Geodetic deviation and geometrical meaning of Riemannian curvature tensor; geodesic congruences; expansion, shear, and twist; Raychaudhuri's equation.
Frobenius theorem; energetic conditions; focusing theorem, FRWL and Schwarzschild example
Null geodesics and their congruences; focusing theorem and Raychaudhuri's equation for null geodesics
Null radial congruences in Schwarschild spacetime; Future oriented normal vectors; trapped surfaces; outline of Penrose's singularity theorem
Review of tetrad formalism: coordinate and non-coordinate bases, orthonormal tetrad basis, covariant derivatives: spin connection and Ricci rotation coefficients; adapted tetrad for observers in gravitational field; acceleration object; static and free-falling observers in Schwarzschild spacetime
A joint course between Bratislava and Tartu
Lecturers: Martin Krššák, María José Guzmán, Sotirios Karamitsos
Schwarzschild black hole reminder; coordinate systems for Schwarzschild black holes; maximally extended solution; Kruskal and Carter-Penrose conformal diagrams
Charged black holes: Reissner-Nordstrom solution; naked singularities; extremal black holes; Kruskal and conformal diagrams
Rotating black holes: Kerr solution; horizon and ergosphere; zero angular momentum and static observers; maximal extension of Kerr spacetime, Penrose diagram
Killing vectors and conserved quantities for particles; stationary observers in Kerr spacetime; Penrose process; irreducible mass; Smarr law; black hole mechanics and the analogy with thermodynamics
Conserved currents and quantities; hypersurfaces, normal vectors, induced metric; Komar mass, Schwarzchild example; covariant conservation laws as a consequence of diffeomorphism symmetry
S. Carroll, Spacetime and Geometry, CUP 2004
E. Poisson, A Relativist’s Toolkit: The Mathematics of Black-Hole Mechanics, CUP 2009
H. Reall, Black Holes, Math Tripos pIII Cambridge notes
C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation, W. H. Freeman,1973
R. Wald, General Relativity, University of Chicago Press, 1984
T. Padmanabhan, Gravitation: Foundations and frontiers, CUP, 2014
P. K. Townsend, Black Holes, ArXiv:gr-qc/9707012