Course Syllabus
- Riemannian Geometry: Vectors and Tensors; parallel transport, covariant differentiation; Geodesics; Riemann-Christoffel curvature tensor - its symmetry properties, Ricci tensor; Bianchi identities; vanishing of the curvature tensor as a condition for flatness, Geodesic deviation equation.
- Principle of general covariance and principle of equivalence; Einstein field equations, derivation from a variational principle.
- Schwarzschild exterior solution, Birkhoff's theorem. Geodesics in a Schwarzschild geometry. Crucial tests of general relativity - perihelion shift, bending of light, gravitational redshift. Schwarzschild blackhole - event horizon and static limit, Kruskal - Szekere's diagram.
- Maxwell's equations in general relativity. Reissner - Nordstrom solutions - charged blackhole. Kerr - Newman solutions, Kerr - Newman blackholes. Ergosphere, Penrose process and extraction of energy from a blackhole.
- Interior solutions for a spherical star; Oppenheimer - Volkoff equation; Chandrasekhar limit and white dwarfs, Oppenheimer - Volkoff limit and neutron stars; pulsars. Oppenheimer - Snider non-static dust model - gravitational collapse.
- Linearised filed equations and gravitational waves.
- Lie derivatives; spacetime symmetries, Killing vectors.
- Cosmological assumptions - Cosmological principle,Hydrodynamics approximation and general relativity; Robertson-Walker metric. Red shift, Hubble's observations. Friedman models, cosmological parameters, age of the Universe, cosmological horizons; models with Λ-term.