The Volume of 2D Black Holes

TL;DR

This paper extends a black hole volume definition to two-dimensional dilaton gravity, showing that infinite volume cannot be achieved without pathologies, and finds a proportionality between area and volume for the Witten black hole.

Contribution

It generalizes the black hole volume concept to D=2 dilaton gravity, including the dilaton in the measure, and analyzes volume-area relations in this context.

Findings

Infinite volume cannot be achieved without pathologies in D=2, similar to higher dimensions.

The area is proportional to the volume for the Witten black hole.

The volume definition is consistent with spherically reduced gravity.

Abstract

It is shown that the definition for the volume of stationary black holes advocated in hep-th/0508108 readily generalizes to the case of dilaton gravity in D=2. The dilaton field is included as part of the measure. A feature observed in D=3 and 4 has been the impossibility to obtain infinite volume while retaining finite area without encountering some kind of pathology. It is demonstrated that this also holds in D=2. Consistency with spherically reduced gravity is shown. For the Witten black hole it is found that the area is proportional to the volume.