Geometry of nonextreme black holes near the extreme state

TL;DR

This paper explores the geometry of nonextreme black holes approaching the extreme state, providing explicit classical and quantum-corrected solutions in various symmetrical cases, and demonstrating their non-singular Lorentzian structures.

Contribution

It derives explicit geometries of nonextreme black holes near extremality, including quantum corrections, and shows their Lorentzian versions are free of singularities.

Findings

Classical geometry approaches Bertotti-Robinson spacetime in the extremal limit.

Quantum corrections modify the metric structure while preserving non-singularity.

Lorentzian counterparts of the near-extreme geometries are free from singularities.

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.