广义相对论 第2版PDF|Epub|txt|kindle电子书版本网盘下载

广义相对论 第2版PDF格式电子书版下载

注意：本站所有压缩包均有解压码： 点击下载压缩包解压工具

Part Ⅰ The General Theory of Relativity3

1 Introduction3

2 Physics in External Gravitational Fields7

2.1 Characteristic Properties of Gravitation7

2.1.1 Strength of the Gravitational Interaction7

2.1.2 Universality of Free Fall8

2.1.3 Equivalence Principle9

2.1.4 Gravitational Red-and Blueshifts10

2.2 Special Relativity and Gravitation12

2.2.1 Gravitational Redshift and Special Relativity12

2.2.2 Global Inertial Systems Cannot Be Realized in the Presence ofGravitational Fields13

2.2.3 Gravitational Deflection of Light Rays14

2.2.4 Theories of Gravity in Flat Spacetime14

2.2.5 Exercises18

2.3 Spacetime as a Lorentzian Manifold19

2.4 Non-gravitational Laws in External Gravitational Fields21

2.4.1 Motion of a Test Body in a Gravitational Field22

2.4.2 World Lines of Light Rays23

2.4.3 Exercises23

2.4.4 Energy and Momentum"Conservation"in the Presence of an External Gravitational Field25

2.4.5 Exercises27

2.4.6 Electrodynamics28

2.4.7 Exercises31

2.5 The Newtonian Limit32

2.5.1 Exercises33

2.6 The Redshift in a Stationary Gravitational Field34

2.7 Fermat's Principle for Static Gravitational Fields35

2.8 Geometric Optics in Gravitational Fields38

2.8.1 Exercises42

2.9 Stationary and Static Spacetimes42

2.9.1 Killing Equation44

2.9.2 The Redshift Revisited45

2.10 Spin Precession and Fermi Transport48

2.10.1 Spin Precession in a Gravitational Field49

2.10.2 Thomas Precession50

2.10.3 Fermi Transport51

2.10.4 The Physical Difference Between Static and Stationary Fields53

2.10.5 Spin Rotation in a Stationary Field55

2.10.6 Adapted Coordinate Systems for Accelerated Observers56

2.10.7 Motion of a Test Body58

2.10.8 Exercises60

2.11 General Relativistic Ideal Magnetohydrodynamics62

2.11.1 Exercises63

3 Einstein's Field Equations65

3.1 Physical Meaning ofthe Curvature Tensor65

3.1.1 Comparison with Newtonian Theory69

3.1.2 Exercises69

3.2 The Gravitational Field Equations72

3.2.1 Heuristic"Derivation"of the Field Equations73

3.2.2 The Question of Uniqueness74

3.2.3 Newtonian Limit,Interpretation of the Constants ？ and K78

3.2.4 On the Cosmological Constant ？79

3.2.5 The Einstein-Fokker Theory82

3.2.6 Exercises82

3.3 Lagrangian Formalism84

3.3.1 Canonical Measure on a Pseudo-Riemannian Manifold84

3.3.2 The Einstein-Hilbert Action85

3.3.3 Reduced Bianchi Identity and General Invariance87

3.3.4 Energy-Momentum Tensor in a Lagrangian Field Theory89

3.3.5 Analogy with Electrodynamics92

3.3.6 Meaning of the Equation ？·T=094

3.3.7 The Equations of Motion and ？·T=094

3.3.8 Variational Principle for the Coupled System95

3.3.9 Exercises95

3.4 Non-localizability ofthe Gravitational Energy97

3.5 On Covariance and Invariance98

3.5.1 Note on Unimodular Gravity101

3.6 The Tetrad Formalism102

3.6.1 Variation of Tetrad Fields103

3.6.2 The Einstein-Hilbert Action104

3.6.3 Consequences of the Invariance Properties of the Lagrangian L107

3.6.4 Lovelock's Theorem in Higher Dimensions110

3.6.5 Exercises111

3.7 Energy,Momentum,and Angular Momentum for Isolated Systems112

3.7.1 Interpretation116

3.7.2 ADM Expressions for Energy and Momentum119

3.7.3 Positive Energy Theorem121

3.7.4 Exercises122

3.8 The Initial Value Problem of General Relativity124

3.8.1 Nature of the Problem124

3.8.2 Constraint Equations125

3.8.3 Analogy with Electrodynamics126

3.8.4 Propagation of Constraints127

3.8.5 Local Existence and Uniqueness Theorems128

3.8.6 Analogy with Electrodynamics128

3.8.7 Harmonic Gauge Condition130

3.8.8 Field Equations in Harmonic Gauge130

3.8.9 Characteristics of Einstein's Field Equations135

3.8.10 Exercises136

3.9 General Relativity in 3+1 Formulation137

3.9.1 Generalities138

3.9.2 Connection Forms139

3.9.3 Curvature Forms,Einstein and Ricci Tensors142

3.9.4 Gaussian Normal Coordinates145

3.9.5 Maximal Slicing146

3.9.6 Exercises147

3.10 Domain of Dependence and Propagation of Matter Disturbances147

3.11 Boltzmann Equation in GR149

3.11.1 One-Particle Phase Space,Liouville Operator for Geodesic Spray149

3.11.2 The General Relativistic Boltzmann Equation153

Part Ⅱ Applications of General Relativity157

4 The Schwarzschild Solution and Classical Tests of General Relativity157

4.1 Derivation of the Schwarzschild Solution157

4.1.1 The Birkhoff Theorem161

4.1.2 Geometric Meaning of the Spatial Part of the Schwarzschild Metric164

4.1.3 Exercises164

4.2 Equation of Motion in a Schwarzschild Field166

4.2.1 Exercises169

4.3 Perihelion Advance170

4.4 Deflection of Light174

4.4.1 Exercises178

4.5 Time Delay of Radar Echoes180

4.6 Geodetic Precession184

4.7 Schwarzschild Black Holes187

4.7.1 The Kruskal Continuation of the Schwarzschild Solution188

4.7.2 Discussion194

4.7.3 Eddington-Finkelstein Coordinates197

4.7.4 Spherically Symmetric Collapse to a Black Hole199

4.7.5 Redshift for a Distant Observer201

4.7.6 Fate of an Observer on the Surface ofthe Star204

4.7.7 Stability of the Schwarzschild Black Hole207

4.8 Penrose Diagram for Kruskal Spacetime207

4.8.1 Conformal Compactification of Minkowski Spacetime208

4.8.2 Penrose Diagram for Schwarzschild-Kruskal Spacetime210

4.9 Charged Spherically Symmetric Black Holes211

4.9.1 Resolution of the Apparent Singularity211

4.9.2 Timelike Radial Geodesics214

4.9.3 Maximal Extension of the Reissner-Nordstr？m Solution216

Appendix:Spherically Symmetric Gravitational Fields220

4.10.1 General Form of the Metric220

4.10.2 The Generalized Birkhoff Theorem224

4.10.3 Spherically Symmetric Metrics for Fluids225

4.10.4 Exercises226

5 Weak Gravitational Fields227

5.1 The Linearized Theory of Gravity227

5.1.1 Generalization230

5.1.2 Exercises233

5.2 Nearly Newtonian Gravitational Fields234

5.2.1 Gravitomagnetic Field and Lense-Thirring Precession235

5.2.2 Exercises236

5.3 Gravitational Waves in the Linearized Theory237

5.3.1 Plane Waves238

5.3.2 Transverse and Traceless Gauge239

5.3.3 Geodesic Deviation in the Metric Field of a Gravitational Wave240

5.3.4 A Simple Mechanical Detector242

5.4 Energy Carried by a Gravitational Wave245

5.4.1 The Short Wave Approximation246

5.4.2 Discussion of the Linearized Equation R？[h]=0248

5.4.3 Averaged Energy-Momentum Tensor for Gravitational Waves250

5.4.4 Effective Energy-Momentum Tensor for a Plane Wave252

5.4.5 Exercises253

5.5 Emission of Gravitational Radiation256

5.5.1 Slow Motion Approximation256

5.5.2 Rapidly Varying Sources259

5.5.3 Radiation Reaction(Preliminary Remarks)261

5.5.4 Simple Examples and Rough Estimates261

5.5.5 Rigidly Rotating Body261

5.5.6 Radiation from Binary Star Systems in Elliptic Orbits266

5.5.7 Exercises269

5.6 Laser Interferometers270

5.7 Gravitational Field at Large Distances from a Stationary Source272

5.7.1 The Komar Formula278

5.7.2 Exercises280

5.8 Gravitational Lensing280

5.8.1 Three Derivations ofthe Effective Refraction Index281

5.8.2 Deflection by an Arbitrary Mass Concentration283

5.8.3 The General Lens Map286

5.8.4 Alternative Derivation of the Lens Equation288

5.8.5 Magnification,Critical Curves and Caustics290

5.8.6 Simple Lens Models292

5.8.7 Axially Symmetric Lenses:Generalities292

5.8.8 The Schwarzschild Lens:Microlensing295

5.8.9 Singular Isothermal Sphere298

5.8.10 Isothermal SpherewithFiniteCoreRadius300

5.8.11 Relation Between Shear and Observable Distortions300

5.8.12 Mass Reconstruction from Weak Lensing301

6 The Post-Newtonian Approximation307

6.1 Motion and Gravitational Radiation(Generalities)307

6.1.1 Asymptotic Flatness308

6.1.2 Bondi-Sachs Energy and Momentum309

6.1.3 The Effacement Property311

6.2 Field Equations in Post-Newtonian Approximation312

6.2.1 Equations of Motion for a Test Particle319

6.3 Stationary Asymptotic Fields in Post-Newtonian Approximation320

6.4 Point-Particle Limit322

6.5 The Einstein-Infeld-Hoffmann Equations326

6.5.1 The Two-Body Problem in the Post-Newtonian Approximation329

6.6 Precession of a Gyroscope in the Post-Newtonian Approximation335

6.6.1 Gyroscope in Orbit Around the Earth338

6.6.2 Precession of Binary Pulsars339

6.7 General Strategies of Approximation Methods340

6.7.1 Radiation Damping345

6.8 Binary Pulsars346

6.8.1 Discovery and Gross Features346

6.8.2 Timing Measurements and Data Reduction351

6.8.3 Arrival Time351

6.8.4 Solar System Corrections352

6.8.5 Theoretical Analysis of the Arrival Times354

6.8.6 Einstein Time Delay355

6.8.7 Roemer and Shapiro Time Delays356

6.8.8 Explicit Expression for the Roemer Delay359

6.8.9 Aberration Correction361

6.8.10 The Timing Formula363

6.8.11Results for Keplerian and Post-Keplerian Parameters366

6.8.12 Masses of the Two Neutron Stars366

6.8.13 Confirmation of the Gravitational Radiation Damping367

6.8.14 Results for the Binary PSR B 1534+12369

6.8.15 Double-Pulsar372

7 White Dwarfs and Neutron Stars375

7.1 Introduction375

7.2 White Dwarfs377

7.2.1 The Free Relativistic Electron Gas378

7.2.2 Thomas-Fermi Approximation for White Dwarfs379

7.2.3 Historical Remarks383

7.2.4 Exercises384

7.3 Formation of Neutron Stars385

7.4 General Relativistic Stellar Structure Equations387

7.4.1 Interpretation of M390

7.4.2 General Relativistic Virial Theorem391

7.4.3 Exercises392

7.5 Linear Stability392

7.6 The Interior of Neutron Stars395

7.6.1 Qualitative Overview395

7.6.2 Ideal Mixture of Neutrons,Protons and Electrons397

7.6.3 Oppenheimer-Volkoff Model399

7.6.4 Pion Condensation400

7.7 Equation of State at High Densities401

7.7.1 Effective Nuclear Field Theories401

7.7.2 Many-Body Theory of Nucleon Matter401

7.8 Gross Structure of Neutron Stars402

7.8.1 Measurements of Neutron Star Masses Using Shapiro Time Delay404

7.9 Bounds for the Mass of Non-rotating Neutron Stars405

7.9.1 Basic Assumptions405

7.9.2 Simple Bounds for Allowed Cores408

7.9.3 Allowed Core Region408

7.9.4 Upper Limit for the Total Gravitational Mass410

7.10 Rotating Neutron Stars412

7.11 Cooling of Neutron Stars415

7.12 Neutron Stars in Binaries416

7.12.1 Some Mechanics in Binary Systems416

7.12.2 Some History of X-Ray Astronomy419

7.12.3 X-Ray Pulsars420

7.12.4 The Eddington Limit422

7.12.5 X-Ray Bursters423

7.12.6 Formation and Evolution of Binary Systems425

7.12.7 Millisecond Pulsars427

8 Black Holes429

8.1 Introduction429

8.2 Proof of Israel's Theorem430

8.2.1 Foliation of ∑,Ricci Tensor,etc431

8.2.2 The Invariant(4)Rαβγδ(4)Rαβγδ434

8.2.3 The Proof(W.Israel,1967)435

8.3 Derivation of the Kerr Solution442

8.3.1 Axisymmetric Stationary Spacetimes443

8.3.2 Ricci Circularity444

8.3.3 Footnote:Derivation of Two Identities446

8.3.4 The Ernst Equation447

8.3.5 Footnote:Derivation of Eq.(8.90)449

8.3.6 Ricci Curvature450

8.3.7 Intermediate Summary455

8.3.8 Weyl Coordinates455

8.3.9 Conjugate Solutions458

8.3.10 Basic Equations in Elliptic Coordinates459

8.3.11 The Kerr Solution462

8.3.12 Kerr Solution in Boyer-Lindquist Coordinates464

8.3.13 Interpretation of the Parameters a and m465

8.3.14 Exercises467

8.4 Discussion ofthe Kerr-Newman Family467

8.4.1 Gyromagnetic Factor of a Charged Black Hole468

8.4.2 Symmetries of the Metric470

8.4.3 Static Limit and Stationary Observers470

8.4.4 Killing Horizon and Ergosphere471

8.4.5 Coordinate Singularity at the Horizon and Kerr Coordinates475

8.4.6 Singularities ofthe Kerr-Newman Metric476

8.4.7 Structure of the Light Cones and Event Horizon476

8.4.8 Penrose Mechanism477

8.4.9 Geodesics of a Kerr Black Hole478

8.4.10 The Hamilton-Jacobi Method478

8.4.11 The Fourth Integral of Motion480

8.4.12 Equatorial Circular Geodesics482

8.5 Accretion Tori Around Kerr Black Holes485

8.5.1 Newtonian Approximation486

8.5.2 General Relativistic Treatment488

8.5.3 Footnote:Derivation of Eq.(8.276)490

8.6 The Four Laws of Black Hole Dynamics491

8.6.1 General Definition of Black Holes491

8.6.2 The Zeroth Law of Black Hole Dynamics492

8.6.3 Surface Gravity492

8.6.4 The First Law495

8.6.5 Surface Area of Kerr-Newman Horizon495

8.6.6 The First Law for the Kerr-Newman Family496

8.6.7 The First Law for Circular Spacetimes497

8.6.8 The Second Law of Black Hole Dynamics502

8.6.9 Applications503

8.7 Evidence for Black Holes505

8.7.1 Black Hole Formation505

8.7.2 Black Hole Candidates in X-Ray Binaries507

8.7.3 The X-Ray Nova XTE J118+480508

8.7.4 Super-Massive Black Holes509

Appendix:Mathematical Appendix on Black Holes511

8.8.1 Proof of the Weak Rigidity Theorem511

8.8.2 The Zeroth Law for Circular Spacetimes512

8.8.3 Geodesic Null Congruences514

8.8.4 Optical Scalars515

8.8.5 Transport Equation516

8.8.6 The Sachs Equations519

8.8.7 Applications520

8.8.8 Change of Area522

8.8.9 Area Law for Black Holes525

9 The Positive Mass Theorem527

9.1 Total Energy and Momentum for Isolated Systems528

9.2 Witten's Proof of the Positive Energy Theorem531

9.2.1 Remarks on the Witten Equation534

9.2.2 Application534

9.3 Generalization to Black Holes535

9.4 Penrose Inequality537

9.4.1 Exercises538

Appendix:Spin Structures and Spinor Analysis in General Relativity538

9.5.1 Spinor Algebra538

9.5.2 Spinor Analysis in GR542

9.5.3 Exercises545

10 Essentials of Friedmann-Lema？tre Models547

10.1 Introduction547

10.2 Friedmann-Lema？tre Spacetimes549

10.2.1 Spaces of Constant Curvature550

10.2.2 Curvature of Friedmann Spacetimes551

10.2.3 Einstein Equations for Friedmann Spacetimes551

10.2.4 Redshift553

10.2.5 Cosmic Distance Measures555

10.3 Thermal History Below 100 MeV557

10.3.1 Overview557

10.3.2 Chemical Potentials of the Leptons558

10.3.3 Constancy of Entropy559

10.3.4 Neutrino Temperature561

10.3.5 Epoch of Matter-Radiation Equality562

10.3.6 Recombination and Decoupling563

10.4 Luminosity-Redshift Relation for Type Ia Supernovae565

10.4.1 Theoretical Redshift-Luminosity Relation566

10.4.2 Type Ia Supernovae as Standard Candles570

10.4.3 Results572

10.4.4 Exercises573

Part Ⅲ Differential Geometry579

11 Differentiable Manifolds579

12 Tangent Vectors,Vector and Tensor Fields585

12.1 The Tangent Space585

12.2 Vector Fields592

12.3 Tensor Fields594

13 The Lie Derivative599

13.1 Integral Curves and Flow of a Vector Field599

13.2 Mappings and Tensor Fields601

13.3 The Lie Derivative603

13.3.1 Local Coordinate Expressions for Lie Derivatives604

14 Differential Forms607

14.1 Exterior Algebra607

14.2 Exterior Differential Forms609

14.2.1 Differential Forms and Mappings610

14.3 Derivations and Antiderivations611

14.4 The Exterior Derivative613

14.4.1 Morphisms and Exterior Derivatives615

14.5 Relations Among the Operators d,ix and Lx615

14.5.1 Formula for the Exterior Derivative616

14.6 The*-Operation and the Codifferential617

14.6.1 Oriented Manifolds617

14.6.2 The*-Operation618

14.6.3 Exercises619

14.6.4 The Codifferential622

14.6.5 Coordinate Expression for the Codifferential623

14.6.6 Exercises624

14.7 The Integral Theorems of Stokes and Gauss624

14.7.1 Integration of Differential Forms624

14.7.2 Stokes' Theorem626

14.7.3 Application627

14.7.4 Expression for divΩX in Local Coordinates628

14.7.5 Exercises629

15 Affine Connections631

15.1 Covariant Derivative of a Vector Field631

15.2 Parallel Transport Along a Curve633

15.3 Geodesics,Exponential Mapping and Normal Coordinates635

15.4 Covariant Derivative of Tensor Fields636

15.4.1 Application638

15.4.2 Local Coordinate Expression for the Covariant Derivative639

15.4.3 Covariant Derivative and Exterior Derivative640

15.5 Curvature and Torsion of an Affine Connection,Bianchi Identities640

15.6 Riemannian Connections643

15.6.1 Local Expressions645

15.6.2 Contracted Bianchi Identity647

15.7 The Cartan Structure Equations649

15.7.1 Solution of the Structure Equations652

15.8 Bianchi Identities for the Curvature and Torsion Forms653

15.8.1 Special Cases655

15.9 Locally Flat Manifolds658

15.9.1 Exercises660

15.10 Wevl Tensor and Conformally Flat Manifolds662

15.11 Covariant Derivatives of Tensor Densities663

16 Some Details and Supplements665

16.1 Proofs of Some Theorems665

16.2 Tangent Bundles670

16.3 Vector Fields Along Maps671

16.3.1 Induced Covariant Derivative672

16.3.2 Exercises675

16.4 Variations of a Smooth Curve675

16.4.1 First Variation Formula676

16.4.2 Jacobi Equation677

Appendix A Fundamental Equations for Hypersurfaces679

A.1 Formulas of Gauss and Weingarten679

A.2 Equations of Gauss and Codazzi-Mainardi681

A.3 Null Hypersurfaces684

A.4 Exercises684

Appendix B Ricci Curvature of Warped Products687

B.1 Application:Friedmann Equations689

Appendix C Frobenius Integrability Theorem691

C.1 Applications695

C.2 Proof of Frobenius'Theorem(in the First Version)698

Appendix D Collection of Important Formulas701

D.1 Vector Fields,Lie Brackets701

D.2 Differential Forms701

D.3 Exterior Differential702

D.4 Poincaré Lemma702

D.5 Interior Product702

D.6 Lie Derivative703

D.7 Relations Between LX,iX and d703

D.8 Volume Form703

D.9 Hodge-Star Operation704

D.10 Codifferential704

D.11 Covariant Derivative704

D.12 Connection Forms705

D.13 Curvature Forms705

D.14 Cartan's Structure Equations705

D.15 Riemannian Connection705

D.16 Coordinate Expressions705

D.17 Absolute Exterior Differential706

D.18 Bianchi Identities706

References709

Textbooks on General Relativity:Classical Texts709

Textbooks on General Relativity:Selection of(Graduate)Textbooks709

Textbooks on General Relativity:Numerical Relativity710

Textbooks on General Physics and Astrophysics710

Mathematical Tools:Modern Treatments of Differential Geometry for Physicists710

Mathematical Tools:Selection of Mathematical Books711

Historical Sources711

Recent Books on Cosmology712

Research Articles,Reviews and Specialized Texts:Chapter 2712

Research Articles,Reviews and Specialized Texts:Chapter 3713

Research Articles,Reviews and Specialized Texts:Chapter 4713

Research Articles,Reviews and Specialized Texts:Chapter 5714

Research Articles,Reviews and Specialized Texts:Chapter 6715

Research Articles,Reviews and Specialized Texts:Chapter 7716

Research Articles,Reviews and Specialized Texts:Chapter 8717

Research Articles,Reviews and Specialized Texts:Chapter 9718

Research Articles,Reviews and Specialized Texts:Chapter 10718

Index721