Negative modes and the thermodynamics of Reissner-Nordstrom black holes

We analyze the problem of negative modes of the Euclidean section of the Reissner-Nordstrom 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 analyzed 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 quantizing Einstein-Maxwell theory, where the so-called conformal factor problem is considerably more intricate. Our method, inspired by Kol [Phys. Rev. D 77, 044039 (2008)], allows us to decouple the divergent gauge volume and treat the metric perturbations sector in a gauge-invariant way.