Equivalence principle and its physical implications. Topological Manifolds. Differentiable manifolds. Vector fields and tangent bundle. Connections. Torsion tensor and Riemann tensor. Jacobi geodetic deviation equation. Geodesics. Geometrical formulation of the newtonian gravity. Lorentzian Manifolds. Killing equation and conservation laws in GR. LL hypersurfaces, generators and dynamics. Physical motivation for the Einstein’s field equations. 2+2 formalism and solutions: RN, Schwarzschild, de Sitter. Event Horizon in static spacetimes. Birkhoff theorem. Eddington-Finkelstein coordinates. Kruskal coordinates. Schwarzschild maximal extension. Penrose conformal mapping. RN maximal extension. Geodesic motion in RN. Thorne’s wormhole. Matter matters: TOV equations. Stationary spacetimes. ZAMO observers. Event horizon in stationary spacetimes. Kerr solution in Boyer-Lindquist coordinates. Ergoregion and Event Horizon. Penrose’s extraction mechanism. Kerr maximal extension. No-Hair theorem. Dynamics of LL hypersurfaces in generic spacetimes: Raychaudhuri equation. BH rigidity. The zeroth Law of BH dynamics. Definition of EH in general spacetime. Non stationary spacetimes: accretion and evaporation. Teleological properties of the EH. Hawking’s area theorem and second Law of BH dynamics. First law of BH (thermo) dynamics. QFT in curved spacetime. Quantum creation of particles in an expanding universe. Thermal emission from a BH. Hamiltonian formalism and WdW equation.

E. Poisson,A Relativist's Toolkit, The Mathematics of Black-Hole Mechanics, Cambridge UP

R. Wald, General Relativity, Chicago UP

IntroducingEintein'sRelativity: from tensors to gravitational waves.Ray d'Inverno. Oxford UP

S. Weinberg, Gravitation and Cosmology,New York: John Wiley and Sons (1972).