Semi-classical thermodynamics of quantum extremal surfaces in Jackiw-Teitelboim gravity

TL;DR

This paper explores the semi-classical thermodynamics of quantum extremal surfaces in Jackiw-Teitelboim gravity, revealing their entropy, extremization conditions, and thermodynamic laws in a two-dimensional AdS black hole context.

Contribution

It provides a detailed analysis of QES thermodynamics in semi-classical JT gravity, including derivation of a Smarr relation and first law incorporating quantum effects.

Findings

QES extremize generalized entropy outside the black hole horizon

The Wald entropy fully captures the generalized entropy including quantum contributions

The thermodynamics of the nested Rindler wedge matches that of the QES in the microcanonical ensemble

Abstract

Quantum extremal surfaces (QES), codimension-2 spacelike regions which extremize the generalized entropy of a gravity-matter system, play a key role in the study of the black hole information problem. The thermodynamics of QESs, however, has been largely unexplored, as a proper interpretation requires a detailed understanding of backreaction due to quantum fields. We investigate this problem in semi-classical Jackiw-Teitelboim (JT) gravity, where the spacetime is the eternal two-dimensional Anti-de Sitter ($\text{AdS}_{2}$) black hole, Hawking radiation is described by a conformal field theory with central charge $c$, and backreaction effects may be analyzed exactly. We show the Wald entropy of the semi-classical JT theory entirely encapsulates the generalized entropy - including time-dependent von Neumann entropy contributions - whose extremization leads to a QES lying just outside of the black hole horizon. Consequently, the QES defines a Rindler wedge nested inside the enveloping black hole. We use covariant phase space techniques on a time-reflection symmetric slice to derive a Smarr relation and first law of nested Rindler wedge thermodynamics, regularized using local counterterms, and including semi-classical effects. Moreover, in the microcanonical ensemble the semi-classical first law implies the generalized entropy of the QES is stationary at fixed energy. Thus, the thermodynamics of the nested Rindler wedge is equivalent to thermodynamics of the QES in the microcanonical ensemble.