Microscopic entropy of the three-dimensional rotating black hole of BHT massive gravity

TL;DR

This paper computes the microscopic entropy of three-dimensional rotating black holes in BHT massive gravity, showing agreement with semiclassical results via holographic duality and Cardy's formula.

Contribution

It extends Strominger's holographic entropy calculation to BHT massive gravity, including the effects of gravitational hair and relaxed asymptotic conditions.

Findings

Microscopic entropy matches semiclassical entropy.

Black holes have a gravitational hair parameter with a lower mass bound.

Holographic duality applies to BHT massive gravity with extended asymptotics.

Abstract

Asymptotically AdS rotating black holes for the Bergshoeff-Hohm-Townsend (BHT) massive gravity theory in three dimensions are considered. In the special case when the theory admits a unique maximally symmetric solution, apart from the mass and the angular momentum, the black hole is described by an independent "gravitational hair" parameter, which provides a negative lower bound for the mass. This bound is saturated at the extremal case and, since the temperature and the semiclassical entropy vanish, it is naturally regarded as the ground state. The absence of a global charge associated with the gravitational hair parameter reflects through the first law of thermodynamics in the fact that the variation of this parameter can be consistently reabsorbed by a shift of the global charges, giving further support to consider the extremal case as the ground state. The rotating black hole fits within relaxed asymptotic conditions as compared with the ones of Brown and Henneaux, such that they are invariant under the standard asymptotic symmetries spanned by two copies of the Virasoro generators, and the algebra of the conserved charges acquires a central extension. Then it is shown that Strominger's holographic computation for general relativity can also be extended to the BHT theory; i.e., assuming that the quantum theory could be consistently described by a dual conformal field theory at the boundary, the black hole entropy can be microscopically computed from the asymptotic growth of the number of states according to Cardy's formula, in exact agreement with the semiclassical result.