Thermodynamics of five-dimensional Schwarzschild black holes in the canonical ensemble

Abstract

We study the thermodynamics of a five-dimensional Schwarzschild black hole in the canonical ensemble using York's formalism. Inside a cavity of fixed size $r$ and fixed temperature $T$, there is a threshold at $\pi r T = 1$ above which a black hole can be in thermal equilibrium. This thermal equilibrium can be achieved for two specific black holes, a small black hole of horizon radius $r_{+1}$, and a large black hole of radius $r_{+2}$. In five dimensions, the radii $r_{+1}$ and $r_{+2}$ have an exact expression. Through the path integral formalism and the partition function, one obtains the action and the free energy. This leads to the thermal energy and entropy of the system, the latter turning out to be given by the Bekenstein-Hawking area law $S = \frac{A_{+}}{4}$, where $A_+$ is the black hole's surface area. The heat capacity is positive when the heat bath is placed at a radius $r$ that is equal or less than the photonic orbit, implying thermodynamic stability. This means that the small black hole is unstable and the large one is stable. A generalized free energy is used to show that it is feasible that classical hot flat space transits through $r_{+1}$ to settle at the stable $r_{+2}$. Remarkably, the free energy of the larger $r_{+2}$ black hole is zero when the cavity radius is equal to the Buchdahl radius. The relation to the instabilities that arise due to perturbations in the path integral in the instanton solution is mentioned. Quantum hot flat space has negative free energy and we find the conditions for which the large black hole, quantum hot flat space, or both are the ground state. The corresponding phase diagram is displayed. Using the density of states $\nu$ at a given energy $E$ we also find that the entropy of the large black hole $r_{+2}$. In addition, we make the connection between the five-dimensional thermodynamics and York's four-dimensional results.