A note on the action with the Schwarzian at the stretched horizon

TL;DR

This paper explores the quantization of a model associating boundary degrees of freedom to a black hole's stretched horizon, revealing issues with finiteness and thermodynamic contributions at one-loop level.

Contribution

It introduces a boundary path integral framework for non-extremal black holes and analyzes its quantum properties, highlighting differences from JT gravity.

Findings

One-loop boundary partition function is divergent.

The boundary degrees of freedom do not contribute to thermodynamics at one-loop.

The model exhibits potential instability or indefiniteness issues.

Abstract

In this paper, we discuss the quantization of an interesting model of Carlip which appeared recently. It shows a way to associate boundary degrees of freedom to the stretched horizon of a stationary non-extremal black hole, as has been done in JT gravity for near-extremal black holes. The path integral now contains an integral over the boundary degrees of freedom, which are time reparametrizations of the stretched horizon keeping its length fixed. These boundary degrees of freedom can be viewed as elements of $Diff(S^1)/S^1$, which is the coadjoint orbit of an ordinary coadjoint vector under the action of the Virasoro group. From the symplectic form on this manifold, we obtain the measure in the boundary path integral. Doing a one-loop computation about the classical solution, we find that the one-loop answer is not finite, signalling that either the classical solution is unstable or there is an indefiniteness problem with this action, similar to the conformal mode problem in quantum gravity. Upon analytically continuing the field, the boundary partition function we get is independent of the inverse temperature and does not contribute to the thermodynamics at least at one-loop. This is in contrast to the study of near-extremal black holes in JT gravity, where the entire contribution to thermodynamics is from boundary degrees of freedom.