Is the semi-classical analysis valid for extreme black holes?

TL;DR

This paper questions the validity of semi-classical analysis for extreme black holes, showing that standard methods fail due to the lack of orthonormal quantum modes near the horizon.

Contribution

It demonstrates that semi-classical analysis breaks down for extreme black holes because of the absence of orthonormal quantum modes at the horizon.

Findings

Standard semi-classical analysis yields ill-defined temperature for extreme black holes.

Bogolubov's coefficients do not satisfy normalization conditions in this case.

Failure attributed to the absence of orthonormal quantum modes near the horizon.

Abstract

The surface gravity for the extreme Reissner-Nordstr\"om black hole is zero suggesting that it has a zero temperature. However, the direct evaluation of the Bogolubov's coefficients, using the standard semi-classical analysis, indicates that the temperature of the extreme black hole is ill definite: the Bogolubov's coefficients obtained by performing the usual analysis of a collapsing model of a thin shell, and employing the geometrical optical approximation, do not obey the normalization conditions. We argue that the failure of the employement of semi-classical analysis for the extreme black hole is due to the absence of orthonormal quantum modes in the vicinity of the event horizon in this particular case.