Geometry of nonextreme black holes near the extreme state

Abstract

Nonextreme black hole in a cavity can achieve the extreme state with a zero surface gravity at a finite temperature on a boundary, the proper distance between the boundary and the horizon being finite. The classical geometry in this state is found explicitly for four-dimensional spherically-symmetrical and 2+1 rotating holes. In the first case the limiting geometry depends only on one scale factor and the whole Euclidean manifold is described by the Bertotti-Robinson spacetime. The general structure of a metric in the limit under consideration is also found with quantum corrections taken into account. Its angular part represents two-sphere of a constant radius. In all cases the Lorentzian counterparts of the metrics are free from singularities.