In 1979 I gave graduate courses at the University of Zurich and lectured in the 'Troisieme Cycle de la Suisse Romande' (a consortium offour uni­ versities in the french-speaking part of Switzerland), and these lectures were the basis of the 'Springer Lecture Notes in Physics', Volume 150, published in 1981. This text appeared in German, because there have been few modern expositions of the general theory of relativity in the mother tongue of its only begetter. Soon after the book appeared, W. Thirring asked me to prepare an English edition for the 'Texts and Mono­ graphs in Physics'. Fortunately E. Borie agreed to translate the original German text into English. An excellent collaboration allowed me to re­ vise and add to the contents of the book. I have updated and improved the original text and have added a number of new sections, mostly on astrophysical topics. In particular, in collaboration with M. Camenzind I have included a chapter on spherical and disk accretion onto compact objects. This book divides into three parts. Part I develops the mathematical tools used in the general theory of relativity. Since I wanted to keep this part short, but reasonably self-contained, I have adopted the dry style of most modern mathematical texts. Readers who have never before been confronted with differential geometry will find the exposition too ab­ stract and will miss motivations of the basic concepts and constructions.

Inhalt

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 C…