ar X iv : g r - qc / 9 90 70 11 v 4 5 O ct 1 99 9 Entropies of general nonextreme stationary axisymmetric black hole : statistical mechanics and thermodynamics ∗

Starting from metric of general nonextreme stationary axisymmetric black hole in four-dimensional spacetime, both statistical-mechanical and thermo-dynamical 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 the Pauli-Villars regularization scheme, an expression for the statistical-mechanical entropy arising from the nonmini-mally 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 nonminimally coupled scalar fields. 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 ξ ≤ 0. After renormal-ization, a relation between the two entropies is given.