General Relativity for the Gifted Amateur: contents
General Relativity for the Gifted Amateur: contents
0 Overture
0.1 What is relativity? 1
0.2 What is general relativity? 3
0.3 What is a metric? 5
0.4 What are we building on? 6
0.5 Who is this book for? 8
0.6 Units in this book 9
Exercises 10
I Geometry and mechanics in at spacetime 11
1 Special relativity 12
1.1 A common sense start 12
1.2 The speed of light 13
1.3 Light cones and the Lorentz transformation 15
1.4 Paths through spacetime 18
1.5 Experiments 19
Exercises 21
2 Vectors in flat spacetime 22
2.1 Vectors 23
2.2 Coordinate transformations 24
2.3 Examples of vectors 27
2.4 Principle of least action 31
Exercises 34
3 Coordinates 36
3.1 Coordinates in Euclidean space 36
3.2 Farewell to the position vector 39
3.3 Non-Euclidean space 40
Exercises 41
4 Linear slot machines 43
4.1 Dot products and down vectors 44
4.2 Vectors and 1-forms 46
4.3 Transformations 49
4.4 Tensors 50
4.5 Energy-momentum tensor 52
Exercises 55
5 The metric 56
5.1 Metrics in general 56
5.2 Meet some metrics 58
5.3 Light and light cones 60
5.4 Lengths, areas, volumes 62
Exercises 65
II Curvature and general relativity 67
6 Finding a theory of gravitation 68
6.1 Free fall and the equivalence principle 68
6.2 Why general relativity? 73
6.3 A differential equation to describe gravity 75
6.4 Local flatness 76
6.5 Time dilation in a gravitational eld 77
Exercises 79
7 Parallel lines and the covariant derivative 81
7.1 Parallelism 82
7.2 Derivatives and connections 83
7.3 The covariant derivative 85
7.4 Parametrized paths 86
7.5 Enter the metric 88
Exercises 89
8 Free fall and geodesics 91
8.1 Extremal intervals 91
8.2 A geodesic equation 95
8.3 Inertial forces 96
8.4 Geodesics for photons 98
Exercises
9 Geodesic equations and connection coefficients 101
9.1 Finding connection coeffcients 101
9.2 The geodesic equation from the action 104
Exercises 105
10 Making measurements in relativity 108
10.1 Observers and their observations 108
10.2 Coordinate and non-coordinate bases 110
10.3 The orthonormal frame 114
10.4 Freely-falling frames 116
Exercises 118
11 Riemann curvature and the Ricci tensor 120
11.1 What is curvature?
11.2 Tidal forces 121
11.3 Riemann curvature 124
11.4 Symmetries of the Riemann tensor 126
11.5 The Ricci tensor and Ricci scalar 127
11.6 Example computations 128
11.7 Geodesic deviation revisited 129
Exercises 130
12 The energy-momentum tensor 131
12.1 Another look at the energy-momentum tensor 131
12.2 Example energy-momentum tensors 132
12.3 Classical particles 134
12.4 Conservation laws 136
Exercises 140
13 The gravitational field equations 141
13.1 Geometry: a recap of the key ingredients 141
13.2 Physics: the key ingredients 142
13.3 An incorrect guess 145
13.4 Einstein’s field equation 147
Exercises 150
14 The triumphs of general relativity 151
14.1 Weak fields and the Newtonian limit 151
14.2 Gravitational waves 153
14.3 Stars, trajectories and orbits 155
14.4 Cosmology 156
III Cosmology 157
15 An introduction to cosmology 158
15.1 The cosmological principle 159
15.2 The Hubble flow 160
15.3 Cosmic time 162
15.4 Universe 0: an empty universe 163
15.5 Universe 1: flat and expanding 164
Exercises 167
16 Robertson-Walker spaces 169
16.1 Spaces with constant curvature 169
16.2 Three Robertson-Walker spaces 173
16.3 Redshift and cosmic expansion 176
16.4 The initial singularity 178
Exercises 179
17 The Friedmann equations 181
17.1 Enter energy-momentum 182
17.2 Enter thermodynamics 183
17.3 Dust and radiation 184
Exercises 187
18 Universes of the past and future 188
18.1 Spatially-flat universes 188
18.2 Curved universes with k = 0 191
18.3 Einstein, Lemaitre and Eddington 192
18.4 A brief history of model universes 196
Exercises 199
19 Causality, infinity and horizons 201
19.1 Penrose diagrams 202
19.2 The de Sitter spacetime 209
19.3 Big-bang singularities 211
Exercises 214
IV Orbits, stars and black holes 217
20 Newtonian orbits 218
20.1 Kepler’s laws 219
20.2 Anatomy of an orbit 220
20.3 Effective potentials 222
20.4 Allowed trajectories 223
20.5 The why? of orbits 225
Exercises 227
21 The Schwarzschild geometry 229
21.1 Justifying the solution 230
21.2 Components of the Riemann tensor 231
21.3 A gravitating object 232
21.4 The meaning of the coordinates 234
Exercises 235
22 Motion in the Schwarzschild geometry 237
22.1 Constants of the motion 238
22.2 Gravitational redshift 239
22.3 Motion in Schwarzschild spacetime 240
22.4 Example: the radial plunge 242
Exercises 244
23 Orbits in the Schwarzschild geometry 246
23.1 Orbits for massive particles 246
23.2 Stable circular orbits 248
23.3 Precession of the perihelion 249
Exercises 252
24 Photons in the Schwarzschild geometry 254
24.1 Photon trajectories 254
24.2 Looking around 258
Exercises 261
25 Black holes 262
25.1 The surface r = 2M 264
25.2 The tortoise coordinate 265
25.3 Death of an astronaut 266
25.4 Looking around near a black hole 267
25.5 Gravitational collapse 268
Exercises 270
26 Black-hole singularities 272
26.1 Singularities 272
26.2 Eddington-Finkelstein coordinates 275
Exercises 279
27 Kruskal-Szekeres coordinates 280
27.1 Enter the Kruskal metric 280
27.2 Wormholes 284
27.3 Another Penrose diagram 285
Exercises 287
28 Hawking radiation 289
28.1 Hawking radiation 289
28.2 Black-hole thermodynamics 292
Exercises 296
29 Charged and rotating black holes 297
29.1 Charged black holes 297
29.2 Kerr black holes 299
29.3 Interacting with the Kerr geometry 304
Exercises 305
V Geometry 307
30 Classical curvature 308
30.1 Curvature of a line 308
30.2 Curvature with vectors 310
30.3 Two-dimensional surfaces 312
30.4 Gauss’ equation 314
30.5 Intrinsic and extrinsic curvature 316
30.6 Riemann’s project 318
Exercises 320
31 A reintroduction to geometry 322
31.1 Old notions of vectors and gradients 323
31.2 Vectors and vector fields 324
31.3 Linear slot machines again 327
31.4 Tensors again 329
31.5 Examples of tensor operations 330
Exercises 332
32 Differential forms 334
32.1 2-forms 334
32.2 p-forms 336
32.3 p-vectors 337
Exercises 339
33 Exterior and Lie derivatives 340
33.1 Exterior calculus 340
33.2 Commutators 342
33.3 Lie derivatives of vectors 344
33.4 Lie derivatives of tensors 347
33.5 Killing vectors 348
Exercises 350
34 Geometry of the connection 351
34.1 Covariant derivative in pictures 352
34.2 Connection and exterior derivative 353
34.3 Covariant derivative of tensors 355
34.4 The metric revisited 358
Exercises 361
35 Riemann curvature revisited 363
35.1 Geodesic deviation (slight return) 363
35.2 Components of the curvature tensor 366
35.3 Parallel transport again 368
35.4 The meaning of the Ricci tensor 370
Exercises 372
36 Cartan’s method 374
36.1 Connection 1-forms 374
36.2 Two rules 377
36.3 Le repere mobile 379
36.4 Example computations 380
Exercises 385
37 Duality and the volume form 386
37.1 Motivation: 2-forms and flux 386
37.2 Hodge star operation 387
37.3 Volume forms 392
Exercises 395
38 Forms, chains and Stokes’ theorem 397
38.1 Integration 397
38.2 Integrating over forms 400
38.3 Anatomy of an integral 401
38.4 Boundaries and chains 404
38.5 Stokes’ theorem 405
Exercises 408
VI Classical and quantum fields 411
39 Fluids as dry water 412
39.1 Euler’s equation 413
39.2 Energy and Bernoulli’s equation 415
39.3 Energy-momentum tensor 418
39.4 Relativistic fluids 420
Exercises 425
40 Lagrangian field theory 427
40.1 Matter fields 428
40.2 Action and equations of motion 429
40.3 Fields in curved spacetime 432
40.4 Motivating the Einstein equation 433
40.5 Energy-momentum tensor 436
40.6 Noether’s theorem 437
40.7 The perfect fluid 439
Exercises 442
41 Inflation 444
41.1 Symmetry breaking 445
41.2 Effective potentials 448
41.3 Why flat? 450
Exercises 451
42 The electromagnetic field 452
42.1 Electric charge in a field 452
42.2 Faraday tensor and Maxwell equations 454
42.3 Gauge freedom 457
42.4 Geometrical electromagnetism 459
Exercises 463
43 Charge conservation and the Bianchi identity 466
43.1 Conserving electric charge 466
43.2 Electromagnetic gauge field 468
43.3 Gravitational curvature 470
Exercises 474
44 Gauge fields 475
44.1 Fibre bundles and gauge invariance 475
44.2 Parallel transport and field strength 479
Exercises 482
45 Weak gravitational fields 484
45.1 The Newtonian limit 484
45.2 Linearized theory of gravitation 486
45.3 Exploiting gauges 487
Exercises 491
46 Gravitational waves 493
46.1 Waves in a gauge theory 493
46.2 Lorenz gauge for gravitational waves 495
46.3 Quadrupolar radiation 500
46.4 Radiated energy and power 502
46.5 An exact solution 504
46.6 The discovery of gravitational waves 505
Exercises 508
47 The properties of gravitons 511
47.1 Force-carrying particles 511
47.2 Photon propagation and polarization 513
47.3 Graviton propagation and polarization 515
Exercises 518
48 Higher-dimensional spacetime 519
48.1 Gauge transformations in five dimensions 520
48.2 Unifying electromagnetism and gravitation 521
Exercises 524
49 From classical to quantum gravity 526
49.1 Extra dimensions 526
49.2 String theory 529
49.3 Parametrizing the string 531
49.4 Strings in relativity 533
49.5 Superspace 535
49.6 Loop quantum gravity 536
49.7 Anti-de Sitter spacetime 538
49.8 Our current best guess 541
Exercises 544
50 The Big-Bang singularity 546
50.1 The set up 546
50.2 Facts about Euclidean geometry 546
50.3 Orthogonal geodesics in spacetime 547
50.4 Our Universe 550
Exercises 551
A Further reading 553
B Conventions and notation 561
B.1 Electromagnetic units 561
B.2 Vectors, 1-forms and tensors 561
B.3 Covariant derivatives
C Manifolds and bundles 564
C.1 Preliminaries 565
C.2 Maps and functions 566
C.3 One-to-one, into and onto 566
C.4 Continuous maps 567
C.5 Manifolds, coordinates and charts 568
C.6 Functions on the manifold 570
C.7 Differentiation on the manifold 571
C.8 Compact regions 574
C.9 Curves 574
C.10 Tangent spaces 575
C.11 Fibre bundles 577
D Embedding 580
Exercises 585
E Answers to selected problems 586
Index 613