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

Starting from the 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. D 54, 3932 (1996)] is extended to the general stationary black hole and the nonminimally coupled scalar field. We last show, by comparing the two results, that the statistical-mechanical entropy and one-loop correction to the thermodynamical entropy are equivalent for coupling xi less than or equal to 0. After renormalization, a relation between the two entropies is given.