General Relativity and Relativistic Astrophysics

  • PART I. DIFFERENTIAL GEOMETRY
    • Differentiable Manifolds
    • Tangent Vectors, Vector and Tensor Fields
      • The Tangent Space
      • Vector Fields
      • Tensor Fields
    • The Lie Derivative
      • Integral Curves and Flow of a Vector Field
      • Mappings and Tensor Fields
      • The Lie Derivative
    • Differential Forms
      • Exterior Algebra
      • Exterior Differential Forms
      • Derivations and Antiderivations
      • The Exterior Derivative
      • Relations Among the Operators d, ix and Lx
      • The *-Operation and the Codifferential
        • Oriented Manifolds
        • The *-Operation
        • The Codifferential
      • The Integral Theorems of Stokes and Gauss
        • Integration of Differential Forms
        • Stokes’ Theorem
    • Affine Connections
      • Covariant Derivative of a Vector Field
      • Parallel Transport Along a Curve
      • Geodesics, Exponential Mapping, Normal Coordinates
      • Covariant Derivative of Tensor Fields
      • Curvature and Torsion of an Affine Connection, Bianchi Identities
    • The Cartan Structure Equations
    • Bianchi Identities for the Curvature and Torsion Forms
    • Locally Flat Manifolds
    • Table of Important Formulae
  • PART II. GENERAL THEORY OF RELATIVITY
    • Introduction
    • The Principle of Equivalence
      • Characteristic Properties of Gravitation
        • Strength of the Gravitational Interaction
        • Universality of the Gravitational Interaction
        • Precise Formulation of the Principle of •Equivalence
        • Gravitational Red Shift as Evidence for the Validity of the Principle of Equivalence
    • Special Relativity and Gravitation
      • The Gravitational Red Shift is not Consistent with Special Relativity
      • Global Inertial Systems Cannot be Realized in the Presence of Gravitational Fields
      • The Deflection of Light Rays
      • Theories of Gravity in Flat Space-Time
    • Space and Time as a Lorentzian Manifold, Mathematical Formulation of the Principle of Equivalence
    • Physical Laws in the Presence of External Gravitational Fields
      • Motion of a Test Body in a Gravitational Field and Paths of light Rays
      • Energy and Momentum Conservation in the Presence of an External Gravitational Field
      • Electrodynamics
      • Ambiguities
    • The Newtonian Limit
    • The Red Shift in a Static Gravitational Field
    • Fermat’s Principle for Static Gravitational Fields
    • Geometric Optics in a Gravitational Field
    • Static and Stationary Fields
    • Local Reference Frames and Fermi Transport
      • Precession of the Spin in a Gravitational Field
      • Fermi Transport
      • The Physical Difference Between Static and Stationary Fields
      • Spin Rotation in a Stationary Field
      • Local Coordinate Systems
    • Einstein’s Field Equations
      • Physical Meaning of the Curvature Tensor**
      • The Gravitational Field Equations
      • Lagrangian Formalism
        • Hamilton’s Principle for the Vacuum Field Equations
        • Another Derivation of the Bianchi Identity and its Meaning
        • Energy-Momentum Tensor in a Lagrangian Field Theory
        • Analogy with Electrodynamics
        • Meaning of the Equation Delta · T = 0
        • Variational Principle for the Coupled System
      • Nonlocalizability of the Gravitational Energy
  • PART III. RELATIVISTIC ASTROPHYSICS
  • Appendix