Negative modes and the thermodynamics of Reissner-Nordström black holes

TL;DR

The paper addresses the existence of negative Euclidean modes for the magnetic Reissner–Nordström black hole in Einstein–Maxwell theory by developing a gauge-invariant, five-dimensional Kaluza–Klein framework. By lifting the four-dimensional problem to a 5D setting and isolating a dynamical, gauge-invariant sector while decoupling nondynamical modes, the authors derive a reduced action whose single remaining mode is governed by a radial potential. They analytically show that a negative mode persists when the charge is small but disappears precisely when the canonical specific heat at fixed charge becomes positive, i.e. at |Q|/M ≥ √3/2, in agreement with thermodynamic stability criteria. The work provides a practical, gauge-invariant method to study perturbative quantum gravity with matter and clarifies the interplay between local thermodynamic stability and the well-definedness of the gravitational partition function in the presence of gauge fields.

Abstract

We analyse the problem of negative modes of the Euclidean section of the Reissner-Nordström 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.

Figures (1)

• Figure 1: Evolution of the negative mode with $|Q|/M$.