Basic differential geometry: Local coordinates on manifolds. Covariant and contravariant vector and tensor fields. (Pseudo-) Riemann metric. Covariant differentiation (Christoffel symbols, Levi-Civita connection). Parallel transport. Curved spaces. Lie derivatives and Killing vector fields.
General theory of relativity: Basic concepts in general relativity. Schwarzschild spacetime. Einstein's field equations. The energy-momentum tensor. Weak field limit. Experimental tests of general relativity. Gravitational lensing. Gravitational waves. Introductory cosmology (including the Friedmann–Lemaître–Robertson–Walker metric), including inflation and dark energy.
After completing the course you should be able to:
- Use differential geometry to describe the properties of a curved space and compute basic quantities in differential geometry.
- Derive and use Einstein's field equations and describe the definition and role of the energy-momentum tensor in those, account for the physical interpretation of its components, and prove that Newton's theory of gravity is recovered in the non-relativistic limit.
- Compute physical quantities for test particles in a given solution to Einstein's field equations, e.g., particle trajectories and proper times.
- Give an account of the experiments with which the general theory of relativity has been tested and compare with predictions from Newton's theory of gravity.
- Use the Friedmann–Lemaître–Robertson–Walker metric to describe the different possibilities for how a homogeneous universe develops with time as well as describe the ideas behind cosmological inflation and dark energy.
