Quantum scalar field in D-dimensional static black hole space-times

An Euclidean approach for investigating quantum aspects of a scalar field living on a class of D-dimensional static black hole space-times, including the extremal ones, is reviewed. The approach makes use of a near-horizon approximation of the metric and zeta-function formalism for evaluating the partition function and the expectation value of the field <phi(2)(x)>. After a review of the nonextreme black hole case, the extreme one is considered in some detail. In this case, there is no conical singularity, but the finite imaginary time compactification introduces a cusp singularity. It is found that the zeta-function regularized partition function can be defined, and the vacuum expectation value of the field, is finite on the horizon, as soon as the cusp singularity is absent, namely, the manifold is smooth and the corresponding temperature is T = 0. It is suggested that the requirement of having a smooth near-horizon geometry always selects the correct black hole equilibrium temperature. (C) 1999 American Institute of Physics. [S0022-2488(99)03910-9].