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

摘要

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 andperforming a Legendre transformation, surface terms are added in order toguarantee a well defined variational principle with which to obtain sensibleequations of motion, and also to allow later on the thermodynamical analysis.Then a Kucha\v{r} canonical transformation is done, which changes from themetric canonical coordinates to the physical parameters coordinates. Again awell defined variational principle is guaranteed through boundary terms. Theseterms influence the fall-off conditions of the variables and at the same timethe form of the new Lagrange multipliers. Reduction to the true degrees offreedom is performed, which are the conserved mass and charge of the blackhole. Upon quantization a Lorentzian partition function $Z$ is written for thegrand canonical ensemble, where the temperature $\bf T$ and the electricpotential $\phi$ are fixed at infinity. After imposing Euclidean boundaryconditions on the partition function, the respective effective action $I_*$,and thus the thermodynamical partition function, is determined for anydimension $d$ and topology $k$. This is a quite general action. Severalprevious results can be then condensed in our single general formula for theeffective action $I_*$. Phase transitions are studied for the spherical case,and it is shown that all the other topologies have no phase transitions. Aparallel with the Bose-Einstein condensation can be established. Finally, theexpected values of energy, charge, and entropy are determined for the blackhole solution.