Some global, analytical and topological properties of regular black holes

TL;DR

This paper explores the global and topological properties of regular black holes, analyzing their geometries and topologies to understand how they evade singularity theorems and the implications of topology change.

Contribution

It provides a comprehensive analysis of the geometrical and topological structures of regular black hole cores, including their classification and the conditions for topology change.

Findings

Cores can have topologies like S^3, H^3, R×S^1, or S^1×S^2 depending on the geometry.

The Euler characteristic of Seifert fiber spaces tracks topology transitions in black hole slices.

Nariai cores can be used to construct regular black holes without topology change.

Abstract

In this work we study regular black holes from a global perspective looking for evading some of the well-known singularity theorems by using their "reverses". Then, model geometries for the slices of typical spherically symmetric, (locally) static four dimensional regular black hole solutions are described from both an analytical and a topological point of view. While the finiteness of both the scalar and Kretchmann curvature of the slices around the regular center determines the geometry of the core, the positive answer to the Poincar\'e conjecture assures that, under two assumptions, its topology is that of a three-sphere. However, in general, the cores are shown to be $S^3$, $H^3$, $\mathbb{R}\times S^1$ or $S^1\times S^2$, depending whether a de Sitter, anti de Sitter, Nariai or Bertotti-Robinson geometries are employed to describe the slices at the regular center. Then, a description of the aforementioned slices in terms of Seifert fibre spaces is given in order to show that the Euler characteristic of the bundle can be used to track the transition between the core of the regular black hole and the rest of the slices in most of the cases considered in the literature. After Geroch and Tipler's theorems are employed to study the consequences of topology change on regular black hole spacetimes, we show that Borde and Vilenkin's singularity theorem is used to restrict their possible types. We end by noting that Nariai cores can be safely used to construct regular black holes without topology changes.