Penrose singularity theorems

Abstract

In this work, we will review Penrose singularity theorems. Following the publication of Einstein’s General Theory of Relativity in 1916, the possible existence of singularities in relativistic space-times garnered attention, which initiated a significant debate where various ideas were proposed in order to provide meaning to singularities. In 1965, Roger Penrose published the first modern singularity theorems, which generalised the existence of singularities in any relativistic space-time under certain topological conditions. To present the theorems in an accessible and coherent way, we will follow the structure of Penrose’s publication (as referenced in [17]). We will first establish the causal and chronological order relations between points in space-time. Subsequently, we will address the physical viability of the space-times. To do so, we will study the Cauchy problem and the necessary conditions that a relativistic space-time must satisfy to avoid containing time loops. In the final section, we will explore geodesic congruences in the space-time by applying general relativity, and we will define the concept of trapped surfaces, which is a key notion in Penrose developments. By following this line of reasoning, we will be able to prove the existence of incomplete geodesics defined within space-time. Therefore, we will observe the existence of singularities at the end and the beginning of geodesics on space-times.