Simple compactifications and Black p-branes in Gauss-Bonnet and Lovelock Theories

TL;DR

This paper investigates simple compactifications and black p-brane solutions in higher-dimensional gravity theories with curvature corrections, revealing special cases where solutions exist and analyzing their thermodynamics and stability.

Contribution

It identifies conditions under which asymptotically flat compactifications and black p-branes exist in Gauss-Bonnet and Lovelock theories, including new solutions in specific cases.

Findings

Only pure Einstein-Hilbert or Gauss-Bonnet Lagrangians admit these solutions.

Existence of new exotic black hole solutions in four-dimensional Gauss-Bonnet theory.

A universal thermodynamical Gregory-Laflamme transition occurs across theories.

Abstract

We look for the existence of asymptotically flat simple compactifications of the form $M_{D-p}\times T^{p}$ in $D$-dimensional gravity theories with higher powers of the curvature. Assuming the manifold $M_{D-p}$ to be spherically symmetric, it is shown that the Einstein-Gauss-Bonnet theory admits this class of solutions only for the pure Einstein-Hilbert or Gauss-Bonnet Lagrangians, but not for an arbitrary linear combination of them. Once these special cases have been selected, the requirement of spherical symmetry is no longer relevant since actually any solution of the pure Einstein or pure Gauss-Bonnet theories can then be toroidally extended to higher dimensions. Depending on $p$ and the spacetime dimension, the metric on $M_{D-p}$ may describe a black hole or a spacetime with a conical singularity, so that the whole spacetime describes a black or a cosmic $p$-brane, respectively. For the purely Gauss-Bonnet theory it is shown that, if $M_{D-p}$ is four-dimensional, a new exotic class of black hole solutions exists, for which spherical symmetry can be relaxed. Under the same assumptions, it is also shown that simple compactifications acquire a similar structure for a wide class of theories among the Lovelock family which accepts this toroidal extension. The thermodynamics of black $p$-branes is also discussed, and it is shown that a thermodynamical analogue of the Gregory-Laflamme transition always occurs regardless the spacetime dimension or the theory considered, hence not only for General Relativity. Relaxing the asymptotically flat behavior, it is also shown that exact black brane solutions exist within a very special class of Lovelock theories.