This page intentionally left blank 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
M. P. HOBSON, G. P. EFSTATHIOU and A. N. LASENBY cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
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Contents
Preface page xv
1 The spacetime of special relativity1 1.1 Inertial frames and the principle of relativity1 1.2 Newtonian geometry of space and time3 1.3 The spacetime geometry of special relativity3 1.4 Lorentz transformations as four-dimensional ‘rotations’5 1.5 The interval and the lightcone6 1.6 Spacetime diagrams8 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
vii 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 Contents ix
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 Contents xi
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 Contents xiii
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 Appendix17A:TheEinstein–Maxwellformulationoflinearisedgravity 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
Bibliography 555 Index 556 Preface
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 cosmology. 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 course. 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
xv 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. Preface xvii
With regard to the presentation of the mathematics, it has to be accepted that equations containing partial and covariant derivatives could be written more a compactly by using the comma and semi-colon notation, e.g. v b for the partial a derivative of a vector and v 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 a out such expressions in their more obvious but longer form, using bv for partial a derivatives and bv 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. 1 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
d2x d2y d2z = = = 0 dt2 dt2 dt2
It follows that, in the absence of gravity,ifS 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
1 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 (1.1) y = y z = z
y y'
v
SS'xx'
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).
1 We will prove this in Chapter 5. 1.3 The spacetime geometry of special relativity 3
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 :