Topological black holes for Einstein-Gauss-Bonnet gravity with a nonminimal scalar field

Abstract

We consider the Einstein-Gauss-Bonnet gravity with a negative cosmological constant together with a source given by a scalar field nonminimally coupled in arbitrary dimension $D$. For a certain election of the cosmological and Gauss-Bonnet coupling constants, we derive two classes of AdS black hole solutions whose horizon is planar. The first family of black holes obtained for a particular value of the nonminimal coupling parameter only depends on a constant $M$, and the scalar field vanishes as $M=0$. The second class of solutions corresponds to a two-parametric (with constants $M$ and $A$) black hole stealth configuration, which is a nontrivial scalar field with a black hole metric such that both sides (gravity and matter parts) of the Einstein equations vanish. In this case, in the vanishing $M$, the solution reduces to a stealth scalar field on the pure AdS metric. We note that the existence of these two classes of solutions is indicative of the particular choice of the coupling constants, and they cannot be promoted to spherical or hyperboloid black hole solutions in a standard fashion. In the last part, we add to the original action some exact ($D\ensuremath{-}1$) forms coupled to the scalar field. The direct benefit of introducing such extra fields is to obtain black hole solutions with a planar horizon for an arbitrary value of the nonminimal coupling parameter.