General Relativity and Einstein’s Equations

  • Lorentz geometry
    • Introduction
    • Manifolds
    • Differentiable mappings
    • Vectors and tensors
      • Tangent and cotangent space
      • Vector fields
      • Tensors and tensor fields
    • Pseudo-Riemannian metrics
      • General properties
      • Riemannian and Lorentzian metrics
    • Riemannian connection
    • Geodesics
    • Curvature
    • Geodesic deviation
    • Maximum of length and conjugate points
    • Linearized Ricci and Einstein tensors
    • Second derivative of the Ricci tensor
  • Special Relativity
  • General relativity and Einstein’s equations
  • Schwarzschild spacetime and black holes
  • Cosmology
  • Local Cauchy problem
  • Constraints
  • Other hyperbolic-elliptic well-posed systems
  • Relativistic fluids
  • Relativistic kinetic theory
  • Progressive waves
  • Global hyperbolicity and causality
  • Singularities
  • Stationary spacetimes and black holes
  • Global existence theorems asymptotically Euclidean data
  • Global existence theorems the cosmological case
  • APPENDICES
    • Sobolev spaces on Riemannian manifolds
    • Second-order elliptic systems on Riemannian manifolds
    • Quasi-diagonal, quasi-linear, second-order hyperbolic systems
    • General hyperbolic systems
    • Cauchy–Kovalevski and Fuchs theorems
    • Conformal methods
    • Kaluza–Klein theories
      • Introduction
      • Isometries
      • Kaluza–Klein metrics
        • Metric in adapted frame
        • Structure coefficients
        • Kaluza–Klein connection
      • 4 Curvature tensor
      • 5 Ricci tensor and K–K equations
      • 6 Equations in conformal spacetime metric