Exact black hole solutions with a conformally coupled scalar field and dynamic Ricci curvature in f(R) gravity theories

Abstract

Abstract We report exact black hole solutions in asymptotically flat or (A)dS four-dimensional spacetime with a conformally coupled self-interacting scalar field in f(R) gravity. We first consider the asymptotically flat model $$f(R) = R -2\alpha \sqrt{R}$$ f ( R ) = R - 2 α R and derive an exact black hole solution. Then, we consider the asymptotically (A)dS model $$f(R) =R -2 \Lambda -2 \alpha \sqrt{R-4 \Lambda }$$ f ( R ) = R - 2 Λ - 2 α R - 4 Λ and derive an exact black hole solution. In both cases the modified gravity parameter $$\alpha $$ α , which has the dimension of the inverse mass, cannot be set to zero and the self-interacting potential is determined from the Klein–Gordon equation, preserving the conformal invariance. The thermodynamics of the solutions is also studied.