Observables for Two-Dimensional Black Holes

TL;DR

This work analyzes the most general 1+1 dimensional dilaton gravity with black hole solutions, using a convenient field redefinition to obtain a simple Killing-vector form and to identify the energy as a conserved quantity associated with translations along the Killing direction. A Hamiltonian (ADM-like) analysis reveals the canonical pair of observables: the energy $E$ (via $E = q/G$) and the Killing-time separation at infinity, with the latter depending on global slicing. The authors derive a universal entropy formula $S = (2π/G) τ0$ that matches Wald's geometric entropy, and show how horizon structure leads to an analytic continuation of a related quantum phase, linking entropy to the imaginary part of the wave functional. The results provide a unified classical-quantum-thermodynamic picture for generic 2D black holes and illuminate the role of the Killing vector in their observables.

Abstract

We consider the most general dilaton gravity theory in 1+1 dimensions. By suitably parametrizing the metric and scalar field we find a simple expression that relates the energy of a generic solution to the magnitude of the corresponding Killing vector. In theories that admit black hole solutions, this relationship leads directly to an expression for the entropy $S=2πτ_0/G$, where $τ_0$ is the value of the scalar field (in this parametrization) at the event horizon. This result agrees with the one obtained using the more general method of Wald. Finally, we point out an intriguing connection between the black hole entropy and the imaginary part of the ``phase" of the exact Dirac quantum wave functionals for the theory.