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

We study the thermodynamics of a five-dimensional Schwarzschild black hole, also known as a five-dimensional Schwarzschild-Tangherlini black hole, in the canonical ensemble using York's formalism. Inside a cavity of fixed size r and fixed temperature T , we find that there is a threshold at π r T =1 above which a black hole can be in thermal equilibrium with the cavity's boundary. Moreover, this thermal equilibrium can only be achieved for two specific types of black holes. One is a small black hole (compared with the cavity size r ) of horizon radius r₊₁, while the other is a large black hole (of the order of the cavity size r ) of horizon radius r₊₂. In five dimensions, the radii r₊₁ and r₊₂ have an exact expression. Through the path integral formalism, which directly yields the partition function of the system, one obtains the action and thus the free energy for the black hole in the canonical ensemble. The procedure leads naturally to the thermal energy and entropy of the canonical system, the latter turning out to be given by the Bekenstein-Hawking area law S =A/₊₄, where the black hole's surface area in five dimensions is A₊=2 π²r₊₃ and r₊ stands for both r₊₁ and r₊₂. We also calculate the heat capacity and find that it is positive when the heat bath is placed at a radius r that is equal or less than the photonic orbit, implying in this case thermodynamic stability, and instability otherwise. This means that the small black hole r₊₁ is unstable and the large one r₊₂ is stable. A generalized free energy is used to compare the possible thermodynamic phase transitions relative to classical hot flat space which has zero free energy, and we show that it is feasible in certain instances that classical hot flat space transits through r₊₁ to settle at the stable r₊₂, with the free energy of the unstable smaller black hole r₊₁ acting as the potential barrier between the two states. It is also shown that, 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. Hot flat space is made of gravitons and it should be treated quantum mechanically rather than classically. Quantum hot flat space has negative free energy and we find the conditions for which the large black hole phase, quantum hot flat space phase, or both are the ground state of the canonical ensemble. The corresponding phase diagram is displayed in a r ×T plot showing clearly the three possible phases. Using the density of states ν at a given energy E we also find that the entropy of the large black hole r₊₂ is S A/₊₂ =4 . In addition, we make the connection between the five-dimensional thermodynamics and York's four-dimensional results.