GENERAL RELATIVITY

The course provides an introduction to the theory of General Relativity, illustrating the ground concepts (such as the equivalence principle and the link between space and matter) and the main computation tools (tensor algebra), and applying them to physical systems such as compact objects and cosmologies. The discussion will also include some open questions and the corresponding research techniques, such as approximation schemes and numerical approaches.

Introduction: General Relativity as a physical theory of the gravitational field; history and cultural influences; General Relativity and observations. The mathematics of curved spaces: Riemannian geometry: tensors, parallel transport, covariant derivatives, connections, curvature; metrics and differentiable manifolds; coordinate systems and diffeomorphisms; geodesics. The physics of curved spaces: the laws of physics on curved spaces; stress-energy tensor and Einstein’s tensor; Einstein’s equation; meaning of the matter component; lagrangian formulation. Exact solutions: self-gravitating systems in General Relativity and inevitability of the gravitational collapse; black holes: the Schwarzschild solution and the Kerr solution; singularities, horizons, ergospheres, stationary orbits, frame dragging; other compact objects; homogeneous and isotropic cosmological models; the cosmological constant. Approximate solutions: Newtonian limit and post-Newtonian expansions; linear approximations and perturbation theory; Einstein’s equation as a Cauchy problem; Numerical Relativity. Open problems: the dark sector; gravitational-wave astronomy; the primordial universe and quantum gravity.

• B. Schutz, Introduction to General Relativity, Cambridge University Press (2009)
• R. D’Inverno, Introducing Einstein's Relativity, Clarendon Press (1992)
• C. Misner, K. Thorne, J. Wheeler, Gravitation, W. H. Freeman (1973)
• E. Poisson, A Relativist's Toolkit, Cambridge University Press (2004)
• R. Wald, General Relativity, University of Chicago Press (1984)
• A. Lightman, W. Press, R. Price, S. Teukolski, Problem Book in Relativity and Gravitation, Princeton University Press (1975)