Grand canonical ensemble of a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:math>-dimensional Reissner-Nordström black hole in a cavity

Abstract

The grand canonical ensemble of a $d$-dimensional Reissner-Nordstr\"om black hole space in a cavity is analyzed in every possible aspect. The analysis starts with the realization of the grand canonical ensemble through the Euclidean path integral approach by giving the Euclidean action for the $d$-dimensional Reissner-Nordstr\"om black hole with the correct topology and boundary conditions corresponding to a cavity. More precisely, the fixed quantities of the ensemble are the temperature and the electric potential at the cavity boundary. One then performs a zero loop approximation to find and analyze the stationary points of the reduced action. This yields two solutions for the electrically charged black hole, the smaller, ${r}_{+1}$, and the larger, ${r}_{+2}$. Through perturbations of the reduced action around the stationary points, one finds stability criteria for the solutions of the black hole that show that ${r}_{+1}$ is unstable and ${r}_{+2}$ is stable. Moreover, one analyzes the most probable configurations for each value of the fixed quantities at the boundary, with the configurations being either a stable charged black hole or hot flat space. One also compares the stable black hole with a nongravitating charged shell, which serves as a model for an electrically charged hot flat space. By making the correspondence between the action already evaluated and the grand canonical ensemble potential of thermodynamics one can get the entropy, the mean charge, the mean energy, and the thermodynamic pressure, as well as the Smarr formula, here shown to be valid only for the unstable black hole ${r}_{+1}$. We make a stability analysis in terms of thermodynamic variables, which yields that thermodynamic stability is related to the positivity of the heat capacity at constant electric potential and constant area of the cavity. We also comment on the most favorable thermodynamic phases and deduce the possible phase transitions. We then pick up a specific dimension, $d=5$, which is singled out naturally from the other higher dimensions as it provides an exact solution for the problem, and apply all the results previously found. The case $d=4$ is concisely put in an appendix where the results are directly equated with previous works. We also compare thermodynamic radii with the photonic orbit radius and the Buchdahl-Andr\'easson-Wright bound radius in $d$-dimensional Reissner-Nordstr\"om spacetimes and find they are unconnected, showing that the connections displayed in the Schwarzschild case are not generic, rather they are very restricted equalities holding only in the pure gravitational situation.

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