On the Entropy of a Quantum Field in the Rotating Black Holes

TL;DR

This paper calculates the entropy of a scalar field in rotating black holes using the brick wall method, revealing divergences at the horizon related to the density of states, especially when the field is comoving with the black hole.

Contribution

It provides a detailed analysis of the entropy and free energy of scalar fields in rotating black holes, highlighting divergence behaviors near the horizon and their dependence on the frame of reference.

Findings

Entropy diverges at the stationary limit surface and horizon.

Divergences occur only when the field is comoving with the black hole.

Leading entropy terms include area-dependent and logarithmic divergences.

Abstract

By using the brick wall method we calculate the free energy and the entropy of the scalar field in the rotating black holes. As one approaches the stationary limit surface rather than the event horizon in comoving frame, those become divergent. Only when the field is comoving with the black hole (i.e. $\Omega_0 = \Omega_H$) those become divergent at the event horizon. In the Hartle-Hawking state the leading terms of the entropy are $ A \frac{1}{h} + B \ln(h) + finite$, where $h$ is the cut-off in the radial coordnate near the horizon. In term of the proper distance cut-off $\epsilon$ it is written as $ S = N A_H/\epsilon^2$. The origin of the divergence is that the density of state on the stationary surface and beyond it diverges.