Topological black holes in higher derivative gravity

Abstract

We study static black holes in quadratic gravity with planar and hyperbolic symmetry and non-extremal horizons. We obtain a solution in terms of an infinite power-series expansion around the horizon, which is characterized by two independent integration constants -- the black hole radius and the strength of the Bach tensor at the horizon. While in Einstein's gravity, such black holes require a negative cosmological constant $Λ$, in quadratic gravity they can exist for any sign of $Λ$ and also for $Λ=0$. Different branches of Schwarzschild-Bach-(A)dS or purely Bachian black holes are identified which admit distinct Einstein limits. Depending on the curvature of the transverse space and the value of $Λ$, these Einstein limits result in (A)dS-Schwarzschild spacetimes with a transverse space of arbitrary curvature (such as black holes and naked singularities) or in Kundt metrics of the (anti-)Nariai type (i.e., dS$_2\times$S$^2$, AdS$_2\times$H$^2$, and flat spacetime). In the special case of toroidal black holes with $Λ=0$, we also discuss how the Bach parameter needs to be fine-tuned to ensure that the metric does not blow up near infinity and instead matches asymptotically a Ricci-flat solution.