Part Ⅰ The General Theory of Relativity 3
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 ofGravitational 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
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 ofthe 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 ？ and K 78
3.2.4 On the Cosmological Constant ？ 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 ofthe 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
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 157
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
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 ofthe 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-Nordstr？m 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？[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
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 ofthe 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 SpherewithFiniteCoreRadius 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
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
6.8.11Results 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
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 ∑,Ricci Tensor,etc 431
8.2.2 The Invariant(4)Rαβγδ(4)Rαβγδ 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 ofthe 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 ofthe 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
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-Lema？tre Models 547
10.1 Introduction 547
10.2 Friedmann-Lema？tre Spacetimes 549
10.2.1 Spaces of Constant Curvature 550
10.2.2 Curvature of Friedmann Spacetimes 551
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 579
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,ix and Lx 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
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 divΩX 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 Wevl 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
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 LX,iX 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