Gauss-Bonnet Gravity and Spacetime Singularities

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.

Figures (7)

• Figure 1: Phase-Space Diagram for the Hubble rate
• Figure 2: Time $\tau$ as a function of Scale Factor $\eta$, showing the two disjoint solution branches with $\tau \geq 0$ and $\tau < 0$.
• Figure 3: $V(r)$ for a timelike object in equation (69), with $m=1$, $\alpha = 0.08$ and $L^2 = 10$
• Figure 4: The negative sign f(r) with $m=1$, $\alpha = 0.08$
• Figure 5: $V(r)$ for a timelike object in equation (71), with $m=1$, $\alpha = 0.08$ and $L = 0$