Canonical ensemble of a $d$-dimensional Reissner-Nordstr\"om black hole in a cavity

Abstract

We construct the canonical ensemble of a $d$-dimensional Reissner-Nordstr\"om black hole spacetime in a cavity surrounded by a heat reservoir through the Euclidean path integral formalism. The heat reservoir is described by the boundary of the cavity with fixed radius $R$, fixed temperature $T$, and fixed electric charge $Q$. We use York's approach to find the reduced action, and then perform a zero loop approximation. We find that the number of solutions for the black hole depends on the electric charge being smaller or larger than a critical $Q_s$, having two stable and one unstable solutions for the former case, and one stable solution for the latter. We obtain the system's thermodynamic properties from the partition function. We analyze thermodynamic stability, controlled by the positivity of the heat capacity at constant area and electric charge. We show that there is a discontinuity in the heat capacity, signaling a turning point. We investigate the favorable stable phases and the phase diagram of the system. We show that the two stable black hole solutions can suffer a first order phase transition from one to the other, and at the critical charge $Q_s$ this turns into a second order one. We introduce a model of charged hot flat space, i.e., hot flat space with charge near the boundary. We find that a first order phase transition between the large stable black hole and charged hot flat space occurs at a horizon radius larger than the Buchdahl bound, and comment on the physics. Finally, we recover the Davies thermodynamic solutions and a Rindler solution in the limit of infinite cavity, and the York solutions in the limit of zero charge. Hence, both York and Davies formalisms are unified and connected in our approach. In all instances we mention carefully the four-dimensional case, for which we accomplish new results, and study in detail all aspects of the five-dimensional case.