Thermodynamics of charged black holes in two-dimensional dilaton gravity
Medved, Allan Joseph Michael
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We consider the most general diffeomorphism invariant action in 1+1 space-time dimensions that is described in terms of the following fields: a metric tensor, dilaton scalar and Abelian gauge vector. Included in this action is a topological term that depends linearly on the Abelian field strength and has been previously unaccounted for in the related literature. The focus of this thesis is on studying the thermodynamics of the black hole solutions of the field equations. The problem is first considered on the classical level, which is followed by the calculation of first-order corrections that arise due to the inclusion of matter fields. This matter is assumed to be quantized scalar fields that are conformally coupled to the metric, non-minimally coupled to the dilaton and not coupled with the gauge vector. We begin the classical study by showing that (up to spacetime diffeomorphisms) every solution of the field equations is static and depends on only two physical parameters: the mass and Abelian charge of the associated black hole. We next extract the thermodynamics by imposing boundary conditions that describe a charged black hole in a "box" (with the box being one dimensional in dilaton space). Both a Hamiltonian and Euclidean action (i.e., path integral) approach are used. The two approaches are formally different, but they are shown to generate equivalent results. The former is aesthetically pleasing, as it leads to a canonical Hamiltonian operator in an elegant free energy form. The latter allows for the incorporation of the quantum matter fields in a straightforward manner. We begin the quantum-corrected analysis by constructing a suitable local form of the quantum effective action. The revised field equations are then solved with an appropriate ansatz for the quantum-corrected metric. These solutions describe the back reaction of the matter fields on the classical geometry. Quantum corrections to the geometry and action lead to modifications in the thermodynamics. The geometrical and thermodynamic corrections are both evaluated to first order in ħ. Finally, we illustrate this general formalism with two specific models. These are the dimensionally reduced forms of the Reissner-Nordstrbm black hole and the rotating BTZ black hole.