Microscopic entropy of the charged BTZ black hole

TL;DR

The paper addresses the microscopic entropy of the charged BTZ black hole, whose charge induces a curvature singularity and logarithmic boundary terms. It shows that these boundary deformations are governed by the 2D conformal group and that a renormalization procedure yields finite boundary charges, enabling a Brown–Henneaux central charge $c=rac{3l}{2G}$ and a renormalized energy $l_0=rac{1}{2}l M_0(r_+)$. Cardy-state counting with $S=4\pi\sqrt{\frac{c\,l_0}{6}}$ reproduces the Bekenstein–Hawking entropy $S=\frac{\pi r_+}{2G}$, incorporating the charge via the logarithmic term. This demonstrates a microscopic CFT interpretation of black hole entropy even in the presence of curvature singularities and suggests broader applicability of AdS$_3$/CFT$_2$ methods to low-dimensional gravity and charged spacetimes.

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.