Topological black holes in higher derivative gravity

TL;DR

This paper investigates static black hole solutions in quadratic gravity, revealing new branches that exist for any cosmological constant and connecting them to known Einstein solutions with various horizon geometries.

Contribution

It provides an analytical solution framework for black holes in quadratic gravity, including novel branches with arbitrary cosmological constant and detailed asymptotic analysis.

Findings

Black holes in quadratic gravity can exist for any sign of the cosmological constant.

Multiple solution branches connect to Einstein limits with different horizon geometries.

Conditions for asymptotic Ricci-flat solutions in toroidal black holes are identified.

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 $\Lambda$, in quadratic gravity they can exist for any sign of $\Lambda$ and also for $\Lambda=0$. 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 $\Lambda$, 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 $\Lambda=0$, 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.