Observables for Two-Dimensional Black Holes

TL;DR

This paper derives a simple relation between energy and Killing vectors in 2D dilaton gravity, providing a new expression for black hole entropy that aligns with Wald's method and reveals a link to quantum wave functionals.

Contribution

It introduces a straightforward parametrization to relate energy and Killing vectors, leading to a novel, simple formula for black hole entropy in 2D dilaton gravity theories.

Findings

Derived a relation between energy and Killing vector magnitude.

Obtained a new entropy formula matching Wald's result.

Connected black hole entropy to quantum wave functional phases.

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 $S=2\pi \tau_0/G$, where $\tau_0$ 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.