Negative modes and the thermodynamics of Reissner-Nordstr\"om black holes

TL;DR

This paper investigates the negative modes of Euclidean Reissner-Nordström black holes, showing their disappearance correlates with positive specific heat, and discusses the implications for black hole thermodynamics and quantization.

Contribution

The study analytically links negative modes to thermodynamic stability and introduces a gauge-invariant method inspired by prior work to analyze perturbations.

Findings

Negative modes vanish when specific heat is positive.

Partition function is well-defined under thermodynamic equilibrium.

Method decouples gauge volume divergences in perturbation analysis.

Abstract

We analyse the problem of negative modes of the Euclidean section of the Reissner-Nordstr\"om black hole in four dimensions. We find analytically that a negative mode disappears when the specific heat at constant charge becomes positive. The sector of perturbations analysed here is included in the canonical partition function of the magnetically charged black hole. The result obeys the usual rule that the partition function is only well-defined when there is local thermodynamical equilibrium. We point out the difficulty in quantising Einstein-Maxwell theory, where the so-called conformal factor problem is considerably more intricate. Our method, inspired by hep-th/0608001, allows us to decouple the divergent gauge volume and treat the metric perturbations sector in a gauge-invariant way.