Microscopic entropy of the charged BTZ black hole

TL;DR

This paper calculates the microscopic entropy of the charged BTZ black hole using boundary conformal field theory techniques, confirming the Bekenstein-Hawking entropy through the Cardy formula.

Contribution

It demonstrates how to properly renormalize boundary charges in the charged BTZ black hole and confirms the microscopic origin of its entropy via the dual CFT.

Findings

The central charge matches Brown-Henneaux value.

Boundary deformations are generated by the conformal group.

Cardy formula reproduces the Bekenstein-Hawking entropy.

Abstract

The charged BTZ black hole is characterized by a power-law curvature singularity generated by the electric charge of the hole. The curvature singularity produces ln r terms in the asymptotic expansion of the gravitational field and divergent contributions to the boundary terms. We show that these boundary deformations can be generated by the action of the conformal group in two dimensions and that an appropriate renormalization procedure allows for the definition of finite boundary charges. In the semiclassical regime the central charge of the dual CFT turns out to be that calculated by Brown and Henneaux, whereas the charge associated with time translation is given by the renormalized black hole mass. We then show that the Cardy formula reproduces exactly the Bekenstein-Hawking entropy of the charged BTZ black hole.