**Section:**001

**Term:**WN 2018

**Subject:**Mathematics (MATH)

**Department:**LSA Mathematics

The theory of general relativity (GR) unifies space, time and gravitation. The fundamental objects of study in GR are the spacetimes, which may have a rich geometric structure. A spacetime in general relativity theory is a Lorentzian manifold where the metric solves the Einstein equations.

These equations can be thought of as the laws of GR. They can be written as a system of nonlinear, second-order, hyperbolic pde. The unknown is the metric. Typical physical questions are formulated as initial value problems for the Einstein equations under specific conditions. The solution will lay open the geometry of the resulting spacetime. GR is a rich interplay between geometry, analysis and physics. Here, mathematical results have direct implications in physics. Today, the methods of geometric analysis have proven to be most effective to investigate these structures. Phenomena like the stability of galaxies, the formation of black holes or even the fate of our Universe are described by solutions of the Einstein equations. Gravitational waves are investigated by means of Lorentzian geometry.

This course shall focus on Lorentzian geometry and the Einstein field equations within general relativity. After brief reviews of special relativity and some fundamental facts from differential geometry (Lorentzian/Riemannian manifolds, curvature, Bianchi identity, etc.), we will introduce the spacetime in GR. Null hypersurfaces in Lorentzian manifolds will be treated. We shall then discuss the Schwarzschild solution and black holes. In view of the latter, we shall prove Penrose's incompleteness theorem and study the extensions of this result by Hawking and Penrose. Those results are better known as the `singularity theorems'. The most recent breakthrough (2008) along this way is certainly Christodoulou's result on the formation of black holes, showing that a closed trapped surface will form through the focusing of gravitational waves. Finally, as time allows, we will investigate gravitational waves. In particular, we will discuss nonlinear effects of these waves. In order to establish these results a combination of geometric and analytic methods will be introduced.

**Text:** General Relativity, Univ.Chicago Press, 1984, Robert Wald Optional

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**ISBN: 9780821849743**

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