General Relativity, Black Holes, and Cosmology

1108 Pages·2017·11.7 MB·English

General Relativity, Black Holes, and Cosmology
Andrew J. S. Hamilton
13 January 2020

Contents
Table of contents page iii
List of illustrations xviii
List of tables xxiii
List of exercises and concept questions xxiv
Legal notice 1
Notation 2
PART ONE FUNDAMENTALS 5
Concept Questions 7
What’s important? 9
1 Special Relativity 10
1.1 Motivation 10
1.2 The postulates of special relativity 11
1.3 The paradox of the constancy of the speed of light 13
1.4 Simultaneity 17
1.5 Time dilation 18
1.6 Lorentz transformation 19
1.7 Paradoxes: Time dilation, Lorentz contraction, and the Twin paradox 22
1.8 The spacetime wheel 26
1.9 Scalar spacetime distance 31
1.10 4-vectors 33
1.11 Energy-momentum 4-vector 35
1.12 Photon energy-momentum 37
1.13 What things look like at relativistic speeds 39
1.14 Occupation number, phase-space volume, intensity, and (cid:29)ux 44
1.15 How to program Lorentz transformations on a computer 46
iii
iv Contents
Concept Questions 48
What’s important? 50
2 Fundamentals of General Relativity 51
2.1 Motivation 52
2.2 The postulates of General Relativity 53
2.3 Implications of Einstein’s principle of equivalence 55
2.4 Metric 57
2.5 Timelike, spacelike, proper time, proper distance 58
2.6 Orthonormal tetrad basis γγ 58
m
2.7 Basis of coordinate tangent vectors e 59
µ
2.8 4-vectors and tensors 60
2.9 Covariant derivatives 63
2.10 Torsion 66
2.11 Connection coe(cid:30)cients in terms of the metric 68
2.12 Torsion-free covariant derivative 68
2.13 Mathematical aside: What if there is no metric? 72
2.14 Coordinate 4-velocity 72
2.15 Geodesic equation 72
2.16 Coordinate 4-momentum 73
2.17 A(cid:30)ne parameter 74
2.18 A(cid:30)ne distance 74
2.19 Riemann tensor 76
2.20 Ricci tensor, Ricci scalar 79
2.21 Einstein tensor 80
2.22 Bianchi identities 80
2.23 Covariant conservation of the Einstein tensor 81
2.24 Einstein equations 81
2.25 Summary of the path from metric to the energy-momentum tensor 82
2.26 Energy-momentum tensor of a perfect (cid:29)uid 82
2.27 Newtonian limit 83
3 More on the coordinate approach 90
3.1 Weyl tensor 90
3.2 Evolution equations for the Weyl tensor, and gravitational waves 91
3.3 Geodesic deviation 93
4 Action principle for point particles 95
4.1 Principle of least action for point particles 95
4.2 Generalized momentum 97
4.3 Lagrangian for a test particle 97
4.4 Massless test particle 99
Contents v
4.5 E(cid:27)ective Lagrangian for a test particle 100
4.6 Nice Lagrangian for a test particle 101
4.7 Action for a charged test particle in an electromagnetic (cid:28)eld 102
4.8 Symmetries and constants of motion 104
4.9 Conformal symmetries 105
4.10 (Super-)Hamiltonian 108
4.11 Conventional Hamiltonian 109
4.12 Conventional Hamiltonian for a test particle 109
4.13 E(cid:27)ective (super-)Hamiltonian for a test particle with electromagnetism 111
4.14 Nice (super-)Hamiltonian for a test particle with electromagnetism 111
4.15 Derivatives of the action 113
4.16 Hamilton-Jacobi equation 114
4.17 Canonical transformations 114
4.18 Symplectic structure 116
4.19 Symplectic scalar product and Poisson brackets 118
4.20 (Super-)Hamiltonian as a generator of evolution 118
4.21 In(cid:28)nitesimal canonical transformations 119
4.22 Constancy of phase-space volume under canonical transformations 120
4.23 Poisson algebra of integrals of motion 120
Concept Questions 122
What’s important? 124
5 Observational Evidence for Black Holes 125
6 Ideal Black Holes 128
6.1 De(cid:28)nition of a black hole 128
6.2 Ideal black hole 129
6.3 No-hair theorem 129
7 Schwarzschild Black Hole 131
7.1 Schwarzschild metric 131
7.2 Stationary, static 132
7.3 Spherically symmetric 133
7.4 Energy-momentum tensor 134
7.5 Birkho(cid:27)’s theorem 134
7.6 Horizon 135
7.7 Proper time 136
7.8 Redshift 137
7.9 (cid:16)Schwarzschild singularity(cid:17) 137
7.10 Weyl tensor 138
7.11 Singularity 138
7.12 Gullstrand-PainlevØ metric 140
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7.13 Embedding diagram 148
7.14 Schwarzschild spacetime diagram 150
7.15 Gullstrand-PainlevØ spacetime diagram 151
7.16 Eddington-Finkelstein spacetime diagram 151
7.17 Kruskal-Szekeres spacetime diagram 153
7.18 Antihorizon 154
7.19 Analytically extended Schwarzschild geometry 156
7.20 Penrose diagrams 157
7.21 Penrose diagrams as guides to spacetime 159
7.22 Future and past horizons 161
7.23 Oppenheimer-Snyder collapse to a black hole 161
7.24 Apparent horizon 162
7.25 True horizon 163
7.26 Penrose diagrams of Oppenheimer-Snyder collapse 164
7.27 Illusory horizon 165
7.28 Collapse of a shell of matter on to a black hole 167
7.29 The illusory horizon and black hole thermodynamics 169
7.30 Rindler space and Rindler horizons 170
7.31 Rindler observers who start at rest, then accelerate 172
7.32 Killing vectors 176
7.33 Killing tensors 180
7.34 Lie derivative 181
8 Reissner-Nordstr(cid:246)m Black Hole 187
8.1 Reissner-Nordstr(cid:246)m metric 187
8.2 Energy-momentum tensor 188
8.3 Weyl tensor 189
8.4 Horizons 189
8.5 Gullstrand-PainlevØ metric 189
8.6 Radial null geodesics 191
8.7 Finkelstein coordinates 192
8.8 Kruskal-Szekeres coordinates 193
8.9 Analytically extended Reissner-Nordstr(cid:246)m geometry 196
8.10 Penrose diagram 196
8.11 Antiverse: Reissner-Nordstr(cid:246)m geometry with negative mass 198
8.12 Outgoing, ingoing 198
8.13 The in(cid:29)ationary instability 199
8.14 The X point 201
8.15 Extremal Reissner-Nordstr(cid:246)m geometry 201
8.16 Super-extremal Reissner-Nordstr(cid:246)m geometry 202
Contents vii
8.17 Reissner-Nordstr(cid:246)m geometry with imaginary charge 204
9 Kerr-Newman Black Hole 207
9.1 Boyer-Lindquist metric 207
9.2 Oblate spheroidal coordinates 208
9.3 Time and rotation symmetries 210
9.4 Ring singularity 210
9.5 Horizons 210
9.6 Angular velocity of the horizon 212
9.7 Ergospheres 212
9.8 Turnaround radius 212
9.9 Antiverse 213
9.10 Sisytube 213
9.11 Extremal Kerr-Newman geometry 213
9.12 Super-extremal Kerr-Newman geometry 215
9.13 Energy-momentum tensor 215
9.14 Weyl tensor 216
9.15 Electromagnetic (cid:28)eld 216
9.16 Principal null congruences 216
9.17 Finkelstein coordinates 217
9.18 Doran coordinates 217
9.19 Penrose diagram 220
Concept Questions 221
What’s important? 223
10 Homogeneous, Isotropic Cosmology 224
10.1 Observational basis 224
10.2 Cosmological Principle 230
10.3 Friedmann-Lema(cid:238)tre-Robertson-Walker metric 230
10.4 Spatial part of the FLRW metric: informal approach 230
10.5 Comoving coordinates 233
10.6 Spatial part of the FLRW metric: more formal approach 234
10.7 FLRW metric 235
10.8 Einstein equations for FLRW metric 235
10.9 Newtonian (cid:16)derivation(cid:17) of Friedmann equations 236
10.10 Hubble parameter 238
10.11 Critical density 239
10.12 Omega 239
10.13 Types of mass-energy 241
10.14 Redshifting 243
10.15 Evolution of the cosmic scale factor 244
viii Contents
10.16 Age of the Universe 246
10.17 Conformal time 247
10.18 Looking back along the lightcone 248
10.19 Hubble diagram 249
10.20 Recombination 251
10.21 Horizon 251
10.22 In(cid:29)ation 254
10.23 Evolution of the size and density of the Universe 256
10.24 Evolution of the temperature of the Universe 257
10.25 Neutrino mass 262
10.26 Occupation number, number density, and energy-momentum 265
10.27 Occupation numbers in thermodynamic equilibrium 267
10.28 Maximally symmetric spaces 275
PART TWO TETRAD APPROACH TO GENERAL RELATIVITY 285
Concept Questions 287
What’s important? 289
11 The tetrad formalism 290
11.1 Tetrad 290
11.2 Vierbein 291
11.3 The metric encodes the vierbein 291
11.4 Tetrad transformations 292
11.5 Tetrad vectors and tensors 293
11.6 Index and naming conventions for vectors and tensors 295
11.7 Gauge transformations 296
11.8 Directed derivatives 296
11.9 Tetrad covariant derivative 297
11.10 Relation between tetrad and coordinate connections 299
11.11 Antisymmetry of the tetrad connections 299
11.12 Torsion tensor 299
11.13 No-torsion condition 300
11.14 Tetrad connections in terms of the vierbein 300
11.15 Torsion-free covariant derivative 301
11.16 Riemann curvature tensor 301
11.17 Ricci, Einstein, Bianchi 304
11.18 Expressions with torsion 305
11.19 General relativity in 2 spacetime dimensions 306
Contents ix
12 Spin and Newman-Penrose tetrads 311
12.1 Spin tetrad formalism 311
12.2 Newman-Penrose tetrad formalism 314
12.3 Weyl tensor 316
12.4 Petrov classi(cid:28)cation of the Weyl tensor 319
13 The geometric algebra 322
13.1 Products of vectors 323
13.2 Geometric product 324
13.3 Reverse 326
13.4 The pseudoscalar and the Hodge dual 326
13.5 Multivector metric 328
13.6 General products of multivectors 328
13.7 Re(cid:29)ection 330
13.8 Rotation 331
13.9 Rotor group 334
13.10 Active and passive rotations 338
13.11 A rotor is a spin-1 object 338
2
13.12 2D rotations and complex numbers 339
13.13 Quaternions 340
13.14 3D rotations and quaternions 342
13.15 Pauli matrices 344
13.16 Pauli spinors as quaternions, or scaled rotors 345
13.17 Spin axis 347
14 The spacetime algebra 349
14.1 Spacetime algebra 349
14.2 Complex quaternions 351
14.3 Lorentz transformations and complex quaternions 353
14.4 Spatial inversion (P) and Time reversal (T) 356
14.5 How to implement Lorentz transformations on a computer 358
14.6 Killing vector (cid:28)elds of Minkowski space 363
14.7 Dirac matrices 367
14.8 Dirac spinors 368
14.9 Dirac spinors as complex quaternions 369
14.10 Non-null Dirac spinor 374
14.11 Null Dirac Spinor 376
15 Geometric Di(cid:27)erentiation and Integration 380
15.1 Covariant derivative of a multivector 381
15.2 Riemann tensor of bivectors 383
15.3 Torsion tensor of vectors 384
x Contents
15.4 Covariant spacetime derivative 384
15.5 Torsion-full and torsion-free covariant spacetime derivative 386
15.6 Di(cid:27)erential forms 387
15.7 Wedge product of di(cid:27)erential forms 388
15.8 Exterior derivative 389
15.9 An alternative notation for di(cid:27)erential forms 390
15.10 Hodge dual form 392
15.11 Relation between coordinate- and tetrad-frame volume elements 393
15.12 Generalized Stokes’ theorem 394
15.13 Exact and closed forms 396
15.14 Generalized Gauss’ theorem 397
15.15 Dirac delta-function 398
15.16 Integration of multivector-valued forms 399
16 Action principle for electromagnetism and gravity 400
16.1 Euler-Lagrange equations for a generic (cid:28)eld 401
16.2 Super-Hamiltonian formalism 403
16.3 Conventional Hamiltonian formalism 403
16.4 Symmetries and conservation laws 404
16.5 Electromagnetic action 405
16.6 Electromagnetic action in forms notation 411
16.7 Gravitational action 418
16.8 Variation of the gravitational action 421
16.9 Trading coordinates and momenta 423
16.10 Matter energy-momentum and the Einstein equations with matter 424
16.11 Spin angular-momentum 425
16.12 Lagrangian as opposed to Hamiltonian formulation 432
16.13 Gravitational action in multivector notation 434
16.14 Gravitational action in multivector forms notation 438
16.15 Space+time (3+1) split in multivector forms notation 454
16.16 Loop Quantum Gravity 464
16.17 Bianchi identities in multivector forms notation 470
17 Conventional Hamiltonian (3+1) approach 476
17.1 ADM formalism 477
17.2 ADM gravitational equations of motion 487
17.3 Conformally scaled ADM 493
17.4 Bianchi spacetimes 495
17.5 Friedmann-Lema(cid:238)tre-Robertson-Walker spacetimes 501
17.6 BKL oscillatory collapse 502
17.7 BSSN formalism 511