An Introduction to General Relativity and Cosmology

  • How the theory of relativity came into being (a brief historical sketch)
    • Special versus general relativity
    • Space and inertia in Newtonian physics
    • Newton’s theory and the orbits of planets
    • The basic assumptions of general relativity
  • A short sketch of 2-dimensional differential geometry
  • Tensors, tensor densities
    • What are tensors good for?
    • Differentiable manifolds
    • Scalars
    • Contravariant vectors
    • Covariant vectors
    • Tensors of second rank
  • Covariant derivatives
  • Parallel transport and geodesic lines
  • The curvature of a manifold; flat manifolds
  • Riemannian geometry
  • Symmetries of Riemann spaces, invariance of tensors
  • Methods to calculate the curvature quickly – Cartan forms and algebraic computer programs
  • The spatially homogeneous Bianchi type spacetimes
  • The Petrov classification by the spinor method
  • The Einstein equations and the sources of a gravitational field
  • The Maxwell and Einstein–Maxwell equations and the Kaluza–Klein theory
  • Spherically symmetric gravitational fields of isolated objects
  • Relativistic hydrodynamics and thermodynamics
  • Relativistic cosmology I: general geometry
  • Relativistic cosmology II: the Robertson–Walker geometry
  • Relativistic cosmology III: the Lemaître–Tolman geometry
  • Relativistic cosmology IV: generalisations of L–T and related geometries
  • The Kerr solution