Planar AdS black holes in Lovelock gravity with a nonminimal scalar field

Abstract

In arbitrary dimension D, we consider a self-interacting scalar field nonminimally coupled with a gravity theory given by a particular Lovelock action indexed by an integer k. To be more precise, the coefficients appearing in the Lovelock expansion are fixed by requiring the theory to have a unique AdS vacuum with a fixed value of the cosmological constant. This yields to k = 1, 2,⋯, $ \left[ {\frac{D-1 }{2}} \right] $ inequivalent possible gravity theories; here the case k = 1 corresponds to the standard Einstein-Hilbert Lagrangian. For each par (D, k), we derive two classes of AdS black hole solutions with planar event horizon topology for particular values of the nonminimal coupling parameter. The first family of solutions depends on a unique constant and is valid only for k ≥ 2. In fact, its GR counterpart k = 1 reduces to the pure AdS metric with a vanishing scalar field. The second family of solutions involves two independent constants and corresponds to a stealth black hole configuration; that is a nontrivial scalar field together with a black hole metric such that both side of the Einstein equations (gravity and matter parts) vanishes identically. In this case, the standard GR case k = 1 reduces to the Schwarzschild-AdS-Tangherlini black hole metric with a trivial scalar field. We show that the two-parametric stealth solution defined in D dimension can be promoted to the uniparametric black hole solution in (D + 1) dimension by fixing one of the two constants in term of the other and by adding a transversal coordinate. In both cases, the existence of these solutions is strongly inherent of the presence of the higher order curvature terms k ≥ 2 of the Lovelock gravity. We also establish that these solutions emerge from a stealth configuration defined on the pure AdS metric through a Kerr-Schild transformation. Finally, in the last part, we include multiple exact (D − 1) − forms homogenously distributed and coupled to the scalar field. For a specific coupling, we obtain black hole solutions for arbitrary value of the nonminimal coupling parameter generalizing those obtained in the pure scalar field case.