Thermodynamics of Dilatonic Black Holes in $n$ Dimensions

TL;DR

This paper develops a general formalism to analyze the thermodynamics of dilatonic black holes in various dimensions, defining variables on a quasilocal surface and deriving the first law for diverse theories.

Contribution

It introduces a unified approach for studying black hole thermodynamics in dilaton gravity across different theories and dimensions, including entropy calculation via path integrals.

Findings

Thermodynamic variables are defined on a quasilocal surface.

The first law of thermodynamics is derived for dilatonic black holes.

Explicit calculations are performed for 1+1 dimensional black hole solutions.

Abstract

We present a formalism for studying the thermodynamics of black holes in dilaton gravity. The thermodynamic variables are defined on a quasilocal surface surrounding the black hole system and are obtained from a general class of Lagrangians involving a dilaton. The formalism thus accommodates a large number of possible theories and black hole spacetimes. Many of the thermodynamic quantities are identified from the contribution of the action on the quasilocal boundary. The entropy is found using path integral techniques, and a first law of thermodynamics is obtained. As an illustration, we calculate the thermodynamic quantities for two black hole solutions in $(1+1)$ dimensions: one obtained from a string inspired theory and the other being a Liouville black hole in the ``$R=\kappa T$'' theory with a Liouville field.