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[资源] Straumann_General Relativity_2nd ed._Springer_2013

Contents
Part I The General Theory of Relativity
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 Physics in External Gravitational Fields . . . . . . . . . . . . . . . . 7
2.1 Characteristic Properties of Gravitation . . . . . . . . . . . . . . . 7
2.1.1 Strength of the Gravitational Interaction . . . . . . . . . . 7
2.1.2 Universality of Free Fall . . . . . . . . . . . . . . . . . . . 8
2.1.3 Equivalence Principle . . . . . . . . . . . . . . . . . . . . 9
2.1.4 Gravitational Red- and Blueshifts . . . . . . . . . . . . . . 10
2.2 Special Relativity and Gravitation . . . . . . . . . . . . . . . . . . 12
2.2.1 Gravitational Redshift and Special Relativity . . . . . . . . 12
2.2.2 Global Inertial Systems Cannot Be Realized in the
Presence of Gravitational Fields . . . . . . . . . . . . . . . 13
2.2.3 Gravitational Deflection of Light Rays . . . . . . . . . . . 14
2.2.4 Theories of Gravity in Flat Spacetime . . . . . . . . . . . . 14
2.2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 Spacetime as a Lorentzian Manifold . . . . . . . . . . . . . . . . . 19
2.4 Non-gravitational Laws in External Gravitational Fields . . . . . . 21
2.4.1 Motion of a Test Body in a Gravitational Field . . . . . . . 22
2.4.2 World Lines of Light Rays . . . . . . . . . . . . . . . . . . 23
2.4.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4.4 Energy and Momentum “Conservation” in the Presence of
an External Gravitational Field . . . . . . . . . . . . . . . 25
2.4.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.4.6 Electrodynamics . . . . . . . . . . . . . . . . . . . . . . . 28
2.4.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.5 The Newtonian Limit . . . . . . . . . . . . . . . . . . . . . . . . 32
2.5.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.6 The Redshift in a Stationary Gravitational Field . . . . . . . . . . 34
2.7 Fermat’s Principle for Static Gravitational Fields . . . . . . . . . . 35
ix
x Contents
2.8 Geometric Optics in Gravitational Fields . . . . . . . . . . . . . . 38
2.8.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.9 Stationary and Static Spacetimes . . . . . . . . . . . . . . . . . . 42
2.9.1 Killing Equation . . . . . . . . . . . . . . . . . . . . . . . 44
2.9.2 The Redshift Revisited . . . . . . . . . . . . . . . . . . . . 45
2.10 Spin Precession and Fermi Transport . . . . . . . . . . . . . . . . 48
2.10.1 Spin Precession in a Gravitational Field . . . . . . . . . . . 49
2.10.2 Thomas Precession . . . . . . . . . . . . . . . . . . . . . 50
2.10.3 Fermi Transport . . . . . . . . . . . . . . . . . . . . . . . 51
2.10.4 The Physical Difference Between Static and Stationary
Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.10.5 Spin Rotation in a Stationary Field . . . . . . . . . . . . . 55
2.10.6 Adapted Coordinate Systems for Accelerated Observers . . 56
2.10.7 Motion of a Test Body . . . . . . . . . . . . . . . . . . . . 58
2.10.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
2.11 General Relativistic Ideal Magnetohydrodynamics . . . . . . . . . 62
2.11.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3 Einstein’s Field Equations . . . . . . . . . . . . . . . . . . . . . . . . 65
3.1 Physical Meaning of the Curvature Tensor . . . . . . . . . . . . . 65
3.1.1 Comparison with Newtonian Theory . . . . . . . . . . . . 69
3.1.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.2 The Gravitational Field Equations . . . . . . . . . . . . . . . . . . 72
3.2.1 Heuristic “Derivation” of the Field Equations . . . . . . . . 73
3.2.2 The Question of Uniqueness . . . . . . . . . . . . . . . . . 74
3.2.3 Newtonian Limit, Interpretation of the Constants Λ and κ . 78
3.2.4 On the Cosmological Constant Λ . . . . . . . . . . . . . . 79
3.2.5 The Einstein–Fokker Theory . . . . . . . . . . . . . . . . 82
3.2.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.3 Lagrangian Formalism . . . . . . . . . . . . . . . . . . . . . . . . 84
3.3.1 Canonical Measure on a Pseudo-Riemannian Manifold . . 84
3.3.2 The Einstein–Hilbert Action . . . . . . . . . . . . . . . . . 85
3.3.3 Reduced Bianchi Identity and General Invariance . . . . . 87
3.3.4 Energy-Momentum Tensor in a Lagrangian Field Theory . 89
3.3.5 Analogy with Electrodynamics . . . . . . . . . . . . . . . 92
3.3.6 Meaning of the Equation ∇ ·T = 0 . . . . . . . . . . . . . 94
3.3.7 The Equations of Motion and ∇ ·T = 0 . . . . . . . . . . . 94
3.3.8 Variational Principle for the Coupled System . . . . . . . . 95
3.3.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.4 Non-localizability of the Gravitational Energy . . . . . . . . . . . 97
3.5 On Covariance and Invariance . . . . . . . . . . . . . . . . . . . . 98
3.5.1 Note on Unimodular Gravity . . . . . . . . . . . . . . . . 101
3.6 The Tetrad Formalism . . . . . . . . . . . . . . . . . . . . . . . . 102
3.6.1 Variation of Tetrad Fields . . . . . . . . . . . . . . . . . . 103
3.6.2 The Einstein–Hilbert Action . . . . . . . . . . . . . . . . . 104
Contents xi
3.6.3 Consequences of the Invariance Properties of the
Lagrangian L . . . . . . . . . . . . . . . . . . . . . . . . 107
3.6.4 Lovelock’s Theorem in Higher Dimensions . . . . . . . . . 110
3.6.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
3.7 Energy, Momentum, and Angular Momentum for Isolated Systems 112
3.7.1 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . 116
3.7.2 ADM Expressions for Energy and Momentum . . . . . . . 119
3.7.3 Positive Energy Theorem . . . . . . . . . . . . . . . . . . 121
3.7.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
3.8 The Initial Value Problem of General Relativity . . . . . . . . . . 124
3.8.1 Nature of the Problem . . . . . . . . . . . . . . . . . . . . 124
3.8.2 Constraint Equations . . . . . . . . . . . . . . . . . . . . . 125
3.8.3 Analogy with Electrodynamics . . . . . . . . . . . . . . . 126
3.8.4 Propagation of Constraints . . . . . . . . . . . . . . . . . 127
3.8.5 Local Existence and Uniqueness Theorems . . . . . . . . . 128
3.8.6 Analogy with Electrodynamics . . . . . . . . . . . . . . . 128
3.8.7 Harmonic Gauge Condition . . . . . . . . . . . . . . . . . 130
3.8.8 Field Equations in Harmonic Gauge . . . . . . . . . . . . . 130
3.8.9 Characteristics of Einstein’s Field Equations . . . . . . . . 135
3.8.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
3.9 General Relativity in 3+1 Formulation . . . . . . . . . . . . . . . 137
3.9.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . 138
3.9.2 Connection Forms . . . . . . . . . . . . . . . . . . . . . . 139
3.9.3 Curvature Forms, Einstein and Ricci Tensors . . . . . . . . 142
3.9.4 Gaussian Normal Coordinates . . . . . . . . . . . . . . . . 145
3.9.5 Maximal Slicing . . . . . . . . . . . . . . . . . . . . . . . 146
3.9.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
3.10 Domain of Dependence and Propagation of Matter Disturbances . . 147
3.11 Boltzmann Equation in GR . . . . . . . . . . . . . . . . . . . . . 149
3.11.1 One-Particle Phase Space, Liouville Operator for Geodesic
Spray . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
3.11.2 The General Relativistic Boltzmann Equation . . . . . . . 153
Part II Applications of General Relativity
4 The Schwarzschild Solution and Classical Tests of General Relativity 157
4.1 Derivation of the Schwarzschild Solution . . . . . . . . . . . . . . 157
4.1.1 The Birkhoff Theorem . . . . . . . . . . . . . . . . . . . . 161
4.1.2 Geometric Meaning of the Spatial Part of the
Schwarzschild Metric . . . . . . . . . . . . . . . . . . . . 164
4.1.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
4.2 Equation of Motion in a Schwarzschild Field . . . . . . . . . . . . 166
4.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
4.3 Perihelion Advance . . . . . . . . . . . . . . . . . . . . . . . . . 170
4.4 Deflection of Light . . . . . . . . . . . . . . . . . . . . . . . . . . 174
4.4.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
xii Contents
4.5 Time Delay of Radar Echoes . . . . . . . . . . . . . . . . . . . . 180
4.6 Geodetic Precession . . . . . . . . . . . . . . . . . . . . . . . . . 184
4.7 Schwarzschild Black Holes . . . . . . . . . . . . . . . . . . . . . 187
4.7.1 The Kruskal Continuation of the Schwarzschild Solution . 188
4.7.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 194
4.7.3 Eddington–Finkelstein Coordinates . . . . . . . . . . . . . 197
4.7.4 Spherically Symmetric Collapse to a Black Hole . . . . . . 199
4.7.5 Redshift for a Distant Observer . . . . . . . . . . . . . . . 201
4.7.6 Fate of an Observer on the Surface of the Star . . . . . . . 204
4.7.7 Stability of the Schwarzschild Black Hole . . . . . . . . . 207
4.8 Penrose Diagram for Kruskal Spacetime . . . . . . . . . . . . . . 207
4.8.1 Conformal Compactification of Minkowski Spacetime . . . 208
4.8.2 Penrose Diagram for Schwarzschild–Kruskal Spacetime . . 210
4.9 Charged Spherically Symmetric Black Holes . . . . . . . . . . . . 211
4.9.1 Resolution of the Apparent Singularity . . . . . . . . . . . 211
4.9.2 Timelike Radial Geodesics . . . . . . . . . . . . . . . . . 214
4.9.3 Maximal Extension of the Reissner–Nordstrøm Solution . . 216
Appendix: Spherically Symmetric Gravitational Fields . . . . . . . 220
4.10.1 General Form of the Metric . . . . . . . . . . . . . . . . . 220
4.10.2 The Generalized Birkhoff Theorem . . . . . . . . . . . . . 224
4.10.3 Spherically Symmetric Metrics for Fluids . . . . . . . . . . 225
4.10.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
5 Weak Gravitational Fields . . . . . . . . . . . . . . . . . . . . . . . . 227
5.1 The Linearized Theory of Gravity . . . . . . . . . . . . . . . . . . 227
5.1.1 Generalization . . . . . . . . . . . . . . . . . . . . . . . . 230
5.1.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
5.2 Nearly Newtonian Gravitational Fields . . . . . . . . . . . . . . . 234
5.2.1 Gravitomagnetic Field and Lense–Thirring Precession . . . 235
5.2.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
5.3 Gravitational Waves in the Linearized Theory . . . . . . . . . . . 237
5.3.1 Plane Waves . . . . . . . . . . . . . . . . . . . . . . . . . 238
5.3.2 Transverse and Traceless Gauge . . . . . . . . . . . . . . . 239
5.3.3 Geodesic Deviation in the Metric Field of a Gravitational
Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
5.3.4 A Simple Mechanical Detector . . . . . . . . . . . . . . . 242
5.4 Energy Carried by a Gravitational Wave . . . . . . . . . . . . . . 245
5.4.1 The Short Wave Approximation . . . . . . . . . . . . . . . 246
5.4.2 Discussion of the Linearized Equation R (1)
μν [h] = 0 . . . . . 248
5.4.3 Averaged Energy-Momentum Tensor for Gravitational
Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
5.4.4 Effective Energy-Momentum Tensor for a Plane Wave . . . 252
5.4.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 253
5.5 Emission of Gravitational Radiation . . . . . . . . . . . . . . . . . 256
5.5.1 Slow Motion Approximation . . . . . . . . . . . . . . . . 256
Contents xiii
5.5.2 Rapidly Varying Sources . . . . . . . . . . . . . . . . . . 259
5.5.3 Radiation Reaction (Preliminary Remarks) . . . . . . . . . 261
5.5.4 Simple Examples and Rough Estimates . . . . . . . . . . . 261
5.5.5 Rigidly Rotating Body . . . . . . . . . . . . . . . . . . . . 261
5.5.6 Radiation from Binary Star Systems in Elliptic Orbits . . . 266
5.5.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
5.6 Laser Interferometers . . . . . . . . . . . . . . . . . . . . . . . . 270
5.7 Gravitational Field at Large Distances from a Stationary Source . . 272
5.7.1 The Komar Formula . . . . . . . . . . . . . . . . . . . . . 278
5.7.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
5.8 Gravitational Lensing . . . . . . . . . . . . . . . . . . . . . . . . 280
5.8.1 Three Derivations of the Effective Refraction Index . . . . 281
5.8.2 Deflection by an Arbitrary Mass Concentration . . . . . . . 283
5.8.3 The General Lens Map . . . . . . . . . . . . . . . . . . . 286
5.8.4 Alternative Derivation of the Lens Equation . . . . . . . . 288
5.8.5 Magnification, Critical Curves and Caustics . . . . . . . . 290
5.8.6 Simple Lens Models . . . . . . . . . . . . . . . . . . . . . 292
5.8.7 Axially Symmetric Lenses: Generalities . . . . . . . . . . 292
5.8.8 The Schwarzschild Lens: Microlensing . . . . . . . . . . . 295
5.8.9 Singular Isothermal Sphere . . . . . . . . . . . . . . . . . 298
5.8.10 Isothermal Sphere with Finite Core Radius . . . . . . . . . 300
5.8.11 Relation Between Shear and Observable Distortions . . . . 300
5.8.12 Mass Reconstruction from Weak Lensing . . . . . . . . . . 301
6 The Post-Newtonian Approximation . . . . . . . . . . . . . . . . . . 307
6.1 Motion and Gravitational Radiation (Generalities) . . . . . . . . . 307
6.1.1 Asymptotic Flatness . . . . . . . . . . . . . . . . . . . . . 308
6.1.2 Bondi–Sachs Energy and Momentum . . . . . . . . . . . . 309
6.1.3 The Effacement Property . . . . . . . . . . . . . . . . . . 311
6.2 Field Equations in Post-Newtonian Approximation . . . . . . . . . 312
6.2.1 Equations of Motion for a Test Particle . . . . . . . . . . . 319
6.3 Stationary Asymptotic Fields in Post-Newtonian Approximation . 320
6.4 Point-Particle Limit . . . . . . . . . . . . . . . . . . . . . . . . . 322
6.5 The Einstein–Infeld–Hoffmann Equations . . . . . . . . . . . . . 326
6.5.1 The Two-Body Problem in the Post-Newtonian
Approximation . . . . . . . . . . . . . . . . . . . . . . . . 329
6.6 Precession of a Gyroscope in the Post-Newtonian Approximation . 335
6.6.1 Gyroscope in Orbit Around the Earth . . . . . . . . . . . . 338
6.6.2 Precession of Binary Pulsars . . . . . . . . . . . . . . . . 339
6.7 General Strategies of Approximation Methods . . . . . . . . . . . 340
6.7.1 Radiation Damping . . . . . . . . . . . . . . . . . . . . . 345
6.8 Binary Pulsars . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346
6.8.1 Discovery and Gross Features . . . . . . . . . . . . . . . . 346
6.8.2 Timing Measurements and Data Reduction . . . . . . . . . 351
6.8.3 Arrival Time . . . . . . . . . . . . . . . . . . . . . . . . . 351
xiv Contents
6.8.4 Solar System Corrections . . . . . . . . . . . . . . . . . . 352
6.8.5 Theoretical Analysis of the Arrival Times . . . . . . . . . . 354
6.8.6 Einstein Time Delay . . . . . . . . . . . . . . . . . . . . . 355
6.8.7 Roemer and Shapiro Time Delays . . . . . . . . . . . . . . 356
6.8.8 Explicit Expression for the Roemer Delay . . . . . . . . . 359
6.8.9 Aberration Correction . . . . . . . . . . . . . . . . . . . . 361
6.8.10 The Timing Formula . . . . . . . . . . . . . . . . . . . . . 363
6.8.11 Results for Keplerian and Post-Keplerian Parameters . . . . 366
6.8.12 Masses of the Two Neutron Stars . . . . . . . . . . . . . . 366
6.8.13 Confirmation of the Gravitational Radiation Damping . . . 367
6.8.14 Results for the Binary PSR B 1534+12 . . . . . . . . . . . 369
6.8.15 Double-Pulsar . . . . . . . . . . . . . . . . . . . . . . . . 372
7 White Dwarfs and Neutron Stars . . . . . . . . . . . . . . . . . . . . 375
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375
7.2 White Dwarfs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377
7.2.1 The Free Relativistic Electron Gas . . . . . . . . . . . . . 378
7.2.2 Thomas–Fermi Approximation for White Dwarfs . . . . . 379
7.2.3 Historical Remarks . . . . . . . . . . . . . . . . . . . . . 383
7.2.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 384
7.3 Formation of Neutron Stars . . . . . . . . . . . . . . . . . . . . . 385
7.4 General Relativistic Stellar Structure Equations . . . . . . . . . . . 387
7.4.1 Interpretation of M . . . . . . . . . . . . . . . . . . . . . 390
7.4.2 General Relativistic Virial Theorem . . . . . . . . . . . . . 391
7.4.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 392
7.5 Linear Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . 392
7.6 The Interior of Neutron Stars . . . . . . . . . . . . . . . . . . . . 395
7.6.1 Qualitative Overview . . . . . . . . . . . . . . . . . . . . 395
7.6.2 Ideal Mixture of Neutrons, Protons and Electrons . . . . . 397
7.6.3 Oppenheimer–Volkoff Model . . . . . . . . . . . . . . . . 399
7.6.4 Pion Condensation . . . . . . . . . . . . . . . . . . . . . . 400
7.7 Equation of State at High Densities . . . . . . . . . . . . . . . . . 401
7.7.1 Effective Nuclear Field Theories . . . . . . . . . . . . . . 401
7.7.2 Many-Body Theory of Nucleon Matter . . . . . . . . . . . 401
7.8 Gross Structure of Neutron Stars . . . . . . . . . . . . . . . . . . 402
7.8.1 Measurements of Neutron Star Masses Using Shapiro
Time Delay . . . . . . . . . . . . . . . . . . . . . . . . . 404
7.9 Bounds for the Mass of Non-rotating Neutron Stars . . . . . . . . 405
7.9.1 Basic Assumptions . . . . . . . . . . . . . . . . . . . . . . 405
7.9.2 Simple Bounds for Allowed Cores . . . . . . . . . . . . . 408
7.9.3 Allowed Core Region . . . . . . . . . . . . . . . . . . . . 408
7.9.4 Upper Limit for the Total Gravitational Mass . . . . . . . . 410
7.10 Rotating Neutron Stars . . . . . . . . . . . . . . . . . . . . . . . . 412
7.11 Cooling of Neutron Stars . . . . . . . . . . . . . . . . . . . . . . 415
7.12 Neutron Stars in Binaries . . . . . . . . . . . . . . . . . . . . . . 416
Contents xv
7.12.1 Some Mechanics in Binary Systems . . . . . . . . . . . . . 416
7.12.2 Some History of X-Ray Astronomy . . . . . . . . . . . . . 419
7.12.3 X-Ray Pulsars . . . . . . . . . . . . . . . . . . . . . . . . 420
7.12.4 The Eddington Limit . . . . . . . . . . . . . . . . . . . . . 422
7.12.5 X-Ray Bursters . . . . . . . . . . . . . . . . . . . . . . . 423
7.12.6 Formation and Evolution of Binary Systems . . . . . . . . 425
7.12.7 Millisecond Pulsars . . . . . . . . . . . . . . . . . . . . . 427
8 Black Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429
8.2 Proof of Israel’s Theorem . . . . . . . . . . . . . . . . . . . . . . 430
8.2.1 Foliation of Σ, Ricci Tensor, etc. . . . . . . . . . . . . . . 431
8.2.2 The Invariant
(4) R αβγδ (4) R αβγδ
. . . . . . . . . . . . . . . 434
8.2.3 The Proof (W. Israel, 1967) . . . . . . . . . . . . . . . . . 435
8.3 Derivation of the Kerr Solution . . . . . . . . . . . . . . . . . . . 442
8.3.1 Axisymmetric Stationary Spacetimes . . . . . . . . . . . . 443
8.3.2 Ricci Circularity . . . . . . . . . . . . . . . . . . . . . . . 444
8.3.3 Footnote: Derivation of Two Identities . . . . . . . . . . . 446
8.3.4 The Ernst Equation . . . . . . . . . . . . . . . . . . . . . 447
8.3.5 Footnote: Derivation of Eq. (8.90) . . . . . . . . . . . . . . 449
8.3.6 Ricci Curvature . . . . . . . . . . . . . . . . . . . . . . . 450
8.3.7 Intermediate Summary . . . . . . . . . . . . . . . . . . . . 455
8.3.8 Weyl Coordinates . . . . . . . . . . . . . . . . . . . . . . 455
8.3.9 Conjugate Solutions . . . . . . . . . . . . . . . . . . . . . 458
8.3.10 Basic Equations in Elliptic Coordinates . . . . . . . . . . . 459
8.3.11 The Kerr Solution . . . . . . . . . . . . . . . . . . . . . . 462
8.3.12 Kerr Solution in Boyer–Lindquist Coordinates . . . . . . . 464
8.3.13 Interpretation of the Parameters a and m . . . . . . . . . . 465
8.3.14 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 467
8.4 Discussion of the Kerr–Newman Family . . . . . . . . . . . . . . 467
8.4.1 Gyromagnetic Factor of a Charged Black Hole . . . . . . . 468
8.4.2 Symmetries of the Metric . . . . . . . . . . . . . . . . . . 470
8.4.3 Static Limit and Stationary Observers . . . . . . . . . . . . 470
8.4.4 Killing Horizon and Ergosphere . . . . . . . . . . . . . . . 471
8.4.5 Coordinate Singularity at the Horizon and Kerr Coordinates 475
8.4.6 Singularities of the Kerr–Newman Metric . . . . . . . . . . 476
8.4.7 Structure of the Light Cones and Event Horizon . . . . . . 476
8.4.8 Penrose Mechanism . . . . . . . . . . . . . . . . . . . . . 477
8.4.9 Geodesics of a Kerr Black Hole . . . . . . . . . . . . . . . 478
8.4.10 The Hamilton–Jacobi Method . . . . . . . . . . . . . . . . 478
8.4.11 The Fourth Integral of Motion . . . . . . . . . . . . . . . . 480
8.4.12 Equatorial Circular Geodesics . . . . . . . . . . . . . . . . 482
8.5 Accretion Tori Around Kerr Black Holes . . . . . . . . . . . . . . 485
8.5.1 Newtonian Approximation . . . . . . . . . . . . . . . . . 486
8.5.2 General Relativistic Treatment . . . . . . . . . . . . . . . 488
xvi Contents
8.5.3 Footnote: Derivation of Eq. (8.276) . . . . . . . . . . . . . 490
8.6 The Four Laws of Black Hole Dynamics . . . . . . . . . . . . . . 491
8.6.1 General Definition of Black Holes . . . . . . . . . . . . . 491
8.6.2 The Zeroth Law of Black Hole Dynamics . . . . . . . . . . 492
8.6.3 Surface Gravity . . . . . . . . . . . . . . . . . . . . . . . 492
8.6.4 The First Law . . . . . . . . . . . . . . . . . . . . . . . . 495
8.6.5 Surface Area of Kerr–Newman Horizon . . . . . . . . . . 495
8.6.6 The First Law for the Kerr–Newman Family . . . . . . . . 496
8.6.7 The First Law for Circular Spacetimes . . . . . . . . . . . 497
8.6.8 The Second Law of Black Hole Dynamics . . . . . . . . . 502
8.6.9 Applications . . . . . . . . . . . . . . . . . . . . . . . . . 503
8.7 Evidence for Black Holes . . . . . . . . . . . . . . . . . . . . . . 505
8.7.1 Black Hole Formation . . . . . . . . . . . . . . . . . . . . 505
8.7.2 Black Hole Candidates in X-Ray Binaries . . . . . . . . . 507
8.7.3 The X-Ray Nova XTE J118+480 . . . . . . . . . . . . . . 508
8.7.4 Super-Massive Black Holes . . . . . . . . . . . . . . . . . 509
Appendix: Mathematical Appendix on Black Holes . . . . . . . . 511
8.8.1 Proof of the Weak Rigidity Theorem . . . . . . . . . . . . 511
8.8.2 The Zeroth Law for Circular Spacetimes . . . . . . . . . . 512
8.8.3 Geodesic Null Congruences . . . . . . . . . . . . . . . . . 514
8.8.4 Optical Scalars . . . . . . . . . . . . . . . . . . . . . . . . 515
8.8.5 Transport Equation . . . . . . . . . . . . . . . . . . . . . 516
8.8.6 The Sachs Equations . . . . . . . . . . . . . . . . . . . . . 519
8.8.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . 520
8.8.8 Change of Area . . . . . . . . . . . . . . . . . . . . . . . 522
8.8.9 Area Law for Black Holes . . . . . . . . . . . . . . . . . . 525
9 The Positive Mass Theorem . . . . . . . . . . . . . . . . . . . . . . . 527
9.1 Total Energy and Momentum for Isolated Systems . . . . . . . . . 528
9.2 Witten’s Proof of the Positive Energy Theorem . . . . . . . . . . . 531
9.2.1 Remarks on the Witten Equation . . . . . . . . . . . . . . 534
9.2.2 Application . . . . . . . . . . . . . . . . . . . . . . . . . 534
9.3 Generalization to Black Holes . . . . . . . . . . . . . . . . . . . . 535
9.4 Penrose Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . 537
9.4.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 538
Appendix: Spin Structures and Spinor Analysis in General
Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538
9.5.1 Spinor Algebra . . . . . . . . . . . . . . . . . . . . . . . . 538
9.5.2 Spinor Analysis in GR . . . . . . . . . . . . . . . . . . . . 542
9.5.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 545
10 Essentials of Friedmann–Lemaître Models . . . . . . . . . . . . . . . 547
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547
10.2 Friedmann–Lemaître Spacetimes . . . . . . . . . . . . . . . . . . 549
10.2.1 Spaces of Constant Curvature . . . . . . . . . . . . . . . . 550
10.2.2 Curvature of Friedmann Spacetimes . . . . . . . . . . . . . 551
Contents xvii
10.2.3 Einstein Equations for Friedmann Spacetimes . . . . . . . 551
10.2.4 Redshift . . . . . . . . . . . . . . . . . . . . . . . . . . . 553
10.2.5 Cosmic Distance Measures . . . . . . . . . . . . . . . . . 555
10.3 Thermal History Below 100 MeV . . . . . . . . . . . . . . . . . . 557
10.3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 557
10.3.2 Chemical Potentials of the Leptons . . . . . . . . . . . . . 558
10.3.3 Constancy of Entropy . . . . . . . . . . . . . . . . . . . . 559
10.3.4 Neutrino Temperature . . . . . . . . . . . . . . . . . . . . 561
10.3.5 Epoch of Matter-Radiation Equality . . . . . . . . . . . . . 562
10.3.6 Recombination and Decoupling . . . . . . . . . . . . . . . 563
10.4 Luminosity-Redshift Relation for Type Ia Supernovae . . . . . . . 565
10.4.1 Theoretical Redshift-Luminosity Relation . . . . . . . . . 566
10.4.2 Type Ia Supernovae as Standard Candles . . . . . . . . . . 570
10.4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572
10.4.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 573
Part III Differential Geometry
11 Differentiable Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . 579
12 Tangent Vectors, Vector and Tensor Fields . . . . . . . . . . . . . . . 585
12.1 The Tangent Space . . . . . . . . . . . . . . . . . . . . . . . . . . 585
12.2 Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 592
12.3 Tensor Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594
13 The Lie Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599
13.1 Integral Curves and Flow of a Vector Field . . . . . . . . . . . . . 599
13.2 Mappings and Tensor Fields . . . . . . . . . . . . . . . . . . . . . 601
13.3 The Lie Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . 603
13.3.1 Local Coordinate Expressions for Lie Derivatives . . . . . 604
14 Differential Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 607
14.1 Exterior Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . 607
14.2 Exterior Differential Forms . . . . . . . . . . . . . . . . . . . . . 609
14.2.1 Differential Forms and Mappings . . . . . . . . . . . . . . 610
14.3 Derivations and Antiderivations . . . . . . . . . . . . . . . . . . . 611
14.4 The Exterior Derivative . . . . . . . . . . . . . . . . . . . . . . . 613
14.4.1 Morphisms and Exterior Derivatives . . . . . . . . . . . . 615
14.5 Relations Among the Operators d, i X and L X . . . . . . . . . . . 615
14.5.1 Formula for the Exterior Derivative . . . . . . . . . . . . . 616
14.6 The ∗-Operation and the Codifferential . . . . . . . . . . . . . . . 617
14.6.1 Oriented Manifolds . . . . . . . . . . . . . . . . . . . . . 617
14.6.2 The ∗-Operation . . . . . . . . . . . . . . . . . . . . . . . 618
14.6.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 619
14.6.4 The Codifferential . . . . . . . . . . . . . . . . . . . . . . 622
14.6.5 Coordinate Expression for the Codifferential . . . . . . . . 623
14.6.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 624
xviii Contents
14.7 The Integral Theorems of Stokes and Gauss . . . . . . . . . . . . 624
14.7.1 Integration of Differential Forms . . . . . . . . . . . . . . 624
14.7.2 Stokes’ Theorem . . . . . . . . . . . . . . . . . . . . . . 626
14.7.3 Application . . . . . . . . . . . . . . . . . . . . . . . . . 627
14.7.4 Expression for div Ω X in Local Coordinates . . . . . . . . 628
14.7.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 629
15 Affine Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 631
15.1 Covariant Derivative of a Vector Field . . . . . . . . . . . . . . . 631
15.2 Parallel Transport Along a Curve . . . . . . . . . . . . . . . . . 633
15.3 Geodesics, Exponential Mapping and Normal Coordinates . . . . 635
15.4 Covariant Derivative of Tensor Fields . . . . . . . . . . . . . . . 636
15.4.1 Application . . . . . . . . . . . . . . . . . . . . . . . . . 638
15.4.2 Local Coordinate Expression for the Covariant Derivative 639
15.4.3 Covariant Derivative and Exterior Derivative . . . . . . . 640
15.5 Curvature and Torsion of an Affine Connection, Bianchi Identities 640
15.6 Riemannian Connections . . . . . . . . . . . . . . . . . . . . . . 643
15.6.1 Local Expressions . . . . . . . . . . . . . . . . . . . . . 645
15.6.2 Contracted Bianchi Identity . . . . . . . . . . . . . . . . 647
15.7 The Cartan Structure Equations . . . . . . . . . . . . . . . . . . 649
15.7.1 Solution of the Structure Equations . . . . . . . . . . . . 652
15.8 Bianchi Identities for the Curvature and Torsion Forms . . . . . . 653
15.8.1 Special Cases . . . . . . . . . . . . . . . . . . . . . . . . 655
15.9 Locally Flat Manifolds . . . . . . . . . . . . . . . . . . . . . . . 658
15.9.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 660
15.10 Weyl Tensor and Conformally Flat Manifolds . . . . . . . . . . . 662
15.11 Covariant Derivatives of Tensor Densities . . . . . . . . . . . . . 663
16 Some Details and Supplements . . . . . . . . . . . . . . . . . . . . . 665
16.1 Proofs of Some Theorems . . . . . . . . . . . . . . . . . . . . . 665
16.2 Tangent Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . 670
16.3 Vector Fields Along Maps . . . . . . . . . . . . . . . . . . . . . 671
16.3.1 Induced Covariant Derivative . . . . . . . . . . . . . . . 672
16.3.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . 675
16.4 Variations of a Smooth Curve . . . . . . . . . . . . . . . . . . . 675
16.4.1 First Variation Formula . . . . . . . . . . . . . . . . . . . 676
16.4.2 Jacobi Equation . . . . . . . . . . . . . . . . . . . . . . 677
Appendix A Fundamental Equations for Hypersurfaces . . . . . . . . . 679
A.1 Formulas of Gauss and Weingarten . . . . . . . . . . . . . . . . 679
A.2 Equations of Gauss and Codazzi–Mainardi . . . . . . . . . . . . 681
A.3 Null Hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . . 684
A.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684
Appendix B Ricci Curvature of Warped Products . . . . . . . . . . . . 687
B.1 Application: Friedmann Equations . . . . . . . . . . . . . . . . . 689
Contents xix
Appendix C Frobenius Integrability Theorem . . . . . . . . . . . . . . . 691
C.1 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695
C.2 Proof of Frobenius’ Theorem (in the First Version) . . . . . . . . 698
Appendix D Collection of Important Formulas . . . . . . . . . . . . . . 701
D.1 Vector Fields, Lie Brackets . . . . . . . . . . . . . . . . . . . . . 701
D.2 Differential Forms . . . . . . . . . . . . . . . . . . . . . . . . . 701
D.3 Exterior Differential . . . . . . . . . . . . . . . . . . . . . . . . 702
D.4 Poincaré Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . 702
D.5 Interior Product . . . . . . . . . . . . . . . . . . . . . . . . . . . 702
D.6 Lie Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . 703
D.7 Relations Between L X , i X and d . . . . . . . . . . . . . . . . . . 703
D.8 Volume Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703
D.9 Hodge-Star Operation . . . . . . . . . . . . . . . . . . . . . . . 704
D.10 Codifferential . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704
D.11 Covariant Derivative . . . . . . . . . . . . . . . . . . . . . . . . 704
D.12 Connection Forms . . . . . . . . . . . . . . . . . . . . . . . . . 705
D.13 Curvature Forms . . . . . . . . . . . . . . . . . . . . . . . . . . 705
D.14 Cartan’s Structure Equations . . . . . . . . . . . . . . . . . . . . 705
D.15 Riemannian Connection . . . . . . . . . . . . . . . . . . . . . . 705
D.16 Coordinate Expressions . . . . . . . . . . . . . . . . . . . . . . 705
D.17 Absolute Exterior Differential . . . . . . . . . . . . . . . . . . . 706
D.18 Bianchi Identities . . . . . . . . . . . . . . . . . . . . . . . . . . 706
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 709
Textbooks on General Relativity: Classical Texts . . . . . . . . . 709
Textbooks on General Relativity: Selection of (Graduate)
Textbooks . . . . . . . . . . . . . . . . . . . . . . . . . . 709
Textbooks on General Relativity: Numerical Relativity . . . . . . 710
Textbooks on General Physics and Astrophysics . . . . . . . . . 710
Mathematical Tools: Modern Treatments of Differential
Geometry for Physicists . . . . . . . . . . . . . . . . . . 710
Mathematical Tools: Selection of Mathematical Books . . . . . . 711
Historical Sources . . . . . . . . . . . . . . . . . . . . . . . . . 711
Recent Books on Cosmology . . . . . . . . . . . . . . . . . . . . 712
Research Articles, Reviews and Specialized Texts: Chapter 2 . . . 712
Research Articles, Reviews and Specialized Texts: Chapter 3 . . . 713
Research Articles, Reviews and Specialized Texts: Chapter 4 . . . 713
Research Articles, Reviews and Specialized Texts: Chapter 5 . . . 714
Research Articles, Reviews and Specialized Texts: Chapter 6 . . . 715
Research Articles, Reviews and Specialized Texts: Chapter 7 . . . 716
Research Articles, Reviews and Specialized Texts: Chapter 8 . . . 717
Research Articles, Reviews and Specialized Texts: Chapter 9 . . . 718
Research Articles, Reviews and Specialized Texts: Chapter 10 . . 718
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 721
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