Geodesics in semi-Riemannian Geometry and links to General Relativity

Abstract

Gravity, as a fundamental force of nature, is very well described by General Relativity. In General Relativity, our universe is thought of as a spacetime manifold. Differential geometry gives us a lot of tools to study all manifolds, including spacetime. In this thesis, we develop tools to study geodesics, the generalization of straight, unaccelerated curves, that determine paths of freely falling particles in spacetime. We go through their equations for Minkowski, Schwarzschild and Misner spacetime, providing solutions for Minkowski and Misner spacetime. We explain how these tools from Differential Geometry play a fundamental role in the theory of General Relativity.