Hamiltonian thermodynamics of two-dimensional vacuum dilatonic black holes

TL;DR

This paper analyzes the Hamiltonian dynamics and thermodynamics of two-dimensional vacuum dilatonic black holes, deriving a quantized reduced theory and examining the behavior of the partition function and entropy across temperature regimes.

Contribution

It introduces a canonical transformation to reduce the classical dynamics to an unconstrained Hamiltonian system and computes the partition function and thermodynamic properties of the model.

Findings

Partition function exists for all dilaton and temperature values.

Heat capacity remains positive, indicating thermodynamic stability.

At high temperatures, the black hole dominates the partition function and entropy aligns with Bekenstein-Hawking entropy.

Abstract

We consider the Hamiltonian dynamics and thermodynamics of the two-dimensional vacuum dilatonic black hole in the presence of a timelike boundary with a fixed value of the dilaton field. A~canonical transformation, previously developed by Varadarajan and Lau, allows a reduction of the classical dynamics into an unconstrained Hamiltonian system with one canonical pair of degrees of freedom. The reduced theory is quantized, and a partition function of a canonical ensemble is obtained as the trace of the analytically continued time evolution operator. The partition function exists for any values of the dilaton field and the temperature at the boundary, and the heat capacity is always positive. For temperatures higher than $\beta_c^{-1} = \hbar\lambda/(2\pi)$, the partition function is dominated by a classical black hole solution, and the dominant contribution to the entropy is the two-dimensional Bekenstein-Hawking entropy. For temperatures lower than~$\beta_c^{-1}$, the partition function remains well-behaved and the heat capacity is positive in the asymptotically flat space limit, in contrast to the corresponding limit in four-dimensional spherically symmetric Einstein gravity; however, in this limit, the partition function is not dominated by a classical black hole solution.