The Hawking-Penrose singularity theorem: a self-contained review (2024)
In this work, the Hawking-Penrose singularity theorem is studied from the perspective of the problem of geodesic inextendibility in Lorentzian manifolds, a problem that provides a formal definition of the notion of space-time singularity in general relativity. First, geodesic curves were studied in the Riemannian sense as solutions to the curve length optimization problem, and it was shown that every geodesic ceases to be a solution to this optimization problem in finite parametrization. Subsequently, the previous result was extended to families of geodesics whose initial point lies on a Lorentzian submanifold and that share their final point. Finally, the causal structure of space-times and the topology of spaces of causal curves were studied to demonstrate that every co-spacelike geodesic whose initial point lies on a trapped surface halts in finite parametrization. That is, if the space-time satisfies certain geometric properties, this result guarantees the existence of singularities.