General relativity: an introduction for physicists

Half-title......Page 2
Title......Page 4
Copyright......Page 5
Dedication......Page 6
Contents......Page 8
Preface......Page 16
1.1 Inertial frames and the principle of relativity......Page 20
1.3 The spacetime geometry of special relativity......Page 22
1.4 Lorentz transformations as four-dimensional ‘rotations’......Page 24
1.5 The interval and the lightcone......Page 25
1.6 Spacetime diagrams......Page 27
Length contraction......Page 29
1.8 Invariant hyperbolae......Page 30
1.9 The Minkowski spacetime line element......Page 31
1.10 Particle worldlines and proper time......Page 33
1.11 The Doppler effect......Page 35
1.12 Addition of velocities in special relativity......Page 37
1.13 Acceleration in special relativity......Page 38
1.14 Event horizons in special relativity......Page 40
On the Electrodynamics of Moving Bodies by A. Einstein......Page 41
Exercises......Page 43
2.1 The concept of a manifold......Page 45
2.3 Curves and surfaces......Page 46
2.4 Coordinate transformations......Page 47
2.5 Summation convention......Page 49
2.6 Geometry of manifolds......Page 50
2.7 Riemannian geometry......Page 51
2.8 Intrinsic and extrinsic geometry......Page 52
2.9 Examples of non-Euclidean geometry......Page 55
2.10 Lengths, areas and volumes......Page 57
2.11 Local Cartesian coordinates......Page 61
2.12 Tangent spaces to manifolds......Page 63
2.13 Pseudo-Riemannian manifolds......Page 64
2.14 Integration over general submanifolds......Page 66
2.15 Topology of manifolds......Page 68
Exercises......Page 69
3.1 Scalar fields on manifolds......Page 72
3.2 Vector fields on manifolds......Page 73
3.3 Tangent vector to a curve......Page 74
3.4 Basis vectors......Page 75
Coordinate basis vectors......Page 76
3.5 Raising and lowering vector indices......Page 78
3.6 Basis vectors and coordinate transformations......Page 79
3.7 Coordinate-independent properties of vectors......Page 80
3.8 Derivatives of basis vectors and the affine connection......Page 81
3.9 Transformation properties of the affine connection......Page 83
3.10 Relationship of the connection and the metric......Page 84
3.11 Local geodesic and Cartesian coordinates......Page 86
3.12 Covariant derivative of a vector......Page 87
Divergence......Page 89
3.14 Intrinsic derivative of a vector along a curve......Page 90
3.15 Parallel transport......Page 92
3.16 Null curves, non-null curves and affine parameters......Page 94
3.17 Geodesics......Page 95
3.18 Stationary property of non-null geodesics......Page 96
3.19 Lagrangian procedure for geodesics......Page 97
Appendix 3A: Vectors as directional derivatives......Page 100
Appendix 3B: Polar coordinates in a plane......Page 101
Appendix 3C: Calculus of variations......Page 106
Exercises......Page 107
4.1 Tensor fields on manifolds......Page 111
4.2 Components of tensors......Page 112
4.3 Symmetries of tensors......Page 113
4.4 The metric tensor......Page 115
4.6 Mapping tensors into tensors......Page 116
Outer product......Page 117
Contraction (and inner product)......Page 118
4.8 Tensors as geometrical objects......Page 119
4.9 Tensors and coordinate transformations......Page 120
4.10 Tensor equations......Page 121
4.11 The quotient theorem......Page 122
4.12 Covariant derivative of a tensor......Page 123
4.13 Intrinsic derivative of a tensor along a curve......Page 126
Exercises......Page 127
5.1 Minkowski spacetime in Cartesian coordinates......Page 130
5.2 Lorentz transformations......Page 131
5.3 Cartesian basis vectors......Page 132
5.4 Four-vectors and the lightcone......Page 134
5.6 Four-velocity......Page 135
5.7 Four-momentum of a massive particle......Page 137
5.8 Four-momentum of a photon......Page 138
5.9 The Doppler effect and relativistic aberration......Page 139
5.10 Relativistic mechanics......Page 141
5.12 Relativistic collisions and Compton scattering......Page 142
5.13 Accelerating observers......Page 144
5.14 Minkowski spacetime in arbitrary coordinates......Page 147
Exercises......Page 150
6.1 The electromagnetic force on a moving charge......Page 154
6.2 The 4-current density......Page 155
6.3 The electromagnetic field equations......Page 157
6.4 Electromagnetism in the Lorenz gauge......Page 158
6.5 Electric and magnetic fields in inertial frames......Page 160
6.6 Electromagnetism in arbitrary coordinates......Page 161
6.7 Equation of motion for a charged particle......Page 163
Exercises......Page 164
7.1 Newtonian gravity......Page 166
7.2 The equivalence principle......Page 167
7.3 Gravity as spacetime curvature......Page 168
7.4 Local inertial coordinates......Page 170
7.5 Observers in a curved spacetime......Page 171
7.6 Weak gravitational fields and the Newtonian limit......Page 172
7.7 Electromagnetism in a curved spacetime......Page 174
7.8 Intrinsic curvature of a manifold......Page 176
7.9 The curvature tensor......Page 177
7.10 Properties of the curvature tensor......Page 178
7.11 The Ricci tensor and curvature scalar......Page 180
7.12 Curvature and parallel transport......Page 182
7.13 Curvature and geodesic deviation......Page 184
7.14 Tidal forces in a curved spacetime......Page 186
Appendix 7A: The surface of a sphere......Page 189
Exercises......Page 191
8.1 The energy–momentum tensor......Page 195
8.2 The energy–momentum tensor of a perfect fluid......Page 197
8.3 Conservation of energy and momentum for a perfect fluid......Page 198
8.4 The Einstein equations......Page 200
8.5 The Einstein equations in empty space......Page 202
8.6 The weak-field limit of the Einstein equations......Page 203
8.7 The cosmological-constant term......Page 204
8.8 Geodesic motion from the Einstein equations......Page 207
8.9 Concluding remarks......Page 209
Brans–Dicke theory......Page 210
Torsion theories......Page 211
Exercises......Page 212
9.1 The general static isotropic metric......Page 215
9.2 Solution of the empty-space field equations......Page 217
9.4 Gravitational redshift for a fixed emitter and receiver......Page 221
9.5 Geodesics in the Schwarzschild geometry......Page 224
9.6 Trajectories of massive particles......Page 226
9.7 Radial motion of massive particles......Page 228
9.8 Circular motion of massive particles......Page 231
9.9 Stability of massive particle orbits......Page 232
9.10 Trajectories of photons......Page 236
9.11 Radial motion of photons......Page 237
9.12 Circular motion of photons......Page 238
9.13 Stability of photon orbits......Page 239
Appendix 9A: General approach to gravitational redshifts......Page 240
Exercises......Page 243
10.1 Precession of planetary orbits......Page 249
10.2 The bending of light......Page 252
10.3 Radar echoes......Page 255
10.4 Accretion discs around compact objects......Page 259
10.5 The geodesic precession of gyroscopes......Page 263
Exercises......Page 265
11.1 The characterisation of coordinates......Page 267
11.2 Singularities in the Schwarzschild metric......Page 268
11.3 Radial photon worldlines in Schwarzschild coordinates......Page 270
11.4 Radial particle worldlines in Schwarzschild coordinates......Page 271
Advanced Eddington–Finkelstein coordinates......Page 273
Retarded Eddington–Finkelstein coordinates......Page 276
11.6 Gravitational collapse and black-hole formation......Page 278
11.7 Spherically symmetric collapse of dust......Page 279
11.8 Tidal forces near a black hole......Page 283
11.9 Kruskal coordinates......Page 285
11.10 Wormholes and the Einstein–Rosen bridge......Page 290
11.11 The Hawking effect......Page 293
Appendix 11A: Compact binary systems......Page 296
Appendix 11B: Supermassive black holes......Page 298
Appendix 11C: Conformal flatness of two-dimensional Riemannian manifolds......Page 301
Exercises......Page 302
12.1 The form of the metric for a stellar interior......Page 307
12.2 The relativistic equations of stellar structure......Page 311
12.3 The Schwarzschild constant-density interior solution......Page 313
12.5 The metric outside a spherically symmetric charged mass......Page 315
12.6 The Reissner–Nordström geometry: charged black holes......Page 319
12.7 Radial photon trajectories in the RN geometry......Page 321
12.8 Radial massive particle trajectories in the RN geometry......Page 323
Exercises......Page 324
13.1 The general stationary axisymmetric metric......Page 329
13.2 The dragging of inertial frames......Page 331
13.3 Stationary limit surfaces......Page 333
13.4 Event horizons......Page 334
13.5 The Kerr metric......Page 336
13.6 Limits of the Kerr metric......Page 338
13.7 The Kerr–Schild form of the metric......Page 340
Singularities and horizons......Page 341
Stationary limit surfaces......Page 343
The ergoregion......Page 344
13.9 The Penrose process......Page 346
13.10 Geodesics in the equatorial plane......Page 349
13.11 Equatorial trajectories of massive particles......Page 351
13.12 Equatorial motion of massive particles with zero angular momentum......Page 352
13.13 Equatorial circular motion of massive particles......Page 354
13.14 Stability of equatorial massive particle circular orbits......Page 356
13.15 Equatorial trajectories of photons......Page 357
13.16 Equatorial principal photon geodesics......Page 358
13.17 Equatorial circular motion of photons......Page 360
13.18 Stability of equatorial photon orbits......Page 361
13.19 Eddington–Finkelstein coordinates......Page 363
13.20 The slow-rotation limit and gyroscope precession......Page 366
Exercises......Page 369
14.1 The cosmological principle......Page 374
14.2 Slicing and threading spacetime......Page 375
14.3 Synchronous coordinates......Page 376
14.4 Homogeneity and isotropy of the universe......Page 377
14.5 The maximally symmetric 3-space......Page 378
14.7 Geometric properties of the FRW metric......Page 381
14.8 Geodesics in the FRW metric......Page 384
14.9 The cosmological redshift......Page 386
14.10 The Hubble and deceleration parameters......Page 387
14.11 Distances in the FRW geometry......Page 390
Luminosity distance......Page 391
Angular diameter distance......Page 392
14.12 Volumes and number densities in the FRW geometry......Page 393
14.13 The cosmological field equations......Page 395
14.14 Equation of motion for the cosmological fluid......Page 398
Exercises......Page 400
15.1 Components of the cosmological fluid......Page 405
Matter......Page 406
Radiation......Page 407
Relative contributions of the components......Page 408
15.2 Cosmological parameters......Page 409
15.3 The cosmological field equations......Page 411
15.4 General dynamical behaviour of the universe......Page 412
15.5 Evolution of the scale factor......Page 416
The Friedmann models......Page 419
The Lemaitre models......Page 423
Einstein’s static universe......Page 426
15.7 Look-back time and the age of the universe......Page 427
15.8 The distance–redshift relation......Page 430
15.9 The volume–redshift relation......Page 432
15.10 Evolution of the density parameters......Page 434
15.11 Evolution of the spatial curvature......Page 436
Particle horizon......Page 437
Hubble distance......Page 439
Exercises......Page 440
16.1 Definition of inflation......Page 447
16.2 Scalar fields and phase transitions in the very early universe......Page 449
16.3 A scalar field as a cosmological fluid......Page 450
16.4 An inflationary epoch......Page 452
16.5 The slow-roll approximation......Page 453
16.7 The amount of inflation......Page 454
16.8 Starting inflation......Page 456
16.9 ‘New’ inflation......Page 457
16.10 Chaotic inflation......Page 459
16.11 Stochastic inflation......Page 460
16.13 Classical evolution of scalar-field perturbations......Page 461
The perturbed Einstein field equations......Page 462
Perturbation equations in Fourier space......Page 464
16.14 Gauge invariance and curvature perturbations......Page 465
16.15 Classical evolution of curvature perturbations......Page 468
16.16 Initial conditions and normalisation of curvature perturbations......Page 471
16.17 Power spectrum of curvature perturbations......Page 475
16.18 Power spectrum of matter-density perturbations......Page 477
16.19 Comparison of theory and observation......Page 478
Exercises......Page 481
17.1 The weak-field metric......Page 486
Global Lorentz transformations......Page 487
Infinitesimal general coordinate transformations......Page 488
17.2 The linearised gravitational field equations......Page 489
17.3 Linearised gravity in the Lorenz gauge......Page 491
17.4 General properties of the linearised field equations......Page 492
17.5 Solution of the linearised field equations in vacuo......Page 493
17.6 General solution of the linearised field equations......Page 494
17.7 Multipole expansion of the general solution......Page 499
17.8 The compact-source approximation......Page 500
17.9 Stationary sources......Page 502
17.10 Static sources and the Newtonian limit......Page 504
17.11 The energy–momentum of the gravitational field......Page 505
Appendix 17A: The Einstein–Maxwell formulation of linearised gravity......Page 509
Exercises......Page 512
18.1 Plane gravitational waves and polarisation states......Page 517
18.2 Analogy between gravitational and electromagnetic waves......Page 520
18.3 Transforming to the transverse-traceless gauge......Page 521
18.4 The effect of a gravitational wave on free particles......Page 523
18.5 The generation of gravitational waves......Page 526
18.6 Energy flow in gravitational waves......Page 530
18.7 Energy loss due to gravitational-wave emission......Page 532
18.8 Spin-up of binary systems: the binary pulsar PSR B1913+16......Page 535
18.9 The detection of gravitational waves......Page 536
Exercises......Page 539
19.1 Hamilton’s principle in Newtonian mechanics......Page 543
19.2 Classical field theory and the action......Page 546
19.3 Euler–Lagrange equations......Page 548
19.4 Alternative form of the Euler–Lagrange equations......Page 550
19.5 Equivalent actions......Page 552
19.6 Field theory of a real scalar field......Page 553
19.7 Electromagnetism from a variational principle......Page 555
19.8 The Einstein–Hilbert action and general relativity in vacuo......Page 558
19.9 An equivalent action for general relativity in vacuo......Page 561
19.10 The Palatini approach for general relativity in vacuo......Page 562
19.11 General relativity in the presence of matter......Page 564
19.12 The dynamical energy–momentum tensor......Page 565
Exercises......Page 568
Bibliography......Page 574
Index......Page 575