Entropies of the general nonextreme stationary axisymmetric black hole: statistical mechanics and thermodynamics

Starting from metric of the general nonextreme stationary axisymmetric black hole in four-dimensional spacetime, both statistical-mechanical and thermodynamical entropies are studied. First, by means of the "brick wall" model in which the Dirichlet condition is replaced by a scattering ansatz for the field functions at the horizon and with Pauli-Villars regularization scheme, an expression for the statistical-mechanical entropy arising from the nonminimally coupled scalar fields is obtained. Then, by using the conical singularity method Mann and Solodukhin's result for the Kerr-Newman black hole (Phys. Rev. D54, 3932(1996)) is extended to the general stationary black hole and the nonminimally coupled scalar field. We last shown by comparing the two results that the statistical-mechanical entropy and one-loop correction to the thermodynamical entropy are equivalent for coupling $\xi\leq 0$. After renormalization, a relation between the two entropies is given.