Spacetime structure of static solutions in Gauss-Bonnet gravity: Charged case

Abstract

We have studied spacetime structures of static solutions in the $n$-dimensional Einstein-Gauss-Bonnet-Maxwell-$\ensuremath{\Lambda}$ system. Especially we focus on effects of the Maxwell charge. We assume that the Gauss-Bonnet coefficient $\ensuremath{\alpha}$ is non-negative and $4\stackrel{\texttildelow{}}{\ensuremath{\alpha}}/{\ensuremath{\ell}}^{2}\ensuremath{\le}1$ in order to define the relevant vacuum state. Solutions have the $(n\ensuremath{-}2)$-dimensional Euclidean submanifold whose curvature is $k=1$, 0, or $\ensuremath{-}1$. In Gauss-Bonnet gravity, solutions are classified into plus and minus branches. In the plus branch all solutions have the same asymptotic structure as those in general relativity with a negative cosmological constant. The charge affects a central region of a spacetime. A branch singularity appears at the finite radius $r={r}_{b}>0$ for any mass parameter. There the Kretschmann invariant behaves as $O((r\ensuremath{-}{r}_{b}{)}^{\ensuremath{-}3})$, which is much milder than the divergent behavior of the central singularity in general relativity $O({r}^{\ensuremath{-}4(n\ensuremath{-}2)})$. In the $k=1$ and 0 cases plus-branch solutions have no horizon. In the $k=\ensuremath{-}1$ case, the radius of a horizon is restricted as ${r}_{h}<\sqrt{2\stackrel{\texttildelow{}}{\ensuremath{\alpha}}}$ (${r}_{h}>\sqrt{2\stackrel{\texttildelow{}}{\ensuremath{\alpha}}}$) in the plus (minus) branch. Some charged black hole solutions have no inner horizon in Gauss-Bonnet gravity. There are topological black hole solutions with zero and negative mass in the plus branch regardless of the sign of the cosmological constant. Although there is a maximum mass for black hole solutions in the plus branch for $k=\ensuremath{-}1$ in the neutral case, no such maximum exists in the charged case. The solutions in the plus branch with $k=\ensuremath{-}1$ and $n\ensuremath{\ge}6$ have an inner black hole and inner and outer black hole horizons. In the $4\stackrel{\texttildelow{}}{\ensuremath{\alpha}}/{\ensuremath{\ell}}^{2}=1$ case, only a positive mass solution is allowed, otherwise the metric function takes a complex value. Considering the evolution of black holes, we briefly discuss a classical discontinuous transition from one black hole spacetime to another.