You can find more information about the history and physical content of Relativity Theory here.
Examples: Minkowski space-time, Schwarzschild solution, Einstein, de Sitter, anti-de Sitter and Friedman-Lemaitre-Robertson-Walker universes; matching and Oppenheimer-Snyder collapse; Penrose diagrams.
Causality: time orientabiliy, chronological and causal past and future, domain of dependence; chronological, stably causal and globally hyperbolic spacetimes.
Singularities: Jacobi equation, conjugate points; energy conditions; existence of maximizing geodesics; Hawking and Penrose singularity theorems.
Cauchy Problem: wave equation; Cauchy problems with constraints; Gauss-Coddazzi relations and 3+1 decomposition of the Einstein equation; Choquet-Bruhat theorem; constraint equations for the initial data.
Positive Mass Theorem: Komar mass; Einstein-Hilbert action; Lagrangian and Hamiltonian formulation of the Einstein equations; mass of an asymptotically flat Riemannian 3-manifold; positive mass theorem; Penrose inequality.
Black Holes: Reissner-Nordstrom and Kerr solutions; cosmic censorship; area theorem; black hole thermodynamics.
The final grade will be the average of the grades of weekly problem sets. Late homework will not be accepted.
For more exercises check the course webpages from previous years (mostly in Portuguese):