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

Abstract

We study the spacetime structures of the static solutions in the $n$-dimensional Einstein-Gauss-Bonnet-$\ensuremath{\Lambda}$ system systematically. We assume the Gauss-Bonnet coefficient $\ensuremath{\alpha}$ is non-negative and a cosmological constant is either positive, zero, or negative. The solutions have the $(n\ensuremath{-}2)$-dimensional Euclidean submanifold, which is the Einstein manifold with the curvature $k=1$, 0, and $\ensuremath{-}1$. We also assume $4\stackrel{\texttildelow{}}{\ensuremath{\alpha}}/{\ensuremath{\ell}}^{2}\ensuremath{\le}1$, where $\ensuremath{\ell}$ is the curvature radius, in order for the sourceless solution ($M=0$) to be defined. The general solutions are classified into plus and minus branches. The structures of the center, horizons, infinity, and the singular point depend on the parameters $\ensuremath{\alpha}$, ${\ensuremath{\ell}}^{2}$, $k$, $M$, and branches complicatedly so that a variety of global structures for the solutions are found. In our analysis, the $\stackrel{\texttildelow{}}{M}\mathrm{\text{\ensuremath{-}}}r$ diagram is used, which makes our consideration clear and enables easy understanding by visual effects. In the plus branch, all the solutions have the same asymptotic structure at infinity as that in general relativity with a negative cosmological constant. For the negative-mass parameter, a new type of singularity called the branch singularity appears at nonzero finite radius $r={r}_{b}>0$. The divergent behavior around the singularity in Gauss-Bonnet gravity is milder than that around the central singularity in general relativity. There are three types of horizons: inner, black hole, and cosmological. In the $k=1,0$ cases, the plus-branch solutions do not have any horizon. In the $k=\ensuremath{-}1$ case, the radius of the 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. The black hole solution with zero or negative mass exists in the plus branch even for the zero or positive cosmological constant. There is also the extreme black hole solution with positive mass. We briefly discuss the effect of the Gauss-Bonnet corrections on black hole formation in a collider and the possibility of the violation of the third law of the black hole thermodynamics.