Study of topological and physical properties of Schwarzschild and Kerr space-time equipped with Zeeman topology

Abstract

The Zeeman topology, first proposed by E. C. Zeeman in 1967, is a topology for Minkowski space that is physically well-motivated, but more intricate than the standard Euclidean topology. Later, it was extended to general relativity in 1976 by R. Göbel. Despite its compatibility with pseudo-Riemannian geometry, the Zeeman topology is rarely used in the literature due to its lack of second-countability. This work aims to explore the implications of the Zeeman topology for Schwarzschild and Kerr space-time. Specifically, we investigate the compactness of trapped surfaces and the validity of the singularity theorem. Our findings suggest that the Zeeman topology may offer alternative views to certain physical phenomena in general relativity.