Hamiltonian Thermodynamics of Black Holes in Generic 2-D Dilaton Gravity

TL;DR

This paper develops a Hamiltonian framework for analyzing the thermodynamics of eternal black holes in 2-D dilaton gravity, deriving a general form of the Hamiltonian and exploring quantum partition functions.

Contribution

It introduces a generic Hamiltonian form for black holes in 2-D dilaton gravity and analyzes their thermodynamics and quantum partition functions.

Findings

Hamiltonian takes a form involving quasilocal energy and entropy.

Partition function constructed by tracing over mass eigenstates.

Applicable to theories from dimensional reduction of higher-dimensional Einstein gravity.

Abstract

We consider the Hamiltonian mechanics and thermodynamics of an eternal black hole in a box of fixed radius and temperature in generic 2-D dilaton gravity. Imposing boundary conditions analoguous to those used by Louko and Whiting for spherically symmetric gravity, we find that the reduced Hamiltonian generically takes the form: $$ H(M,\phi_+) = \sigma_0 E(M,\phi_+) -{N_0\over 2\pi} S(M) $$ where $E(M,\phi_+)$ is the quasilocal energy of a black hole of mass $M$ inside a static box (surface of fixed dilaton field $\phi_+$) and $S(M)$ is the associated classical thermodynamical entropy. $\sigma_0$ and $N_0$ determine time evolution along the world line of the box and boosts at the bifurcation point, respectively. An ansatz for the quantum partition function is obtained by fixing $\sigma_0$ and $N_0$ and then tracing the operator $e^{-\beta H}$ over mass eigenstates. We analyze this partition function in some detail both generically and for the class of dilaton gravity theories that is obtained by dimensional reduction of Einstein gravity in n+2 dimensions with $S^n$ spherical symmetry.