From Geometry to Black Holes

The primary objective of my dissertation was to develop a comprehensive understanding of General Relativity and the mathematical structures underpinning the theory. The document begins with an in-depth exploration of Differential Geometry, constructing the necessary mathematical framework from the definition of a manifold through to the derivation of Christoffel symbols, the geodesic equation, and the various curvature tensors.

The second half of the dissertation focuses on the derivation of the Einstein Field Equations from the Einstein–Hilbert action. This approach yields the familiar form of the field equations, with the proportionality constant to the stress–energy tensor determined using the Newtonian limit. The work concludes with a discussion of black holes, featuring a derivation of the Schwarzschild metric and an analysis of light cone behaviour when approaching and crossing the event horizon.