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

TL;DR

This paper connects the Euclidean path integral approach to black hole thermodynamics with Davies' theory, analyzing charged black holes in various dimensions and identifying phase transition points.

Contribution

It demonstrates that the path integral approach reproduces Davies' thermodynamic theory for charged black holes, including the identification of the Davies point as a phase transition.

Findings

Heat capacity diverges at the Davies point $T_s$.

Stable black hole solutions transition to hot flat space.

Path integral thermodynamics matches Davies' theory in four dimensions.

Abstract

We establish the connection between the Gibbons-Hawking Euclidean path integral approach applied to the canonical ensemble of a Reissner-Nordstr\"om 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 $d=5$.