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Straumann_General Relativity_2nd ed._Springer_2013
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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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