Quantum entropy of the general non-extreme stationary axisymmetric black hole due to minimally coupled quantum scalar field

The statistical-mechanical entropy arising from the minimally coupled scalar fields which rotate with the angular velocity $\Omega_0$ in the four-dimensional non-extreme stationary axisymmetric black hole space-time is investigated by using the 't Hooft's "brick wall" model and regularized with Pauli-Villars scheme. It is first shown that the entropy does not contain a quadratically divergent term and a logarithmically divergent one as usually since the integrand of the total number of the modes is finite on the event horizon if $\Omega_0\neq\Omega_H$ ($\Omega_H$ is angular velocity of the event horizon). And then a formula of the statistical-mechanical entropy is obtained when $\Omega_0=\Omega_H$. It is also demonstrated, for the Kerr-Newman black hole and the Einstein-Maxwell dilaton-axion black hole, that the statistical-mechanical entropy got by the formula and result computed by conical singularity method are equivalent