Hamiltonian thermodynamics of d-dimensional (d=4 and d>4) Reissner-Nordstr\"om anti-de Sitter black holes with spherical, planar, and hyperbolic topology

TL;DR

This paper applies Hamiltonian thermodynamics to analyze the properties and phase transitions of d-dimensional Reissner-Nordström-AdS black holes with various topologies, providing a unified framework for different dimensions and geometries.

Contribution

It develops a general Hamiltonian formalism for d-dimensional Reissner-Nordström-AdS black holes, deriving a unified effective action and analyzing phase transitions across topologies.

Findings

Phase transitions occur only for spherical topology.

A general formula for the effective action is established.

Energy, charge, and entropy are computed for black holes.

Abstract

The Hamiltonian thermodynamics formalism is applied to the general $d$-dimensional Reissner-Nordstr\"om-anti-de Sitter black hole with spherical, planar, and hyperbolic horizon topology. After writing its action and performing a Legendre transformation, surface terms are added in order to guarantee a well defined variational principle with which to obtain sensible equations of motion, and also to allow later on the thermodynamical analysis. Then a Kucha\v{r} canonical transformation is done, which changes from the metric canonical coordinates to the physical parameters coordinates. Again a well defined variational principle is guaranteed through boundary terms. These terms influence the fall-off conditions of the variables and at the same time the form of the new Lagrange multipliers. Reduction to the true degrees of freedom is performed, which are the conserved mass and charge of the black hole. Upon quantization a Lorentzian partition function $Z$ is written for the grand canonical ensemble, where the temperature $\bf T$ and the electric potential $\phi$ are fixed at infinity. After imposing Euclidean boundary conditions on the partition function, the respective effective action $I_*$, and thus the thermodynamical partition function, is determined for any dimension $d$ and topology $k$. This is a quite general action. Several previous results can be then condensed in our single general formula for the effective action $I_*$. Phase transitions are studied for the spherical case, and it is shown that all the other topologies have no phase transitions. A parallel with the Bose-Einstein condensation can be established. Finally, the expected values of energy, charge, and entropy are determined for the black hole solution.