Here I share my notes on general relativity which I learned in graduate courses. please report any errors found to me.

The primary sources are

  • Padmanabhan, T. Gravitation: Foundations and Frontiers. Cambridge University Press, 2010.
  • D’Inverno, Ray, and James Vickers. Introducing Einstein’s Relativity: A Deeper Understanding. Second, Oxford University Press, 2021.
Chapter Section Subsections
1 Review general relativity mathematics Tensor, Manifold, Atlas, Jacobian, Hypersurface, Vector, Covariant vector, Dyadic, Scaler, Chain derivative, Covariant tensor, Mixed tensor, symmetric and antisymmetric tensor, Contraction, Tangant space, Free index tensors, Lee bracet and comutator, Partial derivative, Lee derivative, Covariant derivative, Affine manifold, Absolute derivative, Parallely propagated, Geodesic, Affine geodesic, Riemann tensor, Geodesic coordinate, Integrable connection, Affine flat manifold, Metric, Riemann manifold, Light like curve and geodesic equation of that, Space like curve and geodesic equation of that, Null cerve and geodecis equation of that, Metricity, Levi-Civita connection, Metric connection, Riemann-Chiristofel or curvature tensor, Signituare of metric, Metric flatness manifold, Properties of Riemann tensor, Riecci tensor, Ricci scaler, Einestian tensor, Difrensial properties of Riemann tensor, Weyl tensor, Biholomorphic transformation, Conformally related, Conformally invariant, Scale invariant, Conformally flat, Tensor densities, Levi-Civita symbol, Generalized Kronecker delta, Metric determinant, Principle of general covariance, Invariant volume element, Hypersurface element, Action, Palatini equation, Direct variation, Palatini variation, Isometry and killing vector
2 General relativity Non-local free falling and geodesic deviation equation, Einstein field equation, Lagrangian of Einstein field equation, Energy-momentum tensor in general relativity
3 Some analytical solutions of Einstein’s field equation General metric with spherical symmetry, Birkhoff theorem, Schwarzschild metric, Reissner–Nordström metric, De Sitter metric, FRW metric, De Sitter-Schwarzschild metric, De Sitter-Reissner–Nordström metric
4 Experimental evidence of general relativity Invariance quantities related to symmetries of space-time, Advance of the perihelion of Mercury, Bending of light in the gravitational field of the Sun, Gravitational redshift
5 Black holes Rindler metric, Schwarzschild metric, Kerr metric, Thermodynamics of black holes
6 Gravitational waves Linearize of field equation, The energy of gravitational waves, Solution of gravitational waves, Gravitational waves, and binary system