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

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.