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General Relativity: An Introduction for Physicists provides a clear mathematical
introduction to Einstein™s theory of general relativity. A wide range of applications
of the theory are included, with a concentration on its physical consequences.
After reviewing the basic concepts, the authors present a clear and intuitive
discussion of the mathematical background, including the necessary tools of tensor
calculus and differential geometry. These tools are used to develop the topic
of special relativity and to discuss electromagnetism in Minkowski spacetime.
Gravitation as spacetime curvature is then introduced and the field equations of
general relativity are derived. A wide range of applications to physical situations
follows, and the conclusion gives a brief discussion of classical field theory and
the derivation of general relativity from a variational principle.
Written for advanced undergraduate and graduate students, this approachable
textbook contains over 300 exercises to illuminate and extend the discussion in
the text.
Michael Hobson specialised in theoretical physics as an undergraduate at the
University of Cambridge and remained at the Cavendish Laboratory to complete
a Ph.D. in the physics of star-formation and radiative transfer. As a Research
Fellow at Trinity Hall, Cambridge, and later as an Advanced Fellow of the Particle
Physics and Astronomy Research Council, he developed an interest in cosmology,
in particular in the study of fluctuations in the cosmic microwave background
(CMB) radiation. He is currently a Reader in Astrophysics and Cosmology at the
Cavendish Laboratory, where he is the principal investigator for the Very Small
Array CMB interferometer. He is also joint project scientist for the Arcminute
Microkelvin Imager project and an associate of the European Space Agency
Planck Surveyor CMB satellite mission. In addition to observational and theo-
retical cosmology, his research interests also include Bayesian analysis methods
and theoretical optics and he has published over 100 research papers in a wide
range of areas. He is a Staff Fellow and Director of Studies in Natural Sciences
at Trinity Hall and enjoys an active role in the teaching of undergraduate physics
and mathematics. He is a co-author with Ken Riley and Stephen Bence of the
well-known undergraduate textbook Mathematical Methods for Physics and Engi-
neering (Cambridge, 1998; second edition, 2002; third edition to be published in
2006) and with Ken Riley of the Student™s Solutions Manual accompanying the
third edition.
George Efstathiou is Professor of Astrophysics and Director of the Institute
of Astronomy at the University of Cambridge. After studying physics as an
undergraduate at Keble College, Oxford, he gained his Ph.D. in astronomy from
Durham University. Following some post-doctoral research at the University of
California at Berkeley he returned to work in the UK at the Institute of Astronomy,
Cambridge, where he was appointed Assistant Director of Research in 1987.
He returned to the Department of Physics at Oxford as Savilian Professor of
Astronomy and Head of Astrophysics, before taking on his current posts at the
Institute of Astronomy in 1997 and 2004 respectively. He is a Fellow of the Royal
Society and the recipient of several awards, including the Maxwell Medal and
Prize of the Institute of Physics in 1990 and the Heineman Prize for Astronomy
of the American Astronomical Society in 2005.
Anthony Lasenby is Professor of Astrophysics and Cosmology at the University
of Cambridge and is currently Head of the Astrophysics Group and the Mullard
Radio Astronomy Observatory in the Cavendish Laboratory, as well as being a
Deputy Head of the Laboratory. He began his astronomical career with a Ph.D.
at Jodrell Bank, specializing in the cosmic microwave background, which has
remained a major subject of his research. After a brief period at the National Radio
Astronomy Observatory in America, he moved from Manchester to Cambridge in
1984 and has been at the Cavendish since then. He is the author or co-author of
over 200 papers spanning a wide range of fields and is the co-author of Geometric
Algebra for Physicists (Cambridge, 2003) with Chris Doran.
General Relativity
An Introduction for Physicists

and A . N . L A S E N B Y
cambridge university press
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo

Cambridge University Press
The Edinburgh Building, Cambridge cb2 2ru, UK
Published in the United States of America by Cambridge University Press, New York
Information on this title: www.cambridge.org/9780521829519

© M. P. Hobson, G. P. Efstathiou and A. N. Lasenby 2006

This publication is in copyright. Subject to statutory exception and to the provision of
relevant collective licensing agreements, no reproduction of any part may take place
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To our families

page xv

1 The spacetime of special relativity 1
1.1 Inertial frames and the principle of relativity 1
1.2 Newtonian geometry of space and time 3
1.3 The spacetime geometry of special relativity 3
1.4 Lorentz transformations as four-dimensional ˜rotations™ 5
1.5 The interval and the lightcone 6
1.6 Spacetime diagrams 8
1.7 Length contraction and time dilation 10
1.8 Invariant hyperbolae 11
1.9 The Minkowski spacetime line element 12
1.10 Particle worldlines and proper time 14
1.11 The Doppler effect 16
1.12 Addition of velocities in special relativity 18
1.13 Acceleration in special relativity 19
1.14 Event horizons in special relativity 21
Appendix 1A: Einstein™s route to special relativity 22
Exercises 24
2 Manifolds and coordinates 26
2.1 The concept of a manifold 26
2.2 Coordinates 27
2.3 Curves and surfaces 27
2.4 Coordinate transformations 28
2.5 Summation convention 30
2.6 Geometry of manifolds 31
2.7 Riemannian geometry 32
2.8 Intrinsic and extrinsic geometry 33

viii Contents

2.9 Examples of non-Euclidean geometry 36
2.10 Lengths, areas and volumes 38
2.11 Local Cartesian coordinates 42
2.12 Tangent spaces to manifolds 44
2.13 Pseudo-Riemannian manifolds 45
2.14 Integration over general submanifolds 47
2.15 Topology of manifolds 49
Exercises 50

3 Vector calculus on manifolds 53
3.1 Scalar fields on manifolds 53
3.2 Vector fields on manifolds 54
3.3 Tangent vector to a curve 55
3.4 Basis vectors 56
3.5 Raising and lowering vector indices 59
3.6 Basis vectors and coordinate transformations 60
3.7 Coordinate-independent properties of vectors 61
3.8 Derivatives of basis vectors and the affine connection 62
3.9 Transformation properties of the affine connection 64
3.10 Relationship of the connection and the metric 65
3.11 Local geodesic and Cartesian coordinates 67
3.12 Covariant derivative of a vector 68
3.13 Vector operators in component form 70
3.14 Intrinsic derivative of a vector along a curve 71
3.15 Parallel transport 73
3.16 Null curves, non-null curves and affine parameters 75
3.17 Geodesics 76
3.18 Stationary property of non-null geodesics 77
3.19 Lagrangian procedure for geodesics 78
3.20 Alternative form of the geodesic equations 81
Appendix 3A: Vectors as directional derivatives 81
Appendix 3B: Polar coordinates in a plane 82
Appendix 3C: Calculus of variations 87
Exercises 88

4 Tensor calculus on manifolds 92
4.1 Tensor fields on manifolds 92
4.2 Components of tensors 93
4.3 Symmetries of tensors 94
4.4 The metric tensor 96
4.5 Raising and lowering tensor indices 97

4.6 Mapping tensors into tensors 97
4.7 Elementary operations with tensors 98
4.8 Tensors as geometrical objects 100
4.9 Tensors and coordinate transformations 101
4.10 Tensor equations 102
4.11 The quotient theorem 103
4.12 Covariant derivative of a tensor 104
4.13 Intrinsic derivative of a tensor along a curve 107
Exercises 108
5 Special relativity revisited 111
5.1 Minkowski spacetime in Cartesian coordinates 111
5.2 Lorentz transformations 112
5.3 Cartesian basis vectors 113
5.4 Four-vectors and the lightcone 115
5.5 Four-vectors and Lorentz transformations 116
5.6 Four-velocity 116
5.7 Four-momentum of a massive particle 118
5.8 Four-momentum of a photon 119
5.9 The Doppler effect and relativistic aberration 120
5.10 Relativistic mechanics 122
5.11 Free particles 123
5.12 Relativistic collisions and Compton scattering 123
5.13 Accelerating observers 125
5.14 Minkowski spacetime in arbitrary coordinates 128
Exercises 131
6 Electromagnetism 135
6.1 The electromagnetic force on a moving charge 135
6.2 The 4-current density 136
6.3 The electromagnetic field equations 138
6.4 Electromagnetism in the Lorenz gauge 139
6.5 Electric and magnetic fields in inertial frames 141
6.6 Electromagnetism in arbitrary coordinates 142
6.7 Equation of motion for a charged particle 144
Exercises 145
7 The equivalence principle and spacetime curvature 147
7.1 Newtonian gravity 147
7.2 The equivalence principle 148
7.3 Gravity as spacetime curvature 149
7.4 Local inertial coordinates 151
x Contents

7.5 Observers in a curved spacetime 152
7.6 Weak gravitational fields and the Newtonian limit 153
7.7 Electromagnetism in a curved spacetime 155
7.8 Intrinsic curvature of a manifold 157
7.9 The curvature tensor 158
7.10 Properties of the curvature tensor 159
7.11 The Ricci tensor and curvature scalar 161
7.12 Curvature and parallel transport 163
7.13 Curvature and geodesic deviation 165
7.14 Tidal forces in a curved spacetime 167
Appendix 7A: The surface of a sphere 170
Exercises 172

8 The gravitational field equations 176
8.1 The energy“momentum tensor 176
8.2 The energy“momentum tensor of a perfect fluid 178
8.3 Conservation of energy and momentum for a perfect fluid 179
8.4 The Einstein equations 181
8.5 The Einstein equations in empty space 183
8.6 The weak-field limit of the Einstein equations 184
8.7 The cosmological-constant term 185
8.8 Geodesic motion from the Einstein equations 188
8.9 Concluding remarks 190
Appendix 8A: Alternative relativistic theories of gravity 191
Appendix 8B: Sign conventions 193
Exercises 193

9 The Schwarzschild geometry 196
9.1 The general static isotropic metric 196
9.2 Solution of the empty-space field equations 198
9.3 Birkhoff™s theorem 202
9.4 Gravitational redshift for a fixed emitter and receiver 202
9.5 Geodesics in the Schwarzschild geometry 205
9.6 Trajectories of massive particles 207
9.7 Radial motion of massive particles 209
9.8 Circular motion of massive particles 212
9.9 Stability of massive particle orbits 213
9.10 Trajectories of photons 217
9.11 Radial motion of photons 218
9.12 Circular motion of photons 219
9.13 Stability of photon orbits 220

Appendix 9A: General approach to gravitational redshifts 221
Exercises 224
10 Experimental tests of general relativity 230
10.1 Precession of planetary orbits 230
10.2 The bending of light 233
10.3 Radar echoes 236
10.4 Accretion discs around compact objects 240
10.5 The geodesic precession of gyroscopes 244
Exercises 246
11 Schwarzschild black holes 248
11.1 The characterisation of coordinates 248
11.2 Singularities in the Schwarzschild metric 249
11.3 Radial photon worldlines in Schwarzschild coordinates 251
11.4 Radial particle worldlines in Schwarzschild coordinates 252
11.5 Eddington“Finkelstein coordinates 254
11.6 Gravitational collapse and black-hole formation 259
11.7 Spherically symmetric collapse of dust 260
11.8 Tidal forces near a black hole 264
11.9 Kruskal coordinates 266
11.10 Wormholes and the Einstein“Rosen bridge 271
11.11 The Hawking effect 274
Appendix 11A: Compact binary systems 277
Appendix 11B: Supermassive black holes 279
Appendix 11C: Conformal flatness of two-dimensional Riemannian
manifolds 282
Exercises 283
12 Further spherically symmetric geometries 288
12.1 The form of the metric for a stellar interior 288
12.2 The relativistic equations of stellar structure 292
12.3 The Schwarzschild constant-density interior solution 294
12.4 Buchdahl™s theorem 296
12.5 The metric outside a spherically symmetric
charged mass 296
12.6 The Reissner“Nordström geometry: charged
black holes 300
12.7 Radial photon trajectories in the RN geometry 302
12.8 Radial massive particle trajectories
in the RN geometry 304
Exercises 305
xii Contents

13 The Kerr geometry 310
13.1 The general stationary axisymmetric metric 310
13.2 The dragging of inertial frames 312
13.3 Stationary limit surfaces 314
13.4 Event horizons 315
13.5 The Kerr metric 317
13.6 Limits of the Kerr metric 319
13.7 The Kerr“Schild form of the metric 321
13.8 The structure of a Kerr black hole 322
13.9 The Penrose process 327
13.10 Geodesics in the equatorial plane 330
13.11 Equatorial trajectories of massive particles 332
13.12 Equatorial motion of massive particles with
zero angular momentum 333
13.13 Equatorial circular motion of massive particles 335
13.14 Stability of equatorial massive particle circular orbits 337
13.15 Equatorial trajectories of photons 338
13.16 Equatorial principal photon geodesics 339
13.17 Equatorial circular motion of photons 341
13.18 Stability of equatorial photon orbits 342
13.19 Eddington“Finkelstein coordinates 344
13.20 The slow-rotation limit and gyroscope precession 347
Exercises 350
14 The Friedmann“Robertson“Walker geometry 355
14.1 The cosmological principle 355
14.2 Slicing and threading spacetime 356
14.3 Synchronous coordinates 357
14.4 Homogeneity and isotropy of the universe 358
14.5 The maximally symmetric 3-space 359
14.6 The Friedmann“Robertson“Walker metric 362
14.7 Geometric properties of the FRW metric 362
14.8 Geodesics in the FRW metric 365
14.9 The cosmological redshift 367
14.10 The Hubble and deceleration parameters 368
14.11 Distances in the FRW geometry 371
14.12 Volumes and number densities in the FRW geometry 374
14.13 The cosmological field equations 376
14.14 Equation of motion for the cosmological fluid 379
14.15 Multiple-component cosmological fluid 381
Exercises 381

15 Cosmological models 386
15.1 Components of the cosmological fluid 386
15.2 Cosmological parameters 390
15.3 The cosmological field equations 392
15.4 General dynamical behaviour of the universe 393
15.5 Evolution of the scale factor 397
15.6 Analytical cosmological models 400
15.7 Look-back time and the age of the universe 408
15.8 The distance“redshift relation 411
15.9 The volume“redshift relation 413
15.10 Evolution of the density parameters 415
15.11 Evolution of the spatial curvature 417
15.12 The particle horizon, event horizon and Hubble distance 418
Exercises 421

16 Inflationary cosmology 428
16.1 Definition of inflation 428
16.2 Scalar fields and phase transitions in the very early universe 430
16.3 A scalar field as a cosmological fluid 431
16.4 An inflationary epoch 433
16.5 The slow-roll approximation 434
16.6 Ending inflation 435
16.7 The amount of inflation 435
16.8 Starting inflation 437
16.9 ˜New™ inflation 438
16.10 Chaotic inflation 440
16.11 Stochastic inflation 441
16.12 Perturbations from inflation 442
16.13 Classical evolution of scalar-field perturbations 442
16.14 Gauge invariance and curvature perturbations 446
16.15 Classical evolution of curvature perturbations 449
16.16 Initial conditions and normalisation of curvature perturbations 452
16.17 Power spectrum of curvature perturbations 456
16.18 Power spectrum of matter-density perturbations 458
16.19 Comparison of theory and observation 459
Exercises 462

17 Linearised general relativity 467
17.1 The weak-field metric 467
17.2 The linearised gravitational field equations 470
17.3 Linearised gravity in the Lorenz gauge 472
xiv Contents

17.4 General properties of the linearised field equations 473
17.5 Solution of the linearised field equations in vacuo 474
17.6 General solution of the linearised field equations 475
17.7 Multipole expansion of the general solution 480
17.8 The compact-source approximation 481
17.9 Stationary sources 483
17.10 Static sources and the Newtonian limit 485
17.11 The energy“momentum of the gravitational field 486
Appendix 17A: The Einstein“Maxwell formulation of linearised gravity 490
Exercises 493
18 Gravitational waves 498
18.1 Plane gravitational waves and polarisation states 498
18.2 Analogy between gravitational and electromagnetic waves 501
18.3 Transforming to the transverse-traceless gauge 502
18.4 The effect of a gravitational wave on free particles 504
18.5 The generation of gravitational waves 507
18.6 Energy flow in gravitational waves 511
18.7 Energy loss due to gravitational-wave emission 513
18.8 Spin-up of binary systems: the binary pulsar PSR B1913 + 16 516
18.9 The detection of gravitational waves 517
Exercises 520
19 A variational approach to general relativity 524
19.1 Hamilton™s principle in Newtonian mechanics 524
19.2 Classical field theory and the action 527
19.3 Euler“Lagrange equations 529
19.4 Alternative form of the Euler“Lagrange equations 531
19.5 Equivalent actions 533
19.6 Field theory of a real scalar field 534
19.7 Electromagnetism from a variational principle 536
19.8 The Einstein“Hilbert action and general relativity in vacuo 539
19.9 An equivalent action for general relativity in vacuo 542
19.10 The Palatini approach for general relativity in vacuo 543
19.11 General relativity in the presence of matter 545
19.12 The dynamical energy“momentum tensor 546
Exercises 549


General relativity is one of the cornerstones of classical physics, providing a
synthesis of special relativity and gravitation, and is central to our understanding
of many areas of astrophysics and cosmology. This book is intended to give an
introduction to this important subject, suitable for a one-term course for advanced
undergraduate or beginning graduate students in physics or in related disciplines
such as astrophysics and applied mathematics. Some of the later chapters should
also provide a useful reference for professionals in the fields of astrophysics and
It is assumed that the reader has already been exposed to special relativity and
Newtonian gravitation at a level typical of early-stage university physics courses.
Nevertheless, a summary of special relativity from first principles is given in
Chapter 1, and a brief discussion of Newtonian gravity is presented in Chapter 7.
No previous experience of 4-vector methods is assumed. Some background in
electromagnetism will prove useful, as will some experience of standard vector
calculus methods in three-dimensional Euclidean space. The overall level of math-
ematical expertise assumed is that of a typical university mathematical methods
The book begins with a review of the basic concepts underlying special rela-
tivity in Chapter 1. The subject is introduced in a way that encourages from the
outset a geometrical and transparently four-dimensional viewpoint, which lays the
conceptual foundations for discussion of the more complicated spacetime geome-
tries encountered later in general relativity. In Chapters 2“4 we then present a
mini-course in basic differential geometry, beginning with the introduction of
manifolds, coordinates and non-Euclidean geometry in Chapter 2. The topic of
vector calculus on manifolds is developed in Chapter 3, and these ideas are
extended to general tensors in Chapter 4. These necessary mathematical prelimi-
naries are presented in such a way as to make them accessible to physics students
with a background in standard vector calculus. A reasonable level of mathematical

xvi Preface

rigour has been maintained throughout, albeit accompanied by the occasional
appeal to geometric intuition. The mathematical tools thus developed are then
illustrated in Chapter 5 by re-examining the familiar topic of special relativity in a
more formal manner, through the use of tensor calculus in Minkowski spacetime.
These methods are further illustrated in Chapter 6, in which electromagnetism is
described as a field theory in Minkowski spacetime, serving in some respects as a
˜prototype™ for the later discussion of gravitation. In Chapter 7, the incompatibility
of special relativity and Newtonian gravitation is presented and the equivalence
principle is introduced. This leads naturally to a discussion of spacetime curvature
and the associated mathematics. The field equations of general relativity are then
derived in Chapter 8, and a discussion of their general properties is presented.
The physical consequences of general relativity in a wide variety of astrophys-
ical and cosmological applications are discussed in Chapters 9“18. In particular,
the Schwarzschild geometry is derived in Chapter 9 and used to discuss the physics
outside a massive spherical body. Classic experimental tests of general relativity
based on the exterior Schwarzschild geometry are presented in Chapter 10. The
interior Schwarzschild geometry and non-rotating black holes are discussed in
Chapter 11, together with a brief mention of Kruskal coordinates and wormholes.
In Chapter 12 we introduce two non-vacuum spherically symmetric geometries
with a discussion of relativistic stars and charged black holes. Rotating objects are
discussed in Chapter 13, including an extensive discussion of the Kerr solution. In
Chapters 14“16 we describe the application of general relativity to cosmology and
present a discussion of the Friedmann“Robertson“Walker geometry, cosmologi-
cal models and the theory of inflation, including the generation of perturbations
in the early universe. In Chapter 17 we describe linearised gravitation and weak
gravitational fields, in particular drawing analogies with the theory of electromag-
netism. The equations of linearised gravitation are then applied to the generation,
propagation and detection of weak gravitational waves in Chapter 18. The book
concludes in Chapter 19 with a brief discussion of classical field theory and the
derivation of the field equations of electromagnetism and general relativity from
variational principles.
Each chapter concludes with a number of exercises that are intended to illumi-
nate and extend the discussion in the main text. It is strongly recommended that
the reader attempt as many of these exercises as time permits, as they should give
ample opportunity to test his or her understanding. Occasionally chapters have
appendices containing material that is not central to the development presented in
the main text, but may nevertheless be of interest to the reader. Some appendices
provide historical context, some discuss current astronomical observations and
some give detailed mathematical derivations that might otherwise interrupt the
flow of the main text.

With regard to the presentation of the mathematics, it has to be accepted
that equations containing partial and covariant derivatives could be written more
compactly by using the comma and semi-colon notation, e.g. va b for the partial
derivative of a vector and va b for its covariant derivative. This would certainly
save typographical space, but many students find the labour of mentally unpacking
such equations is sufficiently great that it is not possible to think of an equation™s
physical interpretation at the same time. Consequently, we have decided to write
out such expressions in their more obvious but longer form, using b va for partial
derivatives and b va for covariant derivatives.
It is worth mentioning that this book is based, in large part, on lecture notes
prepared separately by MPH and GPE for two different relativity courses in the
Natural Science Tripos at the University of Cambridge. These courses were first
presented in this form in the academic year 1999“2000 and are still ongoing. The
course presented by MPH consisted of 16 lectures to fourth-year undergraduates
in Part III Physics and Theoretical Physics and covered most of the material
in Chapters 1“11 and 13“14, albeit somewhat rapidly on occasion. The course
given by GPE consisted of 24 lectures to third-year undergraduates in Part II
Astrophysics and covered parts of Chapters 1, 5“11, 14 and 18, with an emphasis
on the less mathematical material. The process of combining the two sets of
lecture notes into a homogeneous treatment of relativistic gravitation was aided
somewhat by the fortuitous choice of a consistent sign convention in the two
courses, and numerous sections have been rewritten in the hope that the reader
will not encounter any jarring changes in presentational style. For many of the
topics covered in the two courses mentioned above, the opportunity has been
taken to include in this book a considerable amount of additional material beyond
that presented in the lectures, especially in the discussion of black holes. Some
of this material draws on lecture notes written by ANL for other courses in Part
II and Part III Physics and Theoretical Physics. Some topics that were entirely
absent from any of the above lecture courses have also been included in the book,
such as relativistic stars, cosmology, inflation, linearised gravity and variational
principles. While every care has been taken to describe these topics in a clear and
illuminating fashion, the reader should bear in mind that these chapters have not
been ˜road-tested™ to the same extent as the rest of the book.
It is with pleasure that we record here our gratitude to those authors from
whose books we ourselves learnt general relativity and who have certainly
influenced our own presentation of the subject. In particular, we acknowledge
(in their current latest editions) S. Weinberg, Gravitation and Cosmology,
Wiley, 1972; R. M. Wald, General Relativity, University of Chicago Press,
1984; B. Schutz, A First Course in General Relativity, Cambridge Univer-
sity Press, 1985; W. Rindler, Relativity: Special, General and Cosmological,
xviii Preface

Oxford University Press, 2001; and J. Foster & J. D. Nightingale, A Short Course
in General Relativity, Springer-Verlag, 1995.
During the writing of this book we have received much help and encourage-
ment from many of our colleagues at the University of Cambridge, especially
members of the Cavendish Astrophysics Group and the Institute of Astronomy.
In particular, we thank Chris Doran, Anthony Challinor, Steve Gull and Paul
Alexander for numerous useful discussions on all aspects of relativity theory, and
Dave Green for a great deal of advice concerning typesetting in LaTeX. We are
also especially grateful to Richard Sword for creating many of the diagrams and
figures used in the book and to Michael Bridges for producing the plots of recent
measurements of the cosmic microwave background and matter power spectra.
We also extend our thanks to the Cavendish and Institute of Astronomy teach-
ing staff, whose examination questions have provided the basis for some of the
exercises included. Finally, we thank several years of undergraduate students for
their careful reading of sections of the manuscript, for pointing out misprints and
for numerous useful comments. Of course, any errors and ambiguities remaining
are entirely the responsibility of the authors, and we would be most grateful to
have them brought to our attention. At Cambridge University Press, we are very
grateful to our editor Vince Higgs for his help and patience and to our copy-editor
Susan Parkinson for many useful suggestions that have undoubtedly improved the
style of the book.
Finally, on a personal note, MPH thanks his wife, Becky, for patiently enduring
many evenings and weekends spent listening to the sound of fingers tapping on
a keyboard, and for her unending encouragement. He also thanks his mother,
Pat, for her tireless support at every turn. MPH dedicates his contribution to this
book to the memory of his father, Ron, and to his daughter, Tabitha, whose early
arrival succeeded in delaying completion of the book by at least three months, but
equally made him realise how little that mattered. GPE thanks his wife, Yvonne,
for her support. ANL thanks all the students who have sat through his various
lectures on gravitation and cosmology and provided useful feedback. He would
also like to thank his family, and particularly his parents, for the encouragement
and support they have offered at all times.
The spacetime of special relativity

We begin our discussion of the relativistic theory of gravity by reviewing some
basic notions underlying the Newtonian and special-relativistic viewpoints of
space and time. In order to specify an event uniquely, we must assign it three
spatial coordinates and one time coordinate, defined with respect to some frame
of reference. For the moment, let us define such a system S by using a set of three
mutually orthogonal Cartesian axes, which gives us spatial coordinates x, y and
z, and an associated system of synchronised clocks at rest in the system, which
gives us a time coordinate t. The four coordinates t x y z thus label events in
space and time.

1.1 Inertial frames and the principle of relativity
Clearly, one is free to label events not only with respect to a frame S but also
with respect to any other frame S , which may be oriented and/or moving with
respect to S in an arbitrary manner. Nevertheless, there exists a class of preferred
reference systems called inertial frames, defined as those in which Newton™s first
law holds, so that a free particle is at rest or moves with constant velocity, i.e. in
a straight line with fixed speed. In Cartesian coordinates this means that

d2 x d2 y d2 z
= 2 = 2 =0
dt2 dt dt

It follows that, in the absence of gravity, if S and S are two inertial frames then
S can differ from S only by (i) a translation, and/or (ii) a rotation and/or (iii) a
motion of one frame with respect to the other at a constant velocity (for otherwise
Newton™s first law would no longer be true). The concept of inertial frames is
fundamental to the principle of relativity, which states that the laws of physics
take the same form in every inertial frame. No exception has ever been found to

2 The spacetime of special relativity

this general principle, and it applies equally well in both Newtonian theory and
special relativity.
The Newtonian and special-relativistic descriptions differ in how the coor-
dinates of an event P in two inertial frames are related. Let us consider two
Cartesian inertial frames S and S in standard configuration, where S is moving
along the x-axis of S at a constant speed v and the axes of S and S coincide at
t = t = 0 (see Figure 1.1). It is clear that the (primed) coordinates of an event
P with respect to S are related to the (unprimed) coordinates in S via a linear
transformation1 of the form
t = At + Bx
x = Dt + Ex
y =y
z =z
Moreover, since we require that x = 0 corresponds to x = vt and that x = 0
corresponds to x = ’vt , we find immediately that D = ’Ev and D = ’Av, so
that A = E. Thus we must have
t = At + Bx
x = A x ’ vt
y =y
z =z

y y'


S x S' x'

z z'

Figure 1.1 Two inertial frames S and S in standard configuration (the origins
of S and S coincide at t = t = 0).

We will prove this in Chapter 5.
1.3 The spacetime geometry of special relativity

1.2 Newtonian geometry of space and time
Newtonian theory rests on the assumption that there exists an absolute time, which
is the same for every observer, so that t = t. Under this assumption A = 1 and
B = 0, and we obtain the Galilean transformation relating the coordinates of an
event P in the two Cartesian inertial frames S and S :

t =t
x = x ’ vt
y =y
z =z

By symmetry, the expressions for the unprimed coordinates in terms of the primed
ones have the same form but with v replaced by ’v.
The first equation in (1.2) is clearly valid for any two inertial frames S and
S and shows that the time coordinate of an event P is the same in all inertial
frames. The second equation leads to the ˜common sense™ notion of the addition
of velocities. If a particle is moving in the x-direction at a speed u in S then its
speed in S is given by
dx dx dx
ux = = = ’ v = ux ’ v
dt dt dt
Differentiating again shows that the acceleration of a particle is the same in both
S and S , i.e. dux /dt = dux /dt.
If we consider two events A and B that have coordinates tA xA yA zA
and tB xB yB zB respectively, it is straightforward to show that both the time
difference t = tB ’ tA and the quantity

r 2 = x2 + y2 + z2

are separately invariant under any Galilean transformation. This leads us to
consider space and time as separate entities. Moreover, the invariance of r 2
suggests that it is a geometric property of space itself. Of course, we recognise
r 2 as the square of the distance between the events in a three-dimensional
Euclidean space. This defines the geometry of space and time in the Newtonian

1.3 The spacetime geometry of special relativity
In special relativity, Einstein abandoned the postulate of an absolute time and
replaced it by the postulate that the speed of light c is the same in all inertial
4 The spacetime of special relativity

frames.2 By applying this new postulate, together with the principle of relativity,
we may obtain the Lorentz transformations connecting the coordinates of an event
P in two different Cartesian inertial frames S and S .
Let us again consider S and S to be in standard configuration (see Figure 1.1),
and consider a photon emitted from the (coincident) origins of S and S at t =
t = 0 and travelling in an arbitrary direction. Subsequently the space and time
coordinates of the photon in each frame must satisfy
2 2 2 2
c2 t2 ’ x2 ’ y2 ’ z2 = c2 t ’ x ’ y ’ z = 0

Substituting the relations (1.1) into this expression and solving for the constants
A and B, we obtain

ct = ct ’ x
x= x ’ ct
y =y
z =z

where = v/c and = 1 ’ 2 ’1/2 . This Lorentz transformation, also known
as a boost in the x-direction, reduces to the Galilean transformation (1.2) when
1. Once again, symmetry demands that the unprimed coordinates are given
in terms of the primed coordinates by an analogous transformation in which v is
replaced by ’v.
From the equations (1.3), we see that the time and space coordinates are in
general mixed by a Lorentz transformation (note, in particular, the symmetry
between ct and x). Moreover, as we shall see shortly, if we consider two events
A and B with coordinates tA xA yA zA and tB xB yB zB in S, it is straight-
forward to show that the interval (squared)

s2 = c2 t2 ’ x2 ’ y2 ’ z2 (1.4)

is invariant under any Lorentz transformation. As advocated by Minkowski, these
observations lead us to consider space and time as united in a four-dimensional
continuum called spacetime, whose geometry is characterised by (1.4). We note
that the spacetime of special relativity is non-Euclidean, because of the minus
signs in (1.4), and is often called the pseudo-Euclidean or Minkowski geometry.
Nevertheless, for any fixed value of t the spatial part of the geometry remains

The reasoning behind Einstein™s proposal is discussed in Appendix 1A.
1.4 Lorentz transformations as four-dimensional ˜rotations™

We have arrived at the familiar viewpoint (to a physicist!) where the physical
world is modelled as a four-dimensional spacetime continuum that possesses
the Minkowski geometry characterised by (1.4). Indeed, many ideas in special
relativity are most simply explained by adopting a four-dimensional point of view.

1.4 Lorentz transformations as four-dimensional ˜rotations™
Adopting a particular (Cartesian) inertial frame S corresponds to labelling events in
the Minkowski spacetime with a given set of coordinates t x y z . If we choose
instead to describe the world with respect to a different Cartesian inertial frame
S then this corresponds simply to relabelling events in the Minkowski spacetime
with a new set of coordinates t x y z ; the primed and unprimed coordinates
are related by the appropriate Lorentz transformation. Thus, describing physics
in terms of different inertial frames is equivalent to performing a coordinate
transformation on the Minkowski spacetime.
Consider, for example, the case where S is related to S via a spatial rotation
through an angle about the x-axis. In this case, we have
ct = ct
x =x
y = y cos ’ z sin
z = y sin + z cos
Clearly the inverse transform is obtained on replacing by ’ .
The close similarity between the ˜boost™ (1.3) and an ordinary spatial rotation
can be highlighted by introducing the rapidity parameter
= tanh’1
As varies from zero to unity, ranges from 0 to . We also note that = cosh
= sinh . If two inertial frames S and S are in standard configuration, we
therefore have

ct = ct cosh ’ x sinh
x = ’ct sinh + x cosh
y =y
z =z

This has essentially the same form as a spatial rotation, but with hyperbolic
functions replacing trigonometric ones. Once again the inverse transformation is
obtained on replacing by ’ .
6 The spacetime of special relativity


z' S'



S x


Figure 1.2 Two inertial frames S and S in general configuration. The broken
line shown the trajectory of the origin of S .

In general, S is moving with a constant velocity v with respect to S in an
arbitrary direction3 and the axes of S are rotated with respect to those of S.
Moreover, at t = t = 0 the origins of S and S need not be coincident and may
be separated by a vector displacement a, as measured in S (see Figure 1.2).4
The corresponding transformation connecting the two inertial frames is most
easily found by decomposing the transformation into a displacement, followed
by a spatial rotation, followed by a boost, followed by a further spatial rotation.
Physically, the displacement makes the origins of S and S coincident at t = t = 0,
and the first rotation lines up the x-axis of S with the velocity v of S . Then a boost
in this direction with speed v transforms S into a frame that is at rest with respect to
S . A final rotation lines up the coordinate frame with that of S . The displacement
and spatial rotations introduce no new physics, and the only special-relativistic
consideration concerns the boost. Thus, without loss of generality, we can restrict
our attention to inertial frames S and S that are in standard configuration, for
which the Lorentz transformation is given by (1.3) or (1.5).

1.5 The interval and the lightcone
If we consider two events A and B having coordinates tA xA yA zA and
tB xB yB zB in S , then, from (1.5), the interval between the events is given by

Throughout this book, the notation v is used specifically to denote three-dimensional vectors, whereas v
denotes a general vector, which is most often a 4-vector.
If a = 0 then the Lorentz transformation connecting the two inertial frames is called homogeneous, while if

a = 0 it is called inhomogeneous. Inhomogeneous transformations are often referred to as Poincar© transfor-
mations, in which case homogeneous transformations are referred to simply as Lorentz transformations.
1.5 The interval and the lightcone
2 2 2 2
s 2 = c2 t ’ x ’ y ’ z
= c t cosh ’ ’ ’ c t sinh +
2 2
x sinh x cosh
’ y2 ’ z2
= c2 t2 ’ x2 ’ y2 ’ z2

Thus the interval is invariant under the boost (1.5) and, from the above discussion,
we may infer that s2 is in fact invariant under any Poincar© transformation. This
suggests that the interval is an underlying geometrical property of the spacetime
itself, i.e. an invariant ˜distance™ between events in spacetime. It also follows that
the sign of s2 is defined invariantly, as follows:

s2 > 0 the interval is timelike
s2 = 0 the interval is null or lightlike
s2 < 0 the interval is spacelike

This embodies the standard lightcone structure shown in Figure 1.3. Events A and
B are separated by a timelike interval, A and C by a lightlike (or null) interval and

Future of A




˜Elsewhere™ of A ˜Elsewhere™ of A

Past of A

Figure 1.3 Spacetime diagram illustrating the lightcone of an event A (the y-
and z- axes have been suppressed). Events A and B are separated by a timelike
interval, A and C by a lightlike (or null) interval and A and D by a spacelike
8 The spacetime of special relativity

A and D by a spacelike interval. The geometrical distinction between timelike and
spacelike intervals corresponds to a physical distinction: if the interval is timelike
then we can find an inertial frame in which the events occur at the same spatial
coordinates and if the interval is spacelike then we can find an inertial frame
in which the events occur at the same time coordinate. This becomes obvious
when we consider the spacetime diagram of a Lorentz transformation; we shall
do this next.

1.6 Spacetime diagrams
Figure 1.3 is an example of a spacetime diagram. Such diagrams are extremely
useful in illustrating directly many special-relativistic effects, in particular coor-
dinate transformations on the Minkowski spacetime between different inertial
frames. The spacetime diagram in Figure 1.4 shows the change of coordinates of
an event A corresponding to the standard-configuration Lorentz transformation
(1.5). The x -axis is simply the line t = 0 and the t -axis is the line x = 0.
From the Lorentz-boost transformation (1.3) we see that the angle between the
x- and x - axes is the same as that between the t- and t - axes and has the value

ct ct'

Event A
t (A)
t' (A)


x' (A)
x (A)

Figure 1.4 Spacetime diagram illustrating the coordinate transformation
between two inertial frames S and S in standard configuration (the y- and z-
axes have been suppressed). The worldlines of the origins of S and S are the
axes ct and ct respectively.
1.6 Spacetime diagrams

tan’1 v/c . Moreover, we note that the t- and t - axes are also the worldlines of
the origins of S and S respectively.
It is important to realise that the coordinates of the event A in the frame S are
not obtained by extending perpendiculars from A to the x - and t - axes. Since
the x -axis is simply the line t = 0, it follows that lines of simultaneity in S are
parallel to the x -axis. Similarly, lines of constant x are parallel to the t -axis. The
same reasoning is equally valid for obtaining the coordinates of A in the frame
S but, since the x- and t- axes are drawn as orthogonal in the diagram, this is
equivalent simply to extending perpendiculars from A to the x- and t- axes in the
more familiar manner.
The concept of simultaneity is simply illustrated using a spacetime diagram.
For example, in Figure 1.5 we replot the events in Figure 1.3, together with the x -
and t - axes corresponding to a Lorentz boost in standard configuration at some
velocity v. We see that the events A and D, which are separated by a spacelike
interval, lie on a line of constant t and so are simultaneous in S . Evidently, A
and D are not simultaneous in S; D occurs at a later time than A. In a similar
way, it is straightforward to find a standard-configuration Lorentz boost such that
the events A and B, which are separated by a timelike interval, lie on a line of
constant x and hence occur at the same spatial location in S .



line of
constant t'



Figure 1.5 The events illustrated in figure 1.3 and a Lorentz boost such that A
and D are simultaneous in S .
10 The spacetime of special relativity

1.7 Length contraction and time dilation
Two elementary (but profound) consequences of the Lorentz transformations
are length contraction and time dilation. Both these effects are easily derived
from (1.3).

Length contraction
Consider a rod of proper length at rest in S (see Figure 1.6); we have

= xB ’ xA

We want to apply the Lorentz transformation formulae and so find what length
an observer in frame S assigns to the rod. Applying the second formula in (1.3),
we obtain
xA = xA ’ vtA
xB = xB ’ vtB
relating the coordinates of the ends of the rod in S to the coordinates in S. The
observer in S measures the length of the rod at a fixed time t = tA = tB as
1 0
= xB ’ xA = xB ’ xA =

Hence in S the rod appears contracted to the length

= 1 ’ v2 /c2

If a rod is moving relative to S in a direction perpendicular to its length,
however, it is straightforward to show that it suffers no contraction. It thus follows
that the volume V of a moving object, as measured by simultaneously noting the
positions of the boundary points in S, is related to its proper volume V0 by V =
V0 1 ’ v2 /c2 1/2 . This fact must be taken into account when considering densities.

y y'


S x S' x'A x'B x'

z z'

Figure 1.6 Two inertial frames S and S in standard configuration. A rod of
proper length 0 is at rest in S .
1.8 Invariant hyperbolae

y y' y'

v v

˜Click 1™ ˜Click 2™

S S' S'
x x' x'
x'A x'A

z z'

Figure 1.7 Two inertial frames S and S in standard configuration. A clock is
at rest in S .

Time dilation
Suppose we have a clock at rest in S , in which two successive ˜clicks™ of the
clock (events A and B) are separated by a time interval T0 (see Figure 1.7). The
times of the clicks as recorded in S are
tA = tA + vxA /c2
tB = tA + T0 + vxB /c2
Since the clock is at rest in S we have xA = xB , and so on subtracting we obtain

T = tB ’ tA = T0 =
1 ’ v2 /c2 1/2

Hence, the moving clock ticks more slowly by a factor of 1 ’ v2 /c2 1/2 (time
Note that an ideal clock is one that is unaffected by acceleration “ external
forces act identically on all parts of the clock (an example is a muon).

1.8 Invariant hyperbolae
Length contraction and time dilation are easily illustrated using spacetime
diagrams. However, while Figure 1.4 illustrates the positions of the x - and t - axes
corresponding to a standard Lorentz boost, we have not yet calibrated the length
scales along them. To perform this calibration, we make use of the fact that the
interval s2 between two events is an invariant, and draw the invariant hyperbolae
2 2
c2 t2 ’ x2 = c2 t ’ x = ±1
on the spacetime diagram, as shown in Figure 1.8. Then, if we first take the
positive sign, setting ct = 0, we obtain x = ±1. It follows that OA is a unit
12 The spacetime of special relativity

line of
constant x'
line of
constant t'





Figure 1.8 The invariant hyperbolae c2 t2 ’ x2 = c2 t 2 ’ x 2 = ±1.

distance along the x-axis. Now setting ct = 0 we find that x = ±1, so that OC is
a unit distance along the x -axis. Similarly, OB and OD are unit distances along
the t- and t - axes respectively. We also note that the tangents to the invariant
hyperbolae at C and D are lines of constant x and t respectively.
The length contraction and time dilation effects can now be read off directly
from the diagram. For example, the worldlines of the end-points of a unit rod
OC in S , namely x = 0 and x = 1, cut the x-axis in less than unit distance.
Similarly, worldlines x = 0 and x = 1 in S cut the x -axis inside OC, illustrating
the reciprocal nature of length contraction. Also, a clock at rest at the origin of
S will move along the t -axis, reaching D with a reading of t = 1. However, the
event D has a t-coordinate that is greater than unity, thereby illustrating the time
dilation effect.

1.9 The Minkowski spacetime line element
Let consider more closely the meaning of the interval between two events A and
B in spacetime. Given that in a particular inertial frame S the coordinates of A
and B are tA xA yA zA and tB xB yB zB , we have so far taken the square of
the interval between A and B to be
s2 = c2 t2 ’ x2 ’ y2 ’ z2
1.9 The Minkowski spacetime line element





Figure 1.9 Two paths in spacetime connecting the events A and B.

where t = tB ’ tA etc. This interval is invariant under Lorentz transformation
and corresponds to the ˜distance™ in spacetime measured along the straight line
in Figure 1.9 connecting A and B. This line may be interpreted as the worldline
of a particle moving at constant velocity relative to S between events A and B.
However, the question naturally arises of what interval is measured between A
and B along some other path in spacetime, for example the ˜wiggly™ path shown
in Figure 1.9.
To address this question, we must express the intrinsic geometry of the
Minkowski spacetime in infinitesimal form. Clearly, if two infinitesimally sepa-
rated events have coordinates t x y z and t + dt x + dx y + dy z + dz in S
then the square of the infinitesimal interval between them is given by5

ds2 = c2 dt2 ’ dx2 ’ dy2 ’ dz2

which is known as the line element of Minkowski spacetime, or the special-
relativistic line element. From our earlier considerations, it is clear that ds2 is
invariant under any Lorentz transformation. The invariant interval between A and
B along an arbitrary path in spacetime is then given by
s= ds

To avoid mathematical ambiguity, one should properly denote the squares of infinitesimal coordinate intervals
by dt 2 etc., but this notation is not in common use in relativity textbooks. We will thus adopt the more
usual form dt2 , but it should be remembered that this is not the differential of t2 .
14 The spacetime of special relativity

where the integral is evaluated along the particular path under consideration.
Clearly, to perform this integral we must have a set of equations describing the
spacetime path.

1.10 Particle worldlines and proper time
Let us now turn to the description of the motion of a particle in spacetime terms.
A particle describes a worldline in spacetime. In general, for two infinitesimally
separated events in spacetime; by analogy with our earlier discussion we have:

for ds2 > 0 the interval is timelike
for ds2 = 0 the interval is null or lightlike
for ds2 < 0 the interval is spacelike

However, relativistic mechanics prohibits the acceleration of a massive particle
to speeds greater than or equal to c, which implies that its worldline must lie
within the lightcone (Figure 1.3) at each event on it. In other words, the interval
between any two infinitesimally separated events on the particle™s worldline must
be timelike (and future-pointing). For a massless particle such as a photon, any
two events on its worldline are separated by a null interval. Figure 1.10 illustrates
general worldlines for a massive particle and for a photon.



Figure 1.10 The worldlines of a photon (solid line) and a massive particle
(broken line). The lightcones at seven events are shown.
1.10 Particle worldlines and proper time

A particle worldline may be described by giving x, y and z as functions of t
in some inertial frame S. However, a more four-dimensional way of describing a
worldline is to give the four coordinates t x y z of the particle in S as functions
of a parameter that varies monotonically along the worldline. Given the four
functions t , x , y and z , each value of determines a point along
the curve. Any such parameter is possible, but a natural one to use for a massive
particle is its proper time.
We define the proper time interval d between two infinitesimally separated
events on the particle™s worldline by

= ds2
c2 d 2

Thus, if the coordinate differences in S between the two events are dt dx dy dz
then we have

= c2 dt2 ’ dx2 ’ dy2 ’ dz2
c2 d 2

Hence the proper time interval between the events is given by

d = 1 ’ v2 /c2 dt = dt/

where v is the speed of the particle with respect to S over this infinitesimal
interval. If we integrate d between two points A and B on the worldline, we
obtain the total elapsed proper time interval:
v2 t
= d= 1’ 2 (1.7)

We see that if the particle is at rest in S then the proper time is just the
coordinate time t measured by clocks at rest in S. If at any instant in the history
of the particle we introduce an instantaneous rest frame S such that the particle
is momentarily at rest in S then we see that the proper time is simply the
time recorded by a clock that moves along with the particle. It is therefore an
invariantly defined quantity, a fact that is clear from (1.6).
Thus the worldline of a massive particle can be described by giving the four
coordinates t x y z as functions of (see Figure 1.11). For example,
t= 1 ’ v2 /c2
x = v 1 ’ v2 /c2
y =z=0
16 The spacetime of special relativity





„ = “1

„ = “2

Figure 1.11 A path in the t x -plane can be specified by giving one coordinate
in terms of the other, for example x = x t , or alternatively by giving both
coordinates as functions of a parameter along the curve: t = t x=x .
For massive particles the natural parameter to use is the proper time .

is the worldline of a particle, moving at constant speed v along the x-axis of S,
which passes through the origin of S at t = 0.

1.11 The Doppler effect
A useful illustration of particle worldlines and the concept of proper time is
provided by deriving the Doppler effect in a transparently four-dimensional
manner. Let us consider an observer at rest in some inertial frame S, and a
radiation-emitting source moving along the positive x-axis of S at a uniform
speed v. Suppose that the source emits the first wavecrest of a photon at an
event A, with coordinates te xe in S, and the next wavecrest at an event B
with coordinates te + te xe + xe . Let us assume that these two wavecrests
reach the observer at the events C and D coordinates to xo and to + to xo
respectively. This situation is illustrated in Figure 1.12. From (1.7), the proper
time interval experienced by between the events A and B is
= 1 ’ v2 /c2 (1.8)

and the proper time interval experienced by between the events C and D is

= to (1.9)
1.11 The Doppler effect






Figure 1.12 Spacetime diagram of the Doppler effect.

Along each of the worldlines representing the photon wavecrests,

ds2 = c2 dt2 ’ dx2 ’ dy2 ’ dz2 = 0

Thus, since we are assuming that dy = dz = 0, along the worldline connecting
the events A and C we have
to xo
c dt = ’ (1.10)
te xe

where the minus sign on the right-hand side arises because the photon is travelling
in the negative x-direction. From (1.10), we obtain the (obvious) result c to ’te =
’ xo ’ xe . Similarly, along the worldline connecting B and D we have
to + to xo
c dt = ’ dx
te + te xe + x e

Rewriting the integrals on each side, we obtain
to + to te + te xe + xe
to xo
+ ’ c dt = ’ ’ dx
te to te xe xe

where the first integrals on each side of the equation cancel by virtue of (1.10).
Thus we find that c to ’ c te = xe , from which we obtain
1 xe v
to = 1 + te = 1 + (1.11)
c te c
18 The spacetime of special relativity

Hence, using (1.8), (1.9) and (1.11), we can derive the ratio of the proper time
intervals CD and AB experienced by and respectively:
1+ 1+ 1+ 1/2

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