Quantum entropy of a nonextreme stationary axisymmetric black hole due to a minimally coupled quantum scalar field

By using the ’t Hooft “brick wall” model and the Pauli-Villars regularization scheme we calculate the statistical-mechanical entropy arising from minimally coupled scalar fields which rotate with azimuthal angular velocity ${\Omega }_0={\Omega }_H$ $({\Omega }_H$ is the angular velocity of the black hole horizon) in general four-dimensional nonextreme stationary axisymmetric black hole space-time. We also show, for the Kerr-Newman and the Einstein-Maxwell dilaton-axion black holes, that the statistical-mechanical entropy obtained from our derivation and the quantum thermodynamical entropy by the conical singularity method are equivalent.