Relativity: An Introduction to Special and General Relativity

Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 7
Preface......Page 17
Notation......Page 21
1.1 Inertial systems......Page 23
1.2 Invariance under translations......Page 24
1.3 Invariance under rotations......Page 25
1.4 Invariance under Galilei transformations......Page 26
1.5 Some remarks on the homogeneity of time......Page 27
Exercises......Page 28
2.1 The Michelson experiment......Page 29
2.2 The Lorentz transformations......Page 30
2.3 Some properties of Lorentz transformations......Page 32
3 Our world as a Minkowski space......Page 36
3.2 Four-vectors and light cones......Page 37
3.3 Measuring length and time in Minkowski space......Page 39
3.4.2 The twin paradox......Page 42
3.5 Causality, and velocities larger than that of light......Page 43
4.1 Kinematics......Page 46
4.2 Equations of motion......Page 48
4.3 Hyperbolic motion......Page 50
4.4 Systems of particles......Page 52
5.1 Invariance of phase and null vectors......Page 56
5.2 The Doppler effect — shift in the frequency of a wave......Page 57
5.3 Aberration – change in the direction of a light ray......Page 58
5.4 The visual shape of moving bodies......Page 59
5.5 Reflection at a moving mirror......Page 61
5.6 Dragging of light within a fluid......Page 62
6.1 Some definitions......Page 63
6.2 Tensor algebra......Page 65
6.3 Symmetries of tensors......Page 66
6.4 Algebraic properties of second rank tensors......Page 68
6.5 Tensor analysis......Page 70
Exercises......Page 71
7.1 The Maxwell equations in three-dimensional notation......Page 72
7.2 Current four-vector, four-potential, and the retarded potentials......Page 73
7.3 Field tensor and the Maxwell equations......Page 74
7.4 Poynting's theorem, Lorentz force, and the energy-momentum tensor......Page 77
7.5 The variational principle for the Maxwell equations......Page 78
8 Transformation properties of electromagnetic fields: examples......Page 80
8.1 Current and four-potential......Page 81
8.2 Field tensor and energy-momentum tensor......Page 82
Exercises......Page 84
9.1 Null tetrads and Lorentz transformations......Page 85
9.2 Self-dual bivectors and the electromagnetic field tensor......Page 87
9.3 The algebraic classification of electromagnetic fields......Page 88
9.4 The physical interpretation of electromagnetic null fields......Page 89
Exercises......Page 90
10.1 The equations of motion of charged test particles......Page 91
10.2 The variational principle for charged particles......Page 92
10.3 Canonical equations......Page 94
10.4 The field of a charged particle in arbitrary motion......Page 96
10.5 The equations of motion of charged particles — the self-force......Page 99
Further reading for Chapter 10......Page 101
11.1 The current density......Page 102
11.2 The dipole term and its field......Page 104
12.1 Field equations and constitutive relations......Page 106
12.3 The energy-momentum tensor......Page 109
13.1 Perfect fluids......Page 111
13.2 Other physical theories – an outlook......Page 114
14.1 Coordinate systems......Page 117
14.2 Equations of motion......Page 119
14.3 The geodesic equation......Page 120
14.4 Geodesic deviation......Page 122
15 Why Riemannian geometry?......Page 125
16.1 The metric......Page 127
16.2 Geodesics and Christoffel symbols......Page 128
16.3 Coordinate transformations......Page 131
16.4 Special coordinate systems......Page 132
16.5 The physical meaning and interpretation of coordinate systems......Page 136
17 Tensor algebra......Page 138
17.1 Scalars and vectors......Page 139
17.2 Tensors and other geometrical objects......Page 140
17.3 Algebraic operations with tensors......Page 143
17.4 Tetrad and spinor components of tensors......Page 144
18.1 Partial and covariant derivatives......Page 148
18.2 The covariant differential and local parallelism......Page 151
18.3 Parallel displacement along a curve and the parallel propagator......Page 153
18.4 Fermi–Walker transport......Page 154
18.5 The Lie derivative......Page 155
Further reading for Chapter 18......Page 157
19.1 Intrinsic geometry and curvature......Page 158
19.2 The curvature tensor and global parallelism of vectors......Page 159
19.3 The curvature tensor and second derivatives of the metric tensor......Page 161
19.4 Properties of the curvature tensor......Page 163
19.5 Spaces of constant curvature......Page 166
Further reading for Chapter 19......Page 170
20.2 Some important differential operators......Page 171
20.3 Volume, surface and line integrals......Page 172
20.4 Integral laws......Page 175
20.5 Integral conservation laws......Page 176
Further reading for Chapter 20......Page 177
21.1 How does one find the fundamental physical laws?......Page 178
21.2 Particle mechanics......Page 180
21.3 Electrodynamics in vacuo......Page 184
21.4 Geometrical optics......Page 187
21.5 Thermodynamics......Page 189
21.6 Perfect fluids and dust......Page 193
Further reading for Chapter 21......Page 194
22.1 The Einstein field equations......Page 195
22.2 The Newtonian limit......Page 198
22.3 The equations of motion of test particles......Page 200
22.4 A variational principle for Einstein's theory......Page 204
23.1 The field equations......Page 207
23.2 The solution of the vacuum field equations......Page 210
23.3 General discussion of the Schwarzschild solution......Page 211
23.4 The motion of the planets and perihelion precession......Page 213
23.5 The propagation of light in the Schwarzschild field......Page 216
23.6 Further aspects of the Schwarzschild solution......Page 220
23.7 The Reissner—Nordström solution......Page 221
24.1 Some general remarks......Page 222
24.2 Perihelion precession and planetary orbits......Page 223
24.3 Light deflection by the Sun......Page 224
24.5 Measurements of the travel time of radar signals (time delay)......Page 225
24.6 Geodesic precession of a top......Page 226
25.1 The spherically symmetric gravitational lens......Page 227
25.2 Galaxies as gravitational lenses......Page 229
Exercise......Page 230
26.1 The field equations......Page 231
26.2 The solution of the field equations......Page 232
26.3 Matching conditions and connection to the exterior Schwarzschild solution......Page 234
26.4 A discussion of the interior Schwarzschild solution......Page 236
Exercises......Page 237
27.1 Justification for a linearized theory and its realm of validity......Page 239
27.2 The fundamental equations of the linearized theory......Page 240
27.3 A discussion of the fundamental equations and a comparison with special-relativistic electrodynamics......Page 242
27.4 The far field due to a time-dependent source......Page 243
27.5 Discussion of the properties of the far field (linearized theory)......Page 247
27.6 Some remarks on approximation schemes......Page 248
28.1 What are far fields?......Page 249
28.2 The energy-momentum pseudotensor for the gravitational field......Page 252
28.3 The balance equations for momentum and angular momentum......Page 256
28.4 Is there an energy law for the gravitational field?......Page 259
29.1 Are there gravitational waves?......Page 260
29.2 Plane gravitational waves in the linearized theory......Page 262
29.3 Plane waves as exact solutions of Einstein's equations......Page 265
29.4 The experimental evidence for gravitational waves......Page 270
30.2 Three-dimensional hypersurfaces and reduction formulae for the curvature tensor......Page 271
30.3 The Cauchy problem for the vacuum field equations......Page 275
30.4 The characteristic initial value problem......Page 277
30.5 Matching conditions at the boundary surface of two metrics......Page 279
31.1 Special simple vector fields......Page 283
31.2 Timelike vector fields......Page 287
31.3 Null vector fields......Page 290
32.1 What is the Petrov classification?......Page 294
32.2 The algebraic classification of gravitational fields......Page 295
32.3 The physical interpretation of degenerate vacuum gravitational fields......Page 298
33.2 Killing vectors......Page 300
33.3 Killing vectors of some simple spaces......Page 302
33.4 Relations between the curvature tensor and Killing vectors......Page 303
33.5 Groups of motion......Page 305
33.6 Killing vectors and conservation laws......Page 310
Exercises......Page 314
34.1 Degenerate vacuum solutions......Page 315
34.2 Vacuum solutions with special symmetry properties......Page 317
34.3 Perfect fluid solutions with special symmetry properties......Page 320
Exercises......Page 321
35.1 How does one examine the singular points of a metric?......Page 323
35.2 Radial geodesics near r = 2M......Page 325
35.3 The Schwarzschild solution in other coordinate systems......Page 326
35.4 The Schwarzschild solution as a black hole......Page 329
36.1 The evolutionary phases of a spherically symmetric star......Page 332
36.2 The critical mass of a star......Page 333
36.3 Gravitational collapse of spherically symmetric dust......Page 337
Further reading for Chapter 36......Page 343
37.1 The Kerr solution......Page 344
37.2 Gravitational collapse — the possible life history of a rotating star......Page 347
37.3 Some properties of black holes......Page 349
37.4 Are there black holes?......Page 350
Further reading for Chapter 37......Page 351
38.1 The problem......Page 352
38.2 Unifled quantum field theory and quantization of the gravitational field......Page 353
38.3 Semiclassical gravity......Page 354
38.4 Quantization in a given classical gravitational field......Page 355
38.5 Black holes are not black — the thermodynamics of black holes......Page 358
39.1 The problem and methods to answer it......Page 363
39.2 Infinity of the three-dimensional Euclidean space (E3)......Page 365
39.3 The conformal structure of Minkowski space......Page 366
39.4 Asymptotically flat gravitational fields......Page 368
39.5 Examples of Penrose diagrams......Page 369
Exercises......Page 371
VII. Cosmology......Page 373
40.1 The cosmological principle and Robertson—Walker metrics......Page 374
40.2 The motion of particles and photons......Page 375
40.3 Distance definitions and horizons......Page 378
40.4 Some remarks on physics in closed universes......Page 382
41.1 The Einstein field equations for Robertson—Walker metrics......Page 385
41.2 The most important Friedmann universes......Page 387
41.3 Consequences of the field equations for models with arbitrary equation of state having positive pressure and positive rest mass density......Page 391
Exercises......Page 392
42.1 Redshift and mass density......Page 393
42.2 The earliest epochs of our universe and the cosmic background radiation......Page 395
42.3 A Schwarzschild cavity in the Friedmann universe......Page 398
43.1 What is a cosmological model?......Page 402
43.2 Solutions of Bianchi type I with dust......Page 403
43.3 The Gödel universe......Page 406
43.4 Singularity theorems......Page 407
Further reading for Chapter 43......Page 409
Monographs and research articles......Page 410
Index......Page 414