Hamiltonian thermodynamics of three-dimensional dilatonic black holes

TL;DR

This paper develops a Hamiltonian formalism for three-dimensional dilatonic black holes with various  parameters, deriving their thermodynamics and quantizing the theory to compute black hole entropies influenced by the dilaton.

Contribution

It introduces a new Hamiltonian formulation for these black holes, simplifying the constraints to boundary terms and enabling quantization and entropy calculation.

Findings

Hamiltonian formalism yields boundary-only Hamiltonian.

Quantization produces a partition function for black holes.

Black hole entropy deviates from quarter-area law due to dilaton effects.

Abstract

The action for a class of three-dimensional dilaton-gravity theories with a cosmological constant can be recast in a Brans-Dicke type action, with its free $\omega$ parameter. These theories have static spherically symmetric black holes. Those with well formulated asymptotics are studied through a Hamiltonian formalism, and their thermodynamical properties are found out. The theories studied are general relativity ($\omega\to\infty$), a dimensionally reduced cylindrical four-dimensional general relativity theory ($\omega=0$), and a theory representing a class of theories ($\omega=-3$). The Hamiltonian formalism is setup in three dimensions through foliations on the right region of the Carter-Penrose diagram, with the bifurcation 1-sphere as the left boundary, and anti-de Sitter infinity as the right boundary. The metric functions on the foliated hypersurfaces are the canonical coordinates. The Hamiltonian action is written, the Hamiltonian being a sum of constraints. One finds a new action which yields an unconstrained theory with one pair of canonical coordinates $\{M,P_M\}$, $M$ being the mass parameter and $P_M$ its conjugate momenta The resulting Hamiltonian is a sum of boundary terms only. A quantization of the theory is performed. The Schr\"odinger evolution operator is constructed, the trace is taken, and the partition function of the canonical ensemble is obtained. The black hole entropies differ, in general, from the usual quarter of the horizon area due to the dilaton.