Contents Part I The General Theory of Relativity Introduction Physics in External Gravitational Fields

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1 Contents Part I The General Theory of Relativity 1 Introduction Physics in External Gravitational Fields Characteristic Properties of Gravitation Strength of the Gravitational Interaction Universality of Free Fall Equivalence Principle Gravitational Red- and Blueshifts Special Relativity and Gravitation Gravitational Redshift and Special Relativity Global Inertial Systems Cannot Be Realized in the Presence of Gravitational Fields Gravitational Deflection of Light Rays Theories of Gravity in Flat Spacetime Exercises Spacetime as a Lorentzian Manifold Non-gravitational Laws in External Gravitational Fields Motion of a Test Body in a Gravitational Field WorldLinesofLightRays Exercises Energy and Momentum Conservation in the Presence of an External Gravitational Field Exercises Electrodynamics Exercises TheNewtonianLimit Exercises The Redshift in a Stationary Gravitational Field Fermat s Principle for Static Gravitational Fields ix

2 x Contents 2.8 Geometric Optics in Gravitational Fields Exercises Stationary and Static Spacetimes Killing Equation TheRedshiftRevisited SpinPrecessionandFermiTransport Spin Precession in a Gravitational Field ThomasPrecession FermiTransport The Physical Difference Between Static and Stationary Fields Spin Rotation in a Stationary Field Adapted Coordinate Systems for Accelerated Observers MotionofaTestBody Exercises General Relativistic Ideal Magnetohydrodynamics Exercises Einstein s Field Equations Physical Meaning of the Curvature Tensor Comparison with Newtonian Theory Exercises The Gravitational Field Equations Heuristic Derivation ofthefieldequations The Question of Uniqueness Newtonian Limit, Interpretation of the Constants Λ and κ On the Cosmological Constant Λ The Einstein Fokker Theory Exercises Lagrangian Formalism Canonical Measure on a Pseudo-Riemannian Manifold TheEinstein HilbertAction Reduced Bianchi Identity and General Invariance Energy-Momentum Tensor in a Lagrangian Field Theory Analogy with Electrodynamics Meaning of the Equation T = The Equations of Motion and T = Variational Principle for the Coupled System Exercises Non-localizability of the Gravitational Energy On Covariance and Invariance Note on Unimodular Gravity TheTetradFormalism VariationofTetradFields TheEinstein HilbertAction...104

3 Contents xi Part II Consequences of the Invariance Properties of the Lagrangian L Lovelock s Theorem in Higher Dimensions Exercises Energy, Momentum, and Angular Momentum for Isolated Systems Interpretation ADMExpressionsforEnergyandMomentum Positive Energy Theorem Exercises The Initial Value Problem of General Relativity NatureoftheProblem ConstraintEquations Analogy with Electrodynamics Propagation of Constraints Local Existence and Uniqueness Theorems Analogy with Electrodynamics Harmonic Gauge Condition Field Equations in Harmonic Gauge CharacteristicsofEinstein sfieldequations Exercises General Relativity in 3 + 1Formulation Generalities Connection Forms CurvatureForms,EinsteinandRicciTensors GaussianNormalCoordinates MaximalSlicing Exercises Domain of Dependence and Propagation of Matter Disturbances BoltzmannEquationinGR One-Particle Phase Space, Liouville Operator for Geodesic Spray The General Relativistic Boltzmann Equation Applications of General Relativity 4 The Schwarzschild Solution and Classical Tests of General Relativity Derivation of the Schwarzschild Solution The Birkhoff Theorem Geometric Meaning of the Spatial Part of the Schwarzschild Metric Exercises Equation of Motion in a Schwarzschild Field Exercises Perihelion Advance Deflection of Light Exercises...178

4 xii Contents 4.5 Time Delay of Radar Echoes Geodetic Precession Schwarzschild Black Holes The Kruskal Continuation of the Schwarzschild Solution Discussion Eddington Finkelstein Coordinates Spherically Symmetric Collapse to a Black Hole RedshiftforaDistantObserver FateofanObserverontheSurfaceoftheStar Stability of the Schwarzschild Black Hole Penrose Diagram for Kruskal Spacetime Conformal Compactification of Minkowski Spacetime Penrose Diagram for Schwarzschild Kruskal Spacetime Charged Spherically Symmetric Black Holes Resolution of the Apparent Singularity Timelike Radial Geodesics Maximal Extension of the Reissner Nordstrøm Solution Appendix: Spherically Symmetric Gravitational Fields General Form of the Metric The Generalized Birkhoff Theorem Spherically Symmetric Metrics for Fluids Exercises Weak Gravitational Fields The Linearized Theory of Gravity Generalization Exercises Nearly Newtonian Gravitational Fields Gravitomagnetic Field and Lense Thirring Precession Exercises Gravitational Waves in the Linearized Theory PlaneWaves Transverse and Traceless Gauge Geodesic Deviation in the Metric Field of a Gravitational Wave A Simple Mechanical Detector Energy Carried by a Gravitational Wave The Short Wave Approximation Discussion of the Linearized Equation R μν (1) [h]= Averaged Energy-Momentum Tensor for Gravitational Waves Effective Energy-Momentum Tensor for a Plane Wave Exercises Emission of Gravitational Radiation SlowMotionApproximation...256

5 Contents xiii Rapidly Varying Sources RadiationReaction(PreliminaryRemarks) Simple Examples and Rough Estimates RigidlyRotatingBody Radiation from Binary Star Systems in Elliptic Orbits Exercises LaserInterferometers Gravitational Field at Large Distances from a Stationary Source TheKomarFormula Exercises Gravitational Lensing ThreeDerivationsoftheEffectiveRefractionIndex Deflection by an Arbitrary Mass Concentration The General Lens Map AlternativeDerivationoftheLensEquation Magnification, Critical Curves and Caustics Simple Lens Models Axially Symmetric Lenses: Generalities The Schwarzschild Lens: Microlensing Singular Isothermal Sphere Isothermal Sphere with Finite Core Radius Relation Between Shear and Observable Distortions Mass Reconstruction from Weak Lensing The Post-Newtonian Approximation Motion and Gravitational Radiation (Generalities) AsymptoticFlatness Bondi Sachs Energy and Momentum The Effacement Property FieldEquationsinPost-NewtonianApproximation EquationsofMotionforaTestParticle Stationary Asymptotic Fields in Post-Newtonian Approximation Point-ParticleLimit TheEinstein Infeld HoffmannEquations The Two-Body Problem in the Post-Newtonian Approximation Precession of a Gyroscope in the Post-Newtonian Approximation Gyroscope in Orbit Around the Earth PrecessionofBinaryPulsars General Strategies of Approximation Methods RadiationDamping BinaryPulsars Discovery and Gross Features Timing Measurements and Data Reduction ArrivalTime...351

6 xiv Contents SolarSystemCorrections Theoretical Analysis of the Arrival Times EinsteinTimeDelay Roemer and Shapiro Time Delays ExplicitExpressionfortheRoemerDelay AberrationCorrection TheTimingFormula ResultsforKeplerianandPost-KeplerianParameters MassesoftheTwoNeutronStars Confirmation of the Gravitational Radiation Damping Results for the Binary PSR B Double-Pulsar White Dwarfs and Neutron Stars Introduction WhiteDwarfs TheFreeRelativisticElectronGas Thomas Fermi Approximation for White Dwarfs HistoricalRemarks Exercises FormationofNeutronStars General Relativistic Stellar Structure Equations Interpretation of M General Relativistic Virial Theorem Exercises Linear Stability TheInteriorofNeutronStars Qualitative Overview Ideal Mixture of Neutrons, Protons and Electrons Oppenheimer Volkoff Model Pion Condensation Equation of State at High Densities Effective Nuclear Field Theories Many-Body Theory of Nucleon Matter GrossStructureofNeutronStars Measurements of Neutron Star Masses Using Shapiro TimeDelay Bounds for the Mass of Non-rotating Neutron Stars BasicAssumptions Simple Bounds for Allowed Cores AllowedCoreRegion Upper Limit for the Total Gravitational Mass RotatingNeutronStars CoolingofNeutronStars NeutronStarsinBinaries...416

7 Contents xv Some Mechanics in Binary Systems Some History of X-Ray Astronomy X-RayPulsars TheEddingtonLimit X-RayBursters FormationandEvolutionofBinarySystems Millisecond Pulsars Black Holes Introduction Proof of Israel s Theorem Foliation of Σ, RicciTensor,etc The Invariant (4) R (4) αβγ δ R αβγ δ The Proof (W. Israel, 1967) DerivationoftheKerrSolution Axisymmetric Stationary Spacetimes RicciCircularity Footnote: Derivation of Two Identities TheErnstEquation Footnote: Derivation of Eq. (8.90) RicciCurvature IntermediateSummary WeylCoordinates ConjugateSolutions Basic Equations in Elliptic Coordinates TheKerrSolution Kerr Solution in Boyer Lindquist Coordinates Interpretation of the Parameters a and m Exercises DiscussionoftheKerr NewmanFamily Gyromagnetic Factor of a Charged Black Hole SymmetriesoftheMetric Static Limit and Stationary Observers Killing Horizon and Ergosphere Coordinate Singularity at the Horizon and Kerr Coordinates Singularities of the Kerr Newman Metric Structure of the Light Cones and Event Horizon Penrose Mechanism Geodesics of a Kerr Black Hole The Hamilton Jacobi Method TheFourthIntegralofMotion Equatorial Circular Geodesics Accretion Tori Around Kerr Black Holes NewtonianApproximation General Relativistic Treatment...488

8 xvi Contents Footnote: Derivation of Eq. (8.276) The Four Laws of Black Hole Dynamics General Definition of Black Holes The Zeroth Law of Black Hole Dynamics SurfaceGravity TheFirstLaw SurfaceAreaofKerr NewmanHorizon TheFirstLawfortheKerr NewmanFamily The First Law for Circular Spacetimes The Second Law of Black Hole Dynamics Applications Evidence for Black Holes BlackHoleFormation Black Hole Candidates in X-Ray Binaries The X-Ray Nova XTE J Super-Massive Black Holes Appendix: Mathematical Appendix on Black Holes Proof of the Weak Rigidity Theorem The Zeroth Law for Circular Spacetimes Geodesic Null Congruences Optical Scalars TransportEquation The Sachs Equations Applications Change of Area AreaLawforBlackHoles The Positive Mass Theorem TotalEnergyandMomentumforIsolatedSystems Witten s Proof of the Positive Energy Theorem Remarks on the Witten Equation Application Generalization to Black Holes Penrose Inequality Exercises Appendix: Spin Structures and Spinor Analysis in General Relativity Spinor Algebra SpinorAnalysisinGR Exercises Essentials of Friedmann Lemaître Models Introduction Friedmann Lemaître Spacetimes Spaces of Constant Curvature Curvature of Friedmann Spacetimes...551

9 Contents xvii Part III Einstein Equations for Friedmann Spacetimes Redshift Cosmic Distance Measures ThermalHistoryBelow100MeV Overview ChemicalPotentialsoftheLeptons ConstancyofEntropy NeutrinoTemperature Epoch of Matter-Radiation Equality Recombination and Decoupling Luminosity-Redshift Relation for Type Ia Supernovae Theoretical Redshift-Luminosity Relation Type Ia Supernovae as Standard Candles Results Exercises Differential Geometry 11 Differentiable Manifolds Tangent Vectors, Vector and Tensor Fields The Tangent Space VectorFields TensorFields The Lie Derivative IntegralCurvesandFlowofaVectorField Mappings and Tensor Fields TheLieDerivative Local Coordinate Expressions for Lie Derivatives Differential Forms Exterior Algebra ExteriorDifferentialForms Differential Forms and Mappings DerivationsandAntiderivations TheExteriorDerivative MorphismsandExteriorDerivatives Relations Among the Operators d, i X and L X FormulafortheExteriorDerivative The -OperationandtheCodifferential OrientedManifolds The -Operation Exercises TheCodifferential CoordinateExpressionfortheCodifferential Exercises...624

10 xviii Contents 14.7 The Integral Theorems of Stokes and Gauss IntegrationofDifferentialForms Stokes Theorem Application Expression for div Ω X in Local Coordinates Exercises Affine Connections CovariantDerivativeofaVectorField ParallelTransportAlongaCurve Geodesics, Exponential Mapping and Normal Coordinates CovariantDerivativeofTensorFields Application Local Coordinate Expression for the Covariant Derivative CovariantDerivativeandExteriorDerivative Curvature and Torsion of an Affine Connection, Bianchi Identities Riemannian Connections Local Expressions Contracted Bianchi Identity TheCartanStructureEquations SolutionoftheStructureEquations Bianchi Identities for the Curvature and Torsion Forms Special Cases LocallyFlatManifolds Exercises WeylTensorandConformallyFlatManifolds Covariant Derivatives of Tensor Densities Some Details and Supplements Proofs of Some Theorems Tangent Bundles VectorFieldsAlongMaps Induced Covariant Derivative Exercises VariationsofaSmoothCurve FirstVariationFormula Jacobi Equation Appendix A Fundamental Equations for Hypersurfaces A.1 FormulasofGaussandWeingarten A.2 Equations of Gauss and Codazzi Mainardi A.3 Null Hypersurfaces A.4 Exercises Appendix B Ricci Curvature of Warped Products B.1 Application:FriedmannEquations...689

11 Contents xix Appendix C Frobenius Integrability Theorem C.1 Applications C.2 Proof of Frobenius Theorem (in the First Version) Appendix D Collection of Important Formulas D.1 VectorFields,LieBrackets D.2 DifferentialForms D.3 ExteriorDifferential D.4 Poincaré Lemma D.5 Interior Product D.6 LieDerivative D.7 Relations Between L X, i X and d D.8 VolumeForm D.9 Hodge-Star Operation D.10 Codifferential D.11 CovariantDerivative D.12 Connection Forms D.13 CurvatureForms D.14 Cartan sstructureequations D.15 Riemannian Connection D.16 CoordinateExpressions D.17 AbsoluteExteriorDifferential D.18 Bianchi Identities References Textbooks on General Relativity: Classical Texts Textbooks on General Relativity: Selection of (Graduate) Textbooks Textbooks on General Relativity: Numerical Relativity Textbooks on General Physics and Astrophysics Mathematical Tools: Modern Treatments of Differential GeometryforPhysicists Mathematical Tools: Selection of Mathematical Books Historical Sources Recent Books on Cosmology Research Articles, Reviews and Specialized Texts: Chapter Research Articles, Reviews and Specialized Texts: Chapter Research Articles, Reviews and Specialized Texts: Chapter Research Articles, Reviews and Specialized Texts: Chapter Research Articles, Reviews and Specialized Texts: Chapter Research Articles, Reviews and Specialized Texts: Chapter Research Articles, Reviews and Specialized Texts: Chapter Research Articles, Reviews and Specialized Texts: Chapter Research Articles, Reviews and Specialized Texts: Chapter Index...721

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