Resolving space-time singularities in spherically symmetric black holes: geodesic completeness, curvature scalars, and tidal forces

TL;DR

This paper investigates methods to resolve singularities in spherically symmetric black holes by analyzing geodesic completeness, curvature scalars, and tidal forces, proposing mechanisms to avoid space-time incompleteness without relying on specific gravitational theories.

Contribution

It identifies conditions on metric functions that restore geodesic completeness and introduces two mechanisms—displacing focal points or preventing geodesic focusing—applicable across various theories.

Findings

Displacement of focal points to infinite affine distance

Prevention of geodesic focusing via a bounce

Explicit examples of complete and incomplete space-times

Abstract

The existence of black holes in the Universe is nowadays established on the grounds of a blench of astrophysical observations, most notably those of gravitational waves from binary mergers and the imaging of supermassive objects at the heart of M87 and Milky Way galaxies. However, this success of Einstein's General Relativity~(GR) to connect theory of black holes with observations is also the source of its doom, since Penrose's theorem proves that, under physically sensible conditions, the development of a space-time singularity (as defined by the existence of a focal point for some geodesic paths in finite affine time) within black holes as described by GR is unavoidable. In this work, we thoroughly study how to resolve space-time singularities in spherically symmetric black holes. To do it so we find the conditions on the metric functions required for the restoration of geodesic completeness without any regards to the specific theory of the gravitational and matter fields supporting the amended metric. Our discussion considers both the usual trivial radial coordinate case and the bouncing radial function case and arrives to two mechanisms for this restoration: either the focal point is displaced to infinite affine distance or a bounce prevents the focusing of geodesics. Several explicit examples of well known (in)complete space-times are given. Furthermore, we consider the connection of geodesic (in)completeness with another criterion frequently used in the literature to monitor singular space-times: the blow up of (some sets of) curvature scalars and the infinite tidal forces they could bring with them, and discuss the conditions required for the harmlessness upon physical observers according to each criterion.