GENERAL RELATIVITY

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.

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