Verlag GmbH & Co. KGaA.

2 Introduction

and are called free indices (that is, these are actually four equations in the first caseand three equations in the second case since α can have the values 0, 1, 2, or 3,while i can have the values 1, 2, or 3).

We distinguish between the covariant derivative, rα , and the three-space gradi-ent operator

Δ

, which is the operator @/@x i in Cartesian coordinates. The Lapla-cian is

Δ2 D @2/@x2 C @2/@y 2 C @2/@z2 in Cartesian coordinates, and the flat-spaced’Alembertian operator is � D �c�2@2/@t2 C Δ2 in Cartesian coordinates.

Our spacetime sign convention is �, C, C, C so that flat spacetime in Cartesiancoordinates has the line element d s2 D �c2 d t2 C dx2 C d y 2 C dz2. The signconventions of common tensors follow that of Misner et al. (1973) and Wald (1984).

The Fourier transform of some time series x (t) is used to find the frequency seriesQx ( f ) according to

Qx ( f ) D1Z

�1x (t)e�2π i f t d t , (0.1)

while

x (t) D1Z

�1Qx ( f )e2π i f t d f (0.2)

is the inverse Fourier transform.

References

Misner, C.W., Thorne, K.S. and Wheeler, J.A.(1973) Gravitation, Freeman, San Francisco.

Wald, R.M. (1984) General Relativity, Universi-ty of Chicago Press.

3

1Prologue

1.1Tides in Newton’s Gravity

A brief review of Newtonian gravity is useful not only as a limit of weak-field rela-tivistic gravity, but also as a reminder of the principles upon which general relativitywas formulated. Newtonian gravity is conveniently formulated in a fixed rectilinearcoordinate system in terms of an absolute time coordinate. In such coordinates asthese, Newton’s laws of motion and gravitation describe the motion of a body ofmass m falling freely about another body of mass M by the force

F D m d2x

d t2D � G M mkx � x 0k3 (x � x

0) , (1.1)

where x is the position of the body with mass m, x 0 is the position of the body withmass M, t is the absolute time coordinate, and G ' 6.673 � 10�11 m3 kg�1 s�2 isNewton’s gravitational constant. Famously, the quantity m cancels and

d2xd t2

D � G Mkx � x 0k3 (x � x0) . (1.2)

If there is a continuous distribution of matter then we can sum up all contributionsto the acceleration from all pieces of the distribution to obtain

d2xd t2

D �GZ

body

x � x 0kx � x 0k3 �(x

0)d3x 0 D Δ

264G Zbody

�(x 0)kx � x 0k d

3x 0

375 , (1.3)where � is the mass distribution (density) and

Δ

is the gradient operator in x .Therefore, the acceleration of the body (with respect to the Newtonian system ofrectilinear coordinates) is

a D d2x

d t2D � ΔΦ (x) , (1.4)

where

Φ (x) WD �GZ

body

�(x 0)kx � x 0k d

3x 0 (1.5)

Gravitational-Wave Physics and Astronomy, First Edition. Jolien D. E. Creighton, Warren G. Anderson.© 2011 WILEY-VCH Verlag GmbH & Co. KGaA. Published 2011 by WILEY-VCH Verlag GmbH & Co. KGaA.

4 1 Prologue

is the Newtonian potential. The Newtonian potential satisfies the Poisson equation

Δ2Φ (x) D �GZ

�(x 0)

Δ2 1kx � x 0k d

3x 0 D 4πG�(x ) , (1.6)

where we have used

Δ2 1kx � x 0k D �4πδ(x � x

0) . (1.7)

Because the mass of the falling body does not enter into the equations of motion,any two bodies will fall the same way. If you can only see nearby free-falling bodies,you cannot tell whether you’re falling or not. You feel the same if you are freelyfalling toward some massive object as you would if you were in no gravitationalfield whatsoever. The gravitational acceleration describes the motion of the fallingbody with respect to the absolute Newtonian coordinates – but is there any way fora freely falling observer to know if they are accelerating or not?

Einstein codified the observation that freely falling objects fall together as a prin-ciple known as the equivalence principle: a freely falling observer could always setup a local (freely falling) frame in which all the laws of physics are the same as theywould be if that observer were not in a gravitational field. The coordinate acceler-ation a does not have any physical importance (as it does in Newtonian gravity)because one can always choose a frame of reference – freely falling with the ob-server – in which the observer is at rest.

Example 1.1 Coordinate acceleration in non-inertial frames of reference

An inertial frame of reference in Newtonian mechanics is any frame of referencethat can be related to the absolute Newtonian frame of reference by a uniformvelocity and a constant translation of position. That is, if x is the location of aparticle in one inertial frame of reference, then another inertial frame of referencewill have x 0 D x � x0 � v t for some constant vectors x0 and v . Inertial framespreserve the form of Newton’s second law since a0 D d2x 0/d t2 D d2x/d t2 D a.

In non-Cartesian coordinates, however, the form of the coordinate acceleration isdifferent. For example, for a two-dimensional system we could express the locationof a particle in polar coordinates r D (x2 C y 2)1/2 and φ D arctan(y/x ). In thesecoordinates, the coordinate velocity of a particle is given by dr/d t D v � e r anddφ/d t D r�1v � eφ where e r and eφ are unit vectors in the r- and φ-directions,and the equations of motion for the particle are Fr D m[d2r/d t2 C r(dφ/d t)2]and Fφ D m[d2φ/d t2 C 2r�1(dr/d t)(dφ/d t)]. Even when there is no force on theparticle, F D 0, there is still a coordinate acceleration in that d2r/d t2 and d2φ/d t2do not vanish except for purely radial motion. This merely arises because of thechoice of non-Cartesian coordinates – the geometrical form of Newton’s secondlaw, F D ma still holds.

A non-inertial frame is a frame that is accelerating relative to an inertial frame.A common example is a uniformly rotating reference frame with angular velocityvector ω. In such a reference frame, Newton’s second law has the form F D ma C

1.1 Tides in Newton’s Gravity 5

mω � (ω � r) C 2mω � v where the two additional terms, the centrifugal force,mω � (ω � r) and the Coriolis force, 2mω � v , arise because the frame of referenceis non-inertial. These are known as fictitious forces.

A freely falling frame of reference in Newtonian theory is a non-inertial frame ofreference because it is accelerating relative to the absolute set of Newtonian coor-dinates. The following coordinate transformation relates a freely falling frame ofreference (primed coordinates) at point x0 with the absolute Newtonian coordi-nates (unprimed): x 0 D x � x0 � 12 g t2, where g D �

Δ

Φ (x0) is a constant. It isstraightforward to see that a0 D d2x 0/d t2 D � Δ[Φ (x) � Φ (x0)] which vanishes atpoint x0.

In fact, there is a way to tell if you are falling. If there is another object that issome small distance away from you then its acceleration will be slightly different.Suppose � is the vector pointing from you to the other object. The acceleration ofthat object is

a(x C � ) D a(x ) C (� � Δ)a(x ) C O(�2) (1.8)and so the relative acceleration or tidal acceleration is

Δai D �� j @2Φ

@x i@x jD �Ei j � j , (1.9)

where

Ei j WD @2 Φ

@x i@x j(1.10)

is known as the tidal tensor field. The tidal acceleration is not really local since itdepends on the separation � between falling bodies. The tidal field, however, is alocal quantity, and it encodes the presence of the gravitational field. We will see laterthat in General Relativity, the tidal field is a measure of the spacetime curvature.

In the above expressions, the indices i and j run over the three spatial coordinatesfx1, x2, x3g or equivalently fx , y , zg and � i is the ith component of the vector � .(The three components of the vector are �1, �2 and �3 so we would write � i D[�1, �2, �3].) The tidal field is a rank-2 tensor having nine components: E11, E12, E13,E21, E22, E23, E31, E32 and E33. It is symmetric: E12 D E21, E13 D E31 and E23 D E32,or, more concisely, Ei j D E j i . Einstein’s summation convention is being used here:there is an implicit summation over repeated indices. That is, the expression

Ei j � j

is short-hand for3X

j D1Ei j � j D Ei1�1 C Ei2�2 C Ei3�3 .

For example, if two objects are separated in the x3- or z-direction, so that �1 and�2 both vanish, then the three components of the tidal acceleration are

Δa1 D �E13�3 , Δa2 D �E23�3 , and Δa3 D �E33�3 .

6 1 Prologue

Example 1.2 Tidal acceleration

Consider a body falling toward the Earth. The Newtonian potential is

Φ D � G M˚(x2 C y 2 C z2)1/2 . (1.11)

The tidal field component E11 is

E11 D @2 Φ

@x2D �G M˚

�3

x2

(x2 C y 2 C z2)5/2 �1

(x2 C y 2 C z2)3/2�

, (1.12)

the tidal field component E12 is

E12 D @2Φ

@x@yD �G M˚

�3

x y(x2 C y 2 C z2)5/2

�, (1.13)

and so forth. The components can be written concisely as

Ei j D � G M˚r5�3xi x j � δ i j r2

�, (1.14)

where r D (x2 C y 2 C z2)1/2 and δ i j is the Kronecker delta,

δ i j WD(

1 i D j0 i ¤ j , (1.15)

and so xi D δ i j x j .Suppose that a reference body is on the z-axis at a distance r D z from the centre

of the Earth. Then the tidal tensor is

Ei j D G M˚r3

241 0 00 1 00 0 �2

35 . (1.16)Consider a nearby second body that is also on the z-axis, a distance Δz farther

from the centre of the Earth. The relative tidal acceleration of this body is

Δai D �Ei j � j D �Ei3Δz . (1.17)The only non-vanishing component is the z-component:

Δa3 D 2 G M˚r3 Δz . (1.18)A third body is next to the reference body, lying a small distance Δx away on the

x-axis. The relative tidal acceleration of this body is

Δai D �Ei j � j D �Ei1Δx (1.19)and the only non-vanishing component is the x-component:

Δa1 D � G M˚r3 Δx . (1.20)Notice that a collection of freely falling objects will be pulled apart along the

direction in which they are falling while being squeezed together in the orthogonaldirections.

1.1 Tides in Newton’s Gravity 7

Unlike the coordinate acceleration, the tidal acceleration has intrinsic physicalmeaning. We witness ocean tides caused by the Moon and the Sun. These tidesdissipate energy on the Earth. That is, tidal forces can do work. To compute thework, consider an extended body (say, the Earth) moving within a tidal field pro-duced by another body (say, the Moon). An element of the extended body, locatedat a position x and having mass �(x )d3x , experiences a tidal force

Fi D �Ei j x j �(x )d3x . (1.21)If the element is moving through the tidal field with velocity v then there is anamount Fi v i of work per unit time done on that element. Summing over all ele-ments that comprise the body yields the total amount of tidal work:

d Wdt

D �Z

body

Ei j v i x j �(x )d3x

D � 12Ei j

dd t

Zbody

x i x j �(x )d3x

D � 12Ei j

d I i j

d t, (1.22)

where

I i j WDZ

body

x i x j �(x )d3x (1.23)

is the quadrupole tensor. Note that this tensor is closely related to the moment ofinertia tensor

Ii j WD�δ i j δk l � δ i k δ j l

�I k l D

Zbody

�r2δ i j � xi x j

��(x )d3x (1.24)

and also to the (traceless) reduced quadrupole tensor

I i j WD�

δ i k δ j l � 13 δ i j δk l

I k l DZ

body

�xi x j � 13 r

2δ i j

�(x )d3x . (1.25)

Here r2 D kxk2 D δ i j x i x j .Tidal work can also be performed by a dynamical system with a time-changing

tidal field Ei j (t). The work performed by such a system on another body with aquadrupole tensor I i j is found by integrating Eq. (1.22) by parts:

W D � 12Ei j I i j

ˇ̌T0 C

12

TZ0

dEi jd t

I i j d t . (1.26)

The first term is bounded, while the second term secularly increases with time andrepresents a transfer of energy from the dynamical system that is producing the

8 1 Prologue

time-changing tidal field to the other body. For example, the source of the time-changing tidal field might be a rotating dumbbell or a binary system of two stars inorbit about each other. Over a long time (large T ) the secularly growing term willdominate, and we can write the work done by the dynamical source on the bodywith moment of inertia tensor I i j as

d Wdt

� 12

dEi jd t

I i j . (1.27)

1.2Relativity

The special theory of relativity postulates that there is no preferred inertial frame:local measurements of physical quantities are the same no matter which inertialframe the measurement is made in. This is the principle of relativity. In particular,measurements of the speed of light in any inertial frame will always yield the samevalue, c WD 299 792 458 m s�1. The consequence of this is that the Newtonian sep-aration of space and time must be abandoned. Consider a spaceship travelling at aconstant speed v in the x-direction relative to the Earth (see Figure 1.1). Within thespaceship, an experimental determination of the speed of light is made in which aphoton is emitted from a source in the y-direction, reflected by a mirror a distance12 Δy away from the source, and received back at the source. The time-of-flight Δτis measured and the speed of light c D Δy/Δτ is computed. For an observer onthe Earth, however, the distance travelled by the photon is [(Δx )2C(Δy )2]1/2, whereΔx D vΔ t and Δ t is the amount of time the observer on the Earth determines ittakes the photon to travel from the emitter to the receiver. Since the observer onEarth must measure the same speed of light, c D [(Δx )2 C (Δy )2]1/2/Δ t, we seethat

c2 D (Δx )2 C (Δy )2(Δ t)2

D (Δx )2 C (cΔτ)2(Δ t)2

, (1.28)

where we have used Δy D cΔτ, and so

c2(Δτ)2 D c2(Δ t)2 � (Δx )2 . (1.29)

y

x = v t

t = t t = tt = 0

Figure 1.1 A measurement of the speed oflight, performed in a rocket moving at speedv relative to the Earth, as seen by an observeron the Earth. A flash of light is produced att D 0. The light travels a vertical distance

12 Δy , reflects off of the mirror and returnsto the source after a time Δ t (as measuredby the observer on the Earth). The rocket hasmoved a horizontal distance Δx D vΔ t inthis time.

1.2 Relativity 9

The usual time dilation formula Δ t D γ Δτ, where γ D (1 � v2/c2)�1/2 is theLorentz factor, follows by setting Δx D vΔ t. This relationship between how timeis measured within the moving frame of the spaceship to how time is measuredon Earth is not particular to the experiment with the photon: time really does movedifferently in the different inertial frames of reference.

Equation (1.29) relates the amount of time Δτ between two events, as recorded inan inertial frame in which the two events occur at the same spatial position (whichis known as the proper time between the two events), to the amount of time Δ tbetween the same two events as seen in an inertial frame in which the two eventsare separated by a spatial distance Δx . Since the notion of an absolute time is lost inspecial relativity, we understand time to simply be a new coordinate which, alongwith the three spatial coordinates, depends on the frame of reference. Together,the time and space coordinates are used to identify points (or events) on a four-dimensional spacetime. For rectilinear coordinates in an inertial frame, we definean invariant interval (Δ s)2 between two points in spacetime, (t, x , y , z) and (t CΔ t, x C Δx , y C Δy , z C Δz), by

(Δ s)2 WD �c2(Δ t)2 C (Δx )2 C (Δy )2 C (Δz)2 , (1.30)

which has the same form as the Pythagorean theorem except for the factor of �c2in front of the square of the time interval. This equation is just a generalization ofEq. (1.29) with (Δ s)2 WD �c2(Δτ)2.

Special relativity is incompatible with Newtonian gravity because Newton’s lawof gravitation defines a force between two distant bodies in terms of their separa-tion at a given instant in time. However, in special relativity, there is no uniquenotion of simultaneity. In addition, different frames of reference will make differ-ent measurements of the Newtonian gravitational force, a result that is at odds withthe principle of relativity.

The general theory of relativity provides a description of gravity in terms of a curvedspacetime. This is discussed in Chapter 2. In general relativity, the inertial framesof reference are freely falling frames, and the principle of relativity is then takento hold in such frames of reference. Tidal acceleration is the physical manifesta-tion of gravitation, but measurement of a tidal field requires a somewhat extendedapparatus.

Of course, Newtonian gravity must be recovered in some limit of general relativ-ity: this limit is when G M/(c2R) � 1 and v/c � 1 where M is the characteristicmass of the system, R is the characteristic size of the system, and v is the charac-teristic speed of bodies in the system. And since in Newtonian gravity a changingtidal field is capable of producing work on distant bodies, this must be true in gen-eral relativity as well. This means that in order to ensure that energy is conserved,energy must be radiated from the gravitating system that is producing the chang-ing tidal field to the rest of the universe, because there is no way that the bodieson which the work is done can create an instantaneous reactive force on the grav-itating system – this would be incompatible with relativity. The radiation is calledgravitational radiation.

11

2A Brief Review of General Relativity

The intent of this chapter is to provide a brief review of General Relativity and tointroduce the concepts and notation that are required for the discussion of grav-itational waves in subsequent chapters. The review will not be comprehensive asthere are many excellent introductory texts on General Relativity: Hartle (2003) is aclear, physics-first introduction to the subject, and Schutz (2009) is another excel-lent text for a first course in General Relativity. The classic Misner et al. (1973) is acomplete reference book. Advanced texts include Wald (1984) and Weinberg (1972)which have very different approaches but are both essential reading.

The principle of relativity – a foundation of Einstein’s theory of Special Relativity –suggests that there is no preferred frame of reference or state of motion. Physicaltheory needs to be formulated in a manner in which physical quantities are invari-ant under a class of transformations known as Poincaré transformations. That is,physics is invariant under translations, rotations and boosts. Special relativity canbe elegantly formulated on a four-dimensional spacetime in which the three normalspatial dimensions and a time dimension are combined.

To describe relativistic gravity, Einstein extended the principle of relativity to anew principle, the principle of general covariance, which demands that there is nopreferred coordinate system at all. For example, a freely falling observer can alwaysconstruct a freely falling frame of reference and any physical experiment carriedout in that frame of reference must give the same results as a similar experimentcarried out by an observer who is not in any gravitational field whatsoever. Einsteindescribed gravity in terms of a curved spacetime in which particles naturally followthe straightest possible lines – not necessarily straight lines in some predeterminedcoordinate system – and the physical effects of gravity can then be understood interms of the curvature of spacetime. For example, the tidal field is related to thecurvature tensor. The curvature is produced, to some extent, by the masses inspacetime.

Gravitational-Wave Physics and Astronomy, First Edition. Jolien D. E. Creighton, Warren G. Anderson.© 2011 WILEY-VCH Verlag GmbH & Co. KGaA. Published 2011 by WILEY-VCH Verlag GmbH & Co. KGaA.

12 2 A Brief Review of General Relativity

2.1Differential Geometry

General relativity is formulated on a four-dimensional manifold – a four dimen-sional surface on which our physical theory is described. The manifold of generalrelativity is called spacetime because three of the dimensions correspond to theobserved three dimensions of space and the fourth dimension of the manifold cor-responds to what we perceive as time. The structure of the manifold can be quitecomplicated in principle, but for our purposes it is not necessary to consider gen-eral situations.

2.1.1Coordinates and Distances

Like the surface of the Earth, the manifold of spacetime can be covered with patch-es or charts on which coordinates can be constructed. The set of overlapping chartsthat covers all of spacetime is called an atlas. Unlike Newtonian theory, there is nointrinsically physical set of coordinates or charts. Physical theory in general rela-tivity is formulated in a covariant way so that the physical quantities are invariantunder changes of coordinates.

There is a particularly useful class of coordinate choices are called normal coordi-nates. Normal coordinates are the closest things to inertial coordinates in flat-space,and so the reference frame described by normal coordinates is called a locally iner-tial frame. Normal coordinates can typically be constructed over a region with a sizecomparable to the curvature scale of spacetime. We will use the fact that, becauseof the equivalence principle, we can always find normal coordinates in the vicinityof any spacetime point and, in these coordinates, much of our flat-space intuitionwill hold.

The distance between two points that are sufficiently close together is a geometricinvariant and so it is the same regardless of what set of coordinates are adopted. Thetwo points need to be close together so that there is a unique notion of what path istaken from one point to the other over which we construct the distance. Thereforewe write Pythagoras’ formula in its differential form: consider two points, P and Qthat are infinitesimally close together. These points are labelled by the coordinatesx αP and x

αQ respectively, and the infinitesimal coordinate difference between the

two points is dx α D x αQ � x αP . The squared distance between the two points, d s2,is computed by

d s2 D gμν(x α)dx μ dx ν , (2.1)where gμν(x α) is the metric tensor of spacetime, which is a function of spacetimecoordinates x α . Note that the index α runs over four values in a four-dimensionalspacetime, and by convention we take values to be f0, 1, 2, 3g so that x1, x2 and x3are the three spatial coordinates and x0 is the single time coordinate. The metricdetermines the distance between any two neighbouring points in spacetime andtherefore determines all of the geometry of the spacetime.

2.1 Differential Geometry 13

In a flat spacetime or Minkowski spacetime, we use the symbol ηα� for the met-ric. In the standard rectilinear coordinates the distance between any two points inMinkowski spacetime is

d s2 D ημν dx μ dx ν D �c2d t2Cdx2Cd y 2Cdz2 (rectilinear coordinates) .(2.2)

A transformation to a new set of coordinates is specified by the four functionsx 0α(x μ) relating the new primed coordinates with the original unprimed coordi-nates. Under this transformation,

dx μ D @xμ

@x 0αdx 0α , (2.3)

where x μ(x 0α) is the inverse transformation. Since the squared distance elementd s2 is invariant under such transformations,

d s2 D gμν dx μ dx ν D gμν @xμ

@x 0α@x ν

@x 0�dx 0α dx 0� D g0α� dx 0α dx 0� , (2.4)

where

g0α� D gμν@x μ

@x 0α@x ν

@x 0�. (2.5)

In fact, any physical quantity does not depend on the choice of the coordinate sys-tem; the freedom of coordinate redefinition x 0α(x μ) therefore represents the gaugefreedom of our geometric description of gravity, and coordinate transformations arealso gauge transformations.

For an infinitesimal coordinate transformation (or infinitesimal gauge transfor-mations) of the form

x α ! x 0α D x α C α(x μ) , (2.6)where is a displacement vector we see that

dx α ! dx 0α D dx α C @α

@x μdx μ (2.7)

and therefore

gα� ! g0α� D gα� � gαμ@ μ

@x �� gμ� @

μ

@x αC O( 2) . (2.8)

Example 2.1 Transformation to polar coordinates

Given the two-dimensional flat-space metric in rectilinear coordinates,

d s2 D gμν dx μ dx ν D dx2 C d y 2 , (2.9)

one can transform into polar coordinates r D (x2 C y 2)1/2 and φ D arctan(y/x ).The inverse transformation is x D r cos φ and y D r sin φ so dx D

14 2 A Brief Review of General Relativity

cos φdr � r sin φdφ and d y D sin φdr C r cos φdφ and hence

d s2 D dx2 C d y 2 D (cos φdr � r sin φdφ)2 C (sin φdr C r cos φdφ)2D dr2 C r2dφ2 D g0μν dx 0μ dx 0ν . (2.10)

Therefore we have g0r r(r, φ) D 1, g0φφ(r, φ) D r2, and g0r φ(r, φ) D g0φ r(r, φ) D 0.

Example 2.2 Volume element

Under the coordinate transformation of Eq. (2.3), the metric transforms accordingto Eq. (2.5). The metric is a 4 � 4 matrix whose determinant is related to the volumeelement of spacetime. To see this, we take the determinant of Eq. (2.5):

det g0 D detˇ̌̌̌

@(x)@(x0)

ˇ̌̌̌2det g , (2.11)

where J D @(x)/@(x0) is the Jacobian matrix J α� D @x α/@x 0� . Recall that the Jacobiandeterminant arises in a change of variables in integral calculus: under the coordinatetransformation x ! x0, the measure changes as

d4x 0 D detˇ̌̌̌@(x0)@(x)

ˇ̌̌̌d4x . (2.12)

Since we can always locally perform a coordinate transformation to a locally inertialCartesian frame in which the metric is g0α� D ηα� D diag[�c2, 1, 1, 1], which hasdeterminant det η D �c2, and for which the volume element is dV D cd4x 0 Dcd t0dx 0d y 0dz0, we see that

dV D cd4x 0 D c detˇ̌̌̌@(x0)@(x)

ˇ̌̌̌d4x D c

sdet gdet η

d4x

D (� det g)1/2d4x . (2.13)

Therefore j det gj1/2d4x is the volume element at a location in spacetime.As an example in two dimensions, consider the metric of two-dimensional flat-

space in polar coordinates that was found in Example 2.1: gα� D diag[1, r2]. Thevolume element is therefore (det g)1/2d2x D r d r dφ.

2.1.2Vectors

Geometric constructs such as vectors and tensors need to be generalized from theirnormal flat-space definition (e.g. a vector as going from one point in space to an-other) to a generalized definition that can be ported to curved manifolds.