Conical geometry and quantum entropy of a charged Kerr black hole

TL;DR

The paper extends the conical singularity method for black hole entropy to the charged, rotating Kerr-Newman black hole, showing that arbitrary Euclidean-periodicity around the horizon induces a conical structure with δ-function curvature at the horizon. Using heat-kernel techniques, it computes the UV-divergent one-loop entropy of a scalar field on this background and demonstrates that these divergences can be absorbed by renormalizing the tree-level gravitational couplings, just as in static cases. The resulting quantum entropy comprises an area term plus curvature-projected and extrinsic-curvature contributions, whose renormalization confirms the consistency of black hole thermodynamics for stationary spacetimes. These findings reinforce the universality of entropy renormalization across static and stationary black holes and connect with known results for Reissner-Nordström and Schwarzschild limits.

Abstract

We apply the method of conical singularities to calculate the tree-level entropy and its one-loop quantum corrections for a charged Kerr black hole. The Euclidean geometry for the Kerr-Newman metric is considered. We show that for an arbitrary periodization in Euclidean space there exists a conical singularity at the horizon. Its $δ$-function like curvatures are calculated and are shown to behave similar to the static case. The heat kernel expansion for a scalar field on this conical space background is derived and the (divergent) quantum correction to the entropy is obtained. It is argued that these divergences can be removed by renormalization of couplings in the tree-level gravitational action in a manner similar to that for a static black hole.