Learning Outcomes:
- Introducing Differential Geometry:
Manifolds, scalars, vectors, covectors, tensors.
- Introducing Riemannian Geometry: The metric; Riemannian and Lorentzian manifolds, connections and the covariant derivative, torsion and curvature, the Cristoffel connection, parallel transport, normal coordinates, geodesic deviation; symmetries of the Riemann tensor; the Ricci tensor and Einstein tensor.
- The Einstein Equations:
The Einstein-Hilbert action, the cosmological constant; diffeomorphisms; Minkowski, de Sitter and anti-de Sitter spacetimes; symmetries and isometries, Killing vectors, conserved quantities; Coupling matter, the energy-momentum tensor; the Newtonian Limit.
- Geodesics:
Non-relativistic particles and the geodesic equation; relativistic particles, Minkowski space; electromagnetism and gravity; the equivalence principle; time dilation; the Schwarzschild metric, planetary orbits and perihelion precession; the pull of Venus and Jupiter; light bending.
- Black Holes: The Schwarzschild solution, Birkhoff's theorem, Rindler space, black hole thermodynamics; The Reissner-Nordstrom solution.