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