A Primer on Spacetime Singularities I: Mathematical Framework

TL;DR

This paper provides a detailed mathematical overview of spacetime singularities in classical General Relativity, including classifications, key theorems, and limitations of current approaches, establishing a foundation for future research.

Contribution

It introduces a comprehensive classification of singularities based on curvature behavior and reviews the mathematical structures and theorems relevant to spacetime singularities.

Findings

Classifies singularities into quasi-regular, non-scalar, and scalar types.

Analyzes the implications of Penrose and Hawking singularity theorems.

Critically examines limitations of cosmic censorship and extensions beyond GR.

Abstract

This article presents a comprehensive and rigorous overview of spacetime singularities within the framework of classical General Relativity. Singularities are defined through the failure of geodesic completeness, reflecting the limits of predictability in spacetime evolution. The paper reviews the mathematical structures involved, including differentiability classes of the metric, and explores key constructions such as Geroch's and Schmidt's formulations of singular boundaries. A detailed classification of singularities - quasi-regular, non-scalar, and scalar - is proposed, based on the behavior of curvature tensors along incomplete curves. The limitations of previous approaches, including the cosmic censorship conjecture and extensions beyond General Relativity, are critically examined. The work also surveys the major singularity theorems of Penrose and Hawking, emphasizing their implications for gravitational collapse and cosmology. By focusing exclusively on the classical regime, the article lays a solid foundation for the systematic study of singular structures in relativistic spacetimes.