General Relativity and Relativistic Astrophysics

I. Differential Geometry.- 1. Differentiable Manifolds.- 2. Tangent Vectors, Vector and Tensor Fields.- 2.1 The Tangent Space.- 2.2 Vector Fields.- 2.3 Tensor Fields.- 3. The Lie Derivative.- 3.1 Integral Curves and Flow of a Vector Field.- 3.2 Mappings and Tensor Fields.- 3.3 The Lie Derivative.- 4. Differential Forms.- 4.1 Exterior Algebra.- 4.2 Exterior Differential Forms.- 4.3 Derivations and Antiderivations.- 4.4 The Exterior Derivative.- 4.5 Relations Among the Operators d,iX and LX.- 4.6 The *-Operation and the Codifferential.- 4.6.1 Oriented Manifolds.- 4.6.2 The *-Operation.- 4.6.3 The Codifferential.- 4.7 The Integral Theorems of Stokes and Gauss.- 4.7.1 Integration of Differential Forms.- 4.7.2 Stokes' Theorem.- 5. Affine Connections.- 5.1 Covariant Derivative of a Vector Field.- 5.2 Parallel Transport Along a Curve.- 5.3 Geodesics, Exponential Mapping, Normal Coordinates.- 5.4 Covariant Derivative of Tensor Fields.- 5.5 Curvature and Torsion of an Affine Connection, Bianchi Identities.- 5.6 Riemannian Connections.- 5.7 The Cartan Structure Equations.- 5.8 Bianchi Identities for the Curvature and Torsion Forms.- 5.9 Locally Flat Manifolds.- 5.10 Table of Important Formulae.- II. General Theory Of Relativity.- 1. The Principle of Equivalence.- 1.1 Characteristic Properties of Gravitation.- 1.1.1 Strength of the Gravitational Interaction.- 1.1.2 Universality of the Gravitational Interaction.- 1.1.3 Precise Formulation of the Principle of Equivalence.- 1.1.4 Gravitational Red Shift as Evidence for the Validity of the Principle of Equivalence.- 1.2 Special Relativity and Gravitation.- 1.2.1 The Gravitational Red Shift is not Consistent with Special Relativity.- 1.2.2 Global Inertial Systems Cannot be Realized in the Presence of Gravitational Fields.- 1.2.3 The Deflection of Light Rays.- 1.2.4 Theories of Gravity in Flat Space-Time.- 1.3 Space and Time as a Lorentzian Manifold, Mathematical Formulation of the Principle of Equivalence.- 1.4 Physical Laws in the Presence of External Gravitational Fields.- 1.4.1 Motion of a Test Body in a Gravitational Field and Paths of Light Rays.- 1.4.2 Energy and Momentum Conservation in the Presence of an External Gravitational Field.- 1.4.3 Electrodynamics.- 1.4.4 Ambiguities.- 1.5 The Newtonian Limit.- 1.6 The Red Shift in a Static Gravitational Field.- 1.7 Fermat's Principle for Static Gravitational Fields.- 1.8 Geometric Optics in a Gravitational Field.- 1.9 Static and Stationary Fields.- 1.10 Local Reference Frames and Fermi Transport.- 1.10.1 Precession of the Spin in a Gravitational Field.- 1.10.2 Fermi Transport.- 1.10.3 The Physical Difference Between Static and Stationary Fields.- 1.10.4 Spin Rotation in a Stationary Field.- 1.10.5 Local Coordinate Systems.- 2. Einstein's Field Equations.- 2.1 Physical Meaning of the Curvature Tensor.- 2.2 The Gravitational Field Equations.- 2.3 Lagrangian Formalism.- 2.3.1 Hamilton's Principle for the Vacuum Field Equations.- 2.3.2 Another Derivation of the Bianchi Identity and its Meaning.- 2.3.3 Energy-Momentum Tensor in a Lagrangian Field Theory.- 2.3.4 Analogy with Electrodynamics.- 2.3.5 Meaning of the Equation ? · T=0.- 2.3.6 Variational Principle for the Coupled System.- 2.4 Nonlocalizability of the Gravitational Energy.- 2.5 The Tetrad Formalism.- 2.6 Energy, Momentum, and Angular Momentum of Gravity for Isolated Systems.- 2.7 Remarks on the Cauchy Problem.- 2.8 Characteristics of the Einstein Field Equations.- 3. The Schwarzschild Solution and Classical Tests of General Relativity.- 3.1 Derivation of the Schwarzschild Solution.- 3.2 Equation of Motion in a Schwarzschild Field.- 3.3 Advance of the Perihelion of a Planet.- 3.4 Bending of Light Rays.- 3.5 Time Delay of Radar Echoes.- 3.6 Geodetic Precession.- 3.7 Gravitational Collapse and Black Holes (Part 1).- 3.7.1 The Kruskal Continuation of the Schwarzschild Solution.- 3.7.2 Spherically Symmetric Collapse to a Black Hole.- Appendix: Spherically Symmetric Gravitational Fields.- 4. Weak Gravitational Fields.- 4.1 The Linearized Theory of Gravity.- 4.2 Nearly Newtonian Gravitational Fields.- 4.3 Gravitational Waves in the Linearized Theory.- 4.4 The Gravitational Field at Large Distances from the Source.- 4.5 Emission of Gravitational Radiation.- 5. The Post-Newtonian Approximation.- 5.1 The Field Equations in the Post-Newtonian Approximation.- 5.2 Asymptotic Fields.- 5.3 The Post-Newtonian Potentials for a System of Point Particles.- 5.4 The Einstein-Infeld-Hoffmann Equations.- 5.5 Precession of a Gyroscope in the PN-Approximation.- 5.6 The Binary Pulsar.- 5.6.1 Pulse Arrival-Time Data and their Analysis.- 5.6.2 Relativistic Effects.- 5.6.3 Gravitational Radiation.- 5.6.4 The Companion Star.- III. Relativistic Astrophysics.- 6. Neutron Stars.- 6.1 Order-of-Magnitude Estimates.- 6.2 Relativistic Equations for Stellar Structure.- 6.3 Stability.- 6.4 The Interior of Neutron Stars.- 6.4.1 Qualitative Overview.- 6.4.2 Ideal Mixture of Neutrons, Protons, and Electrons.- 6.5 Models for Neutron Stars.- 6.6 Bounds on the Mass of Nonrotating Neutron Stars.- 6.6.1 Basic Assumptions.- 6.6.2 Simple Bounds for Allowed Cores.- 6.6.3 Allowed Core Region.- 6.6.4 Upper Limit for the Total Gravitational Mass.- 6.7 Cooling of Neutron Stars.- 6.7.1 Introduction.- 6.7.2 Thermodynamic Properties of Neutron Stars.- 6.7.3 Neutrino Emissivities.- 6.7.4 Cooling Curves.- 6.8 Addendum 1: Ground State Energy of Macroscopic Matter.- 6.8.1 Stability of Matter with Negligible Self-Gravity.- 6.8.2 Nonsaturation of Gravitational Forces.- 6.8.3 Newton vs. Coulomb.- 6.8.4 Semirelativistic Systems.- 6.9 Addendum 2: Core Collapse Models of Type II Supernova Explosions.- 6.9.1 Some Observational Facts.- 6.9.2 Presupernova Evolution of Massive Stars.- 6.9.3 The Physics of Stellar Collapse.- 6.9.4 Numerical Studies.- 6.10 Addendum 3: Magnetic Fields of Neutron Stars, Pulsars.- 6.10.1 Introduction.- 6.10.2 Magnetic Dipole Radiation.- 6.10.3 Synchrotron Radiation from the Crab Nebula.- 6.10.4 The Pulsar Magnetosphere.- 6.10.5 Matter in Strong Magnetic Fields.- 7. Rotating Black Holes.- 7.1 Analytic Form of the Kerr-Newman Family.- 7.2 Asymptotic Field and g-Factor of a Black Hole.- 7.3 Symmetries of g.- 7.4 Static Limit and Stationary Observers.- 7.5 Horizon and Ergosphere.- 7.6 Coordinate Singularity at the Horizon, Kerr Coordinates.- 7.7 Singularities of the Kerr-Newman Metric.- 7.8 Structure of the Light Cones.- 7.9 Penrose Mechanism.- 7.10 The Second Law of Black Hole Dynamics.- 7.11 Remarks on the Realistic Collapse.- 8. Binary X-Ray Sources.- 8.1 Brief History of X-Ray Astronomy.- 8.2 Mechanics of Binary Systems.- 8.3 X-Ray Pulsars.- 8.4 Bursters.- 8.5 CygX-1: A Black Hole Candidate.- 8.6 Evolution of Binary Systems.- 9. Accretion onto Black Holes and Neutron Stars.- 9.1 Spherically Symmetric Accretion onto a Black Hole.- 9.1.1 Adiabatic Flow.- 9.1.2 Thermal Bremsstrahlung from the Accreting Gas.- 9.2 Disk Accretion onto Black Holes and Neutron Stars.- 9.2.1 Introduction.- 9.2.2 Basic Equations for Thin Accretion Disks (Non-Relativistic Theory).- 9.2.3 Steady Keplerian Disks.- 9.2.4 Standard Disks.- 9.2.5 Stability Analysis of Thin Accretion Disks.- 9.2.6 Relativistic Keplerian Disks.- Appendix: Nonrelativistic and General Relativistic Hydrodynamics of Viscous Fluids.- A. Nonrelativistic Theory.- B. Relativistic Theory.- References.