Gibbons–Hawking action for electrically charged black holes in the canonical ensemble and Davies’ thermodynamic theory of black holes

We establish the connection between the Gibbons–Hawking Euclidean path integral approach applied to the canonical ensemble of a Reissner–Nordström black hole and the thermodynamic theory of black holes of Davies. We build the ensemble, characterized by a reservoir at infinity at temperature T and electric charge Q , in d dimensions. The Euclidean path integral yields the action and partition function. In zero loop, we uncover two solutions, one with horizon radius r + 1 the least massive, the other with r + 2 , both meeting at a saddle point with radius r + s at temperature T s . We derive the thermodynamics, finding that the heat capacity diverges at the turning point T s for each solution. The free energy of the stable solution is positive, so if the system is a black hole, it makes a transition to hot flat space with charge at infinity. For a given Q and T > T s , there is only hot space. An interpretation of the results as energy wavelengths is attempted. For d = 4 , the thermodynamics from the path integral applied to the canonical ensemble is precisely the Davies thermodynamics theory of black holes, with T s being the Davies point. We sketch the case d = 5 .