General theory of relativity: mathematical elements and Hawking’s singularity theorem

Abstract

[en] In this work, we study Riemannian and pseudo-Riemannian manifolds and their main properties. From them, we examine the special and general theories of relativity, and see how they arise from modelling space-time as special kinds of pseudoRiemannian manifolds, the Lorentzian manifolds. Within this theory, we are able to give a rigorous formulation of the fundamental properties of cosmology and the Schwarzschild space-time. We also wish to relate the behaviour of geodesics in a manifold with the intrinsic structure of the manifold. This results in the formulation of the Hopf-Rinow theorem in the case of Riemannian manifolds, and the Hawking singularity theorem, in Lorentzian manifolds.