Extremal Black Holes and Holographic C-Theorem

TL;DR

This paper derives first and second order differential equations in three-dimensional gravity theories with scalar fields, enabling analytic extremal black hole solutions and supporting a holographic c-theorem connecting two AdS spaces.

Contribution

It introduces a Bogomol'nyi-type approach to find extremal black hole solutions in Einstein and new massive gravity, extending domain wall equations to more general cases.

Findings

Analytic extremal black hole solutions with AdS asymptotics.

Perturbative extremal solutions in new massive gravity.

Construction of a holographic c-theorem linking two AdS spaces.

Abstract

We found Bogomol'nyi type of the first order differential equations in three dimensional Einstein gravity and the effective second order ones in new massive gravity when an interacting scalar field is minimally coupled. Using these equations in Einstein gravity, we obtain analytic solutions corresponding to extremally rotating hairy black holes. We also obtain perturbatively extremal black hole solutions in new massive gravity using these lower order differential equations. All these solutions have the anti de-Sitter spaces as their asymptotic geometries and as the near horizon ones. This feature of solutions interpolating two anti de-Sitter spaces leads to the construction of holographic c-theorem in these cases. Since our lower order equations reduce naturally to the well-known equations for domain walls, our results can be regarded as the natural extension of domain walls to more generic cases.