One-loop quantum corrections to the thermodynamics of charged black holes

Quantum corrections are studied for a char ed black hole in a two-dimensional (2D) model obtained by spherisymmetric reduction of the 4D Einstein-Maxwell theory. The classical (tree-level) thermodynamics is reformulated in the framework of the off-shell approach, considering systems at arbitrary temperature. This implies a conical singularity at the horizon and modifies the gravitational action by terms defined on the horizon. A consistent variational procedure for the action functional is formulated. It is shown that the free energy reaches an extremum on the regular manifold with T = Tx. The one-loop contribution to the action in the Liouville-Polyakov form is reexamined. All the boundary terms are taken into account and the dependence on the state of the quantum field is established. The modification of the Liouville-Polyakov term for a 2D space with a conical defect is derived. The back reaction of the Hawking radiation on the geometry is studied and the quantum-corrected black hole metric is calculated perturbatively. Within the off-shell approach the one-loop thermodynamical quantities, energy, and entropy, are found. They are shown to contain a par? due to hot gas surrounding the black hole and a part due to the hole itself. It is noted that the contribution of the hot gas can be eliminated by appropriate choice of the (generally, nonflat) reference geometry. The deviation of the ''entropy - horizon area'' relation for the quantum-corrected black hole from the classical law is discovered and possible physical consequences are discussed.