Gauss-Bonnet Gravity and Spacetime Singularities

TL;DR

This paper explores how Gauss-Bonnet higher-order curvature terms influence spacetime singularities, revealing they can soften or alter singularities in cosmological and black hole solutions, with implications for geodesic extendibility and the nature of singularities.

Contribution

It demonstrates that Gauss-Bonnet terms can replace traditional singularities with weaker or modified ones, and analyzes their effects on geodesic behavior and solution extensions in five-dimensional spacetimes.

Findings

Gauss-Bonnet terms replace Big Bang/Crunch with a sudden singularity.

Black hole singularities are softened, making radial geodesics weakly singular.

Cosmological singularities are modified, with finite expansion parameters.

Abstract

We investigate the effect of higher-order curvature terms, specifically Gauss-Bonnet terms, on spacetime singularities in five dimensions. For FLRW cosmologies, we demonstrate that Gauss-Bonnet terms can replace the Big Bang/Crunch with a "sudden" singularity, characterized by a finite scale factor and Hubble rate but diverging higher-order derivatives. Investigating various branches of solutions shows the possibility of explicit extension of non-spacelike geodesics beyond the singular point. Furthermore, we employ the Gauss-Bonnet junction conditions to verify the consistency of the extension with the field equations. The whole solution describes a contracting phase prior to the expansion phase with a well-defined surface stress-energy tensor. Regarding the Boulware-Deser black hole, we find that Gauss-Bonnet terms soften the central singularity for radial geodesics--rendering them "weak" according to the Tipler and Krolak criteria--whereas non-radial geodesics remain strongly singular. Junction condition analysis of this solution shows that although higher-curvature corrections alter the nature of the singularity, geodesics are still inextendible as a result of divergent extrinsic curvature. Our results are consistent with the Penrose-Hawking singularity theorems since in Gauss-Bonnet black holes, geodesics suffer from focusing (expansion parameter diverges), while in cosmology, there is no focusing since the expansion parameter remains finite at the singularity.