Asymptotically (anti)-de Sitter solutions in Gauss-Bonnet gravity without a cosmological constant

Abstract

In this paper I show that one can have asymptotically de Sitter, anti-de Sitter (AdS), and flat solutions in Gauss-Bonnet gravity without a cosmological constant term in field equations. First, I introduce static solutions whose three surfaces at fixed $r$ and $t$ have constant positive ($k=1$), negative ($k=\ensuremath{-}1$), or zero ($k=0$) curvature. I show that for $k=\ifmmode\pm\else\textpm\fi{}1$ one can have asymptotically de Sitter, AdS, and flat spacetimes, while for the case of $k=0$, one has only asymptotically AdS solutions. Some of these solutions present naked singularities, while some others are black hole or topological black hole solutions. I also find that the geometrical mass of these five-dimensional spacetimes is $m+2\ensuremath{\alpha}|k|$, which is different from the geometrical mass $m$ of the solutions of Einstein gravity. This feature occurs only for the five-dimensional solutions, and is not repeated for the solutions of Gauss-Bonnet gravity in higher dimensions. Second, I add angular momentum to the static solutions with $k=0$, and introduce the asymptotically AdS charged rotating solutions of Gauss-Bonnet gravity. Finally, I introduce a class of solutions which yields an asymptotically AdS spacetime with a longitudinal magnetic field, which presents a naked singularity, and generalize it to the case of magnetic rotating solutions with two rotation parameters.