Three-Dimensional Gravity with Conformal Scalar and Asymptotic Virasoro Algebra

TL;DR

This paper extends Strominger's method for deriving black hole entropy via asymptotic Virasoro algebra to a three-dimensional gravity solution with a conformal scalar field, revealing agreement in entropy form but not in numerical coefficient.

Contribution

It applies the asymptotic Virasoro algebra approach to a non-$AdS_3$ black hole with a conformal scalar, exploring entropy calculation limitations.

Findings

The asymptotic Virasoro algebra applies to the MZ black hole.

Cardy's formula matches the entropy's functional form.

Numerical coefficients of entropy do not agree.

Abstract

Strominger has derived the Bekenstein-Hawking entropy of the BTZ black hole using asymptotic Virasoro algebra. We apply Strominger's method to a black hole solution found by Martinez and Zanelli (MZ). This is a solution of three-dimensional gravity with a conformal scalar field. The solution is not $AdS_3$, but it is asymptotically $AdS_3$; therefore, it has the asymptotic Virasoro algebra. We compute the central charge for the theory and compares Cardy's formula with the Bekenstein-Hawking entropy. It turns out that the functional form does agree, but the overall numerical coefficient does not. This is because this approach gives the "maximum possible entropy" for the numerical coefficient.