General Relativity An Introduction For Physicists

Cover......Page 1
General Relativity: An Introduction for Physicists......Page 3
Title Page......Page 5
Copyright......Page 6
Dedication......Page 7
2 Manifolds and coordinates......Page 9
4 Tensor calculus on manifolds......Page 10
7 The equivalence principle and spacetime curvature......Page 11
9 The Schwarzschild geometry......Page 12
12 Further spherically symmetric geometries......Page 13
14 The Friedmann–Robertson–Walker geometry......Page 14
17 Linearised general relativity......Page 15
Index......Page 16
Preface......Page 17
1.1 Inertial frames and the principle of relativity......Page 21
1.3 The spacetime geometry of special relativity......Page 23
1.4 Lorentz transformations as four-dimensional ‘rotations’......Page 25
1.5 The interval and the lightcone......Page 26
1.6 Spacetime diagrams......Page 28
Length contraction......Page 30
1.8 Invariant hyperbolae......Page 31
1.9 The Minkowski spacetime line element......Page 32
1.10 Particle worldlines and proper time......Page 34
1.11 The Doppler effect......Page 36
1.12 Addition of velocities in special relativity......Page 38
1.13 Acceleration in special relativity......Page 39
1.14 Event horizons in special relativity......Page 41
On the Electrodynamics of Moving Bodies by A. Einstein......Page 42
Exercises......Page 44
2.1 The concept of a manifold......Page 46
2.3 Curves and surfaces......Page 47
2.4 Coordinate transformations......Page 48
2.5 Summation convention......Page 50
2.6 Geometry of manifolds......Page 51
2.7 Riemannian geometry......Page 52
2.8 Intrinsic and extrinsic geometry......Page 53
2.9 Examples of non-Euclidean geometry......Page 56
2.10 Lengths, areas and volumes......Page 58
2.11 Local Cartesian coordinates......Page 62
2.12 Tangent spaces to manifolds......Page 64
2.13 Pseudo-Riemannian manifolds......Page 65
2.14 Integration over general submanifolds......Page 67
2.15 Topology of manifolds......Page 69
Exercises......Page 70
3.1 Scalar fields on manifolds......Page 73
3.2 Vector fields on manifolds......Page 74
3.3 Tangent vector to a curve......Page 75
3.4 Basis vectors......Page 76
Coordinate basis vectors......Page 77
3.5 Raising and lowering vector indices......Page 79
3.6 Basis vectors and coordinate transformations......Page 80
3.7 Coordinate-independent properties of vectors......Page 81
3.8 Derivatives of basis vectors and the affine connection......Page 82
3.9 Transformation properties of the affine connection......Page 84
3.10 Relationship of the connection and the metric......Page 85
3.11 Local geodesic and Cartesian coordinates......Page 87
3.12 Covariant derivative of a vector......Page 88
Divergence......Page 90
3.14 Intrinsic derivative of a vector along a curve......Page 91
3.15 Parallel transport......Page 93
3.16 Null curves, non-null curves and affine parameters......Page 95
3.17 Geodesics......Page 96
3.18 Stationary property of non-null geodesics......Page 97
3.19 Lagrangian procedure for geodesics......Page 98
Appendix 3A: Vectors as directional derivatives......Page 101
Appendix 3B: Polar coordinates in a plane......Page 102
Appendix 3C: Calculus of variations......Page 107
Exercises......Page 108
4.1 Tensor fields on manifolds......Page 112
4.2 Components of tensors......Page 113
4.3 Symmetries of tensors......Page 114
4.4 The metric tensor......Page 116
4.6 Mapping tensors into tensors......Page 117
Outer product......Page 118
Contraction (and inner product)......Page 119
4.8 Tensors as geometrical objects......Page 120
4.9 Tensors and coordinate transformations......Page 121
4.10 Tensor equations......Page 122
4.11 The quotient theorem......Page 123
4.12 Covariant derivative of a tensor......Page 124
4.13 Intrinsic derivative of a tensor along a curve......Page 127
Exercises......Page 128
5.1 Minkowski spacetime in Cartesian coordinates......Page 131
5.2 Lorentz transformations......Page 132
5.3 Cartesian basis vectors......Page 133
5.4 Four-vectors and the lightcone......Page 135
5.6 Four-velocity......Page 136
5.7 Four-momentum of a massive particle......Page 138
5.8 Four-momentum of a photon......Page 139
5.9 The Doppler effect and relativistic aberration......Page 140
5.10 Relativistic mechanics......Page 142
5.12 Relativistic collisions and Compton scattering......Page 143
5.13 Accelerating observers......Page 145
5.14 Minkowski spacetime in arbitrary coordinates......Page 148
Exercises......Page 151
6.1 The electromagnetic force on a moving charge......Page 155
6.2 The 4-current density......Page 156
6.3 The electromagnetic field equations......Page 158
6.4 Electromagnetism in the Lorenz gauge......Page 159
6.5 Electric and magnetic fields in inertial frames......Page 161
6.6 Electromagnetism in arbitrary coordinates......Page 162
6.7 Equation of motion for a charged particle......Page 164
Exercises......Page 165
7.1 Newtonian gravity......Page 167
7.2 The equivalence principle......Page 168
7.3 Gravity as spacetime curvature......Page 169
7.4 Local inertial coordinates......Page 171
7.5 Observers in a curved spacetime......Page 172
7.6 Weak gravitational fields and the Newtonian limit......Page 173
7.7 Electromagnetism in a curved spacetime......Page 175
7.8 Intrinsic curvature of a manifold......Page 177
7.9 The curvature tensor......Page 178
7.10 Properties of the curvature tensor......Page 179
7.11 The Ricci tensor and curvature scalar......Page 181
7.12 Curvature and parallel transport......Page 183
7.13 Curvature and geodesic deviation......Page 185
7.14 Tidal forces in a curved spacetime......Page 187
Appendix 7A: The surface of a sphere......Page 190
Exercises......Page 192
8.1 The energy–momentum tensor......Page 196
8.2 The energy–momentum tensor of a perfect fluid......Page 198
8.3 Conservation of energy and momentum for a perfect fluid......Page 199
8.4 The Einstein equations......Page 201
8.5 The Einstein equations in empty space......Page 203
8.6 The weak-field limit of the Einstein equations......Page 204
8.7 The cosmological-constant term......Page 205
8.8 Geodesic motion from the Einstein equations......Page 208
8.9 Concluding remarks......Page 210
Brans–Dicke theory......Page 211
Torsion theories......Page 212
Exercises......Page 213
9.1 The general static isotropic metric......Page 216
9.2 Solution of the empty-space field equations......Page 218
9.4 Gravitational redshift for a fixed emitter and receiver......Page 222
9.5 Geodesics in the Schwarzschild geometry......Page 225
9.6 Trajectories of massive particles......Page 227
9.7 Radial motion of massive particles......Page 229
9.8 Circular motion of massive particles......Page 232
9.9 Stability of massive particle orbits......Page 233
9.10 Trajectories of photons......Page 237
9.11 Radial motion of photons......Page 238
9.12 Circular motion of photons......Page 239
9.13 Stability of photon orbits......Page 240
Appendix 9A: General approach to gravitational redshifts......Page 241
Exercises......Page 244
10.1 Precession of planetary orbits......Page 250
10.2 The bending of light......Page 253
10.3 Radar echoes......Page 256
10.4 Accretion discs around compact objects......Page 260
10.5 The geodesic precession of gyroscopes......Page 264
Exercises......Page 266
11.1 The characterisation of coordinates......Page 268
11.2 Singularities in the Schwarzschild metric......Page 269
11.3 Radial photon worldlines in Schwarzschild coordinates......Page 271
11.4 Radial particle worldlines in Schwarzschild coordinates......Page 272
Advanced Eddington–Finkelstein coordinates......Page 274
Retarded Eddington–Finkelstein coordinates......Page 277
11.6 Gravitational collapse and black-hole formation......Page 279
11.7 Spherically symmetric collapse of dust......Page 280
11.8 Tidal forces near a black hole......Page 284
11.9 Kruskal coordinates......Page 286
11.10 Wormholes and the Einstein–Rosen bridge......Page 291
11.11 The Hawking effect......Page 294
Appendix 11A: Compact binary systems......Page 297
Appendix 11B: Supermassive black holes......Page 299
Appendix 11C: Conformal flatness of two-dimensional Riemannian manifolds......Page 302
Exercises......Page 303
12.1 The form of the metric for a stellar interior......Page 308
12.2 The relativistic equations of stellar structure......Page 312
12.3 The Schwarzschild constant-density interior solution......Page 314
12.5 The metric outside a spherically symmetric charged mass......Page 316
12.6 The Reissner–Nordström geometry: charged black holes......Page 320
12.7 Radial photon trajectories in the RN geometry......Page 322
12.8 Radial massive particle trajectories in the RN geometry......Page 324
Exercises......Page 325
13.1 The general stationary axisymmetric metric......Page 330
13.2 The dragging of inertial frames......Page 332
13.3 Stationary limit surfaces......Page 334
13.4 Event horizons......Page 335
13.5 The Kerr metric......Page 337
13.6 Limits of the Kerr metric......Page 339
13.7 The Kerr–Schild form of the metric......Page 341
Singularities and horizons......Page 342
Stationary limit surfaces......Page 344
The ergoregion......Page 345
13.9 The Penrose process......Page 347
13.10 Geodesics in the equatorial plane......Page 350
13.11 Equatorial trajectories of massive particles......Page 352
13.12 Equatorial motion of massive particles with zero angular momentum......Page 353
13.13 Equatorial circular motion of massive particles......Page 355
13.14 Stability of equatorial massive particle circular orbits......Page 357
13.15 Equatorial trajectories of photons......Page 358
13.16 Equatorial principal photon geodesics......Page 359
13.17 Equatorial circular motion of photons......Page 361
13.18 Stability of equatorial photon orbits......Page 362
13.19 Eddington–Finkelstein coordinates......Page 364
13.20 The slow-rotation limit and gyroscope precession......Page 367
Exercises......Page 370
14.1 The cosmological principle......Page 375
14.2 Slicing and threading spacetime......Page 376
14.3 Synchronous coordinates......Page 377
14.4 Homogeneity and isotropy of the universe......Page 378
14.5 The maximally symmetric 3-space......Page 379
14.7 Geometric properties of the FRW metric......Page 382
14.8 Geodesics in the FRW metric......Page 385
14.9 The cosmological redshift......Page 387
14.10 The Hubble and deceleration parameters......Page 388
14.11 Distances in the FRW geometry......Page 391
Luminosity distance......Page 392
Angular diameter distance......Page 393
14.12 Volumes and number densities in the FRW geometry......Page 394
14.13 The cosmological field equations......Page 396
14.14 Equation of motion for the cosmological fluid......Page 399
Exercises......Page 401
15.1 Components of the cosmological fluid......Page 406
Matter......Page 407
Radiation......Page 408
Relative contributions of the components......Page 409
15.2 Cosmological parameters......Page 410
15.3 The cosmological field equations......Page 412
15.4 General dynamical behaviour of the universe......Page 413
15.5 Evolution of the scale factor......Page 417
The Friedmann models......Page 420
The Lemaitre models......Page 424
Einstein’s static universe......Page 427
15.7 Look-back time and the age of the universe......Page 428
15.8 The distance–redshift relation......Page 431
15.9 The volume–redshift relation......Page 433
15.10 Evolution of the density parameters......Page 435
15.11 Evolution of the spatial curvature......Page 437
Particle horizon......Page 438
Hubble distance......Page 440
Exercises......Page 441
16.1 Definition of inflation......Page 448
16.2 Scalar fields and phase transitions in the very early universe......Page 450
16.3 A scalar field as a cosmological fluid......Page 451
16.4 An inflationary epoch......Page 453
16.5 The slow-roll approximation......Page 454
16.7 The amount of inflation......Page 455
16.8 Starting inflation......Page 457
16.9 ‘New’ inflation......Page 458
16.10 Chaotic inflation......Page 460
16.11 Stochastic inflation......Page 461
16.13 Classical evolution of scalar-field perturbations......Page 462
The perturbed Einstein field equations......Page 463
Perturbation equations in Fourier space......Page 465
16.14 Gauge invariance and curvature perturbations......Page 466
16.15 Classical evolution of curvature perturbations......Page 469
16.16 Initial conditions and normalisation of curvature perturbations......Page 472
16.17 Power spectrum of curvature perturbations......Page 476
16.18 Power spectrum of matter-density perturbations......Page 478
16.19 Comparison of theory and observation......Page 479
Exercises......Page 482
17.1 The weak-field metric......Page 487
Global Lorentz transformations......Page 488
Infinitesimal general coordinate transformations......Page 489
17.2 The linearised gravitational field equations......Page 490
17.3 Linearised gravity in the Lorenz gauge......Page 492
17.4 General properties of the linearised field equations......Page 493
17.5 Solution of the linearised field equations in vacuo......Page 494
17.6 General solution of the linearised field equations......Page 495
17.7 Multipole expansion of the general solution......Page 500
17.8 The compact-source approximation......Page 501
17.9 Stationary sources......Page 503
17.10 Static sources and the Newtonian limit......Page 505
17.11 The energy–momentum of the gravitational field......Page 506
Appendix 17A: The Einstein–Maxwell formulation of linearised gravity......Page 510
Exercises......Page 513
18.1 Plane gravitational waves and polarisation states......Page 518
18.2 Analogy between gravitational and electromagnetic waves......Page 521
18.3 Transforming to the transverse-traceless gauge......Page 522
18.4 The effect of a gravitational wave on free particles......Page 524
18.5 The generation of gravitational waves......Page 527
18.6 Energy flow in gravitational waves......Page 531
18.7 Energy loss due to gravitational-wave emission......Page 533
18.8 Spin-up of binary systems: the binary pulsar PSR B1913+16......Page 536
18.9 The detection of gravitational waves......Page 537
Exercises......Page 540
19.1 Hamilton’s principle in Newtonian mechanics......Page 544
19.2 Classical field theory and the action......Page 547
19.3 Euler–Lagrange equations......Page 549
19.4 Alternative form of the Euler–Lagrange equations......Page 551
19.5 Equivalent actions......Page 553
19.6 Field theory of a real scalar field......Page 554
19.7 Electromagnetism from a variational principle......Page 556
19.8 The Einstein–Hilbert action and general relativity in vacuo......Page 559
19.9 An equivalent action for general relativity in vacuo......Page 562
19.10 The Palatini approach for general relativity in vacuo......Page 563
19.11 General relativity in the presence of matter......Page 565
19.12 The dynamical energy–momentum tensor......Page 566
Exercises......Page 569
Bibliography......Page 575
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C......Page 577
E......Page 578
F......Page 580
G......Page 581
H......Page 582
L......Page 583
M......Page 584
N......Page 585
O,P......Page 586
Q,R......Page 587
S......Page 588
T......Page 590
U,V......Page 591
W,X,Z......Page 592