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

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 (AdS2) 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 intrinsically 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 the thermodynamics of the QES in the microcanonical ensemble.

The universality of islands outside the horizon

“…Meanwhile, higher dimensional black hole cases are considered in [41][42][43][44][45][46][47][48] as well as higher derivative gravity [49,50]. Interestingly, the page curve can be realized in the moving mirror scenario [51][52][53], and other quantum information or thermodynamic quantities except entanglement entropy are investigated within island formula [54][55][56][57][58][59][60][61][62].…”

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

“…Moreover, at least in eternal backgrounds, we would then be able to attribute the thermodynamics of quantum extremal surfaces to the associated islands. Studying this exciting avenue is currently underway [141]. For dynamical settings, such as an evaporating black hole, perhaps the Iyer-Wald definition of dynamical entropy [84] can be used.…”

Quasi-local energy and microcanonical entropy in two-dimensional nearly de Sitter gravity

“…While we explicitly computed (3.71), the result S vN = − c 6 χ holds for any 2D gravity theory coupled to a large c CFT and with respect to any vacuum state [30]. This primarily follows from the fact that generically χ = −ρ + ξ, as in (3.54), and imposing χ obeys Dirichlet boundary conditions.…”

“…In such toy models many conceptual issues of horizon thermodynamics can be resolved. Recently, for example, the authors in [30] showed that for conformal matter in an eternal AdS 2 black hole, the Wald entropy is equal to the generalized entropy S gen [31], the sum of the classical gravitational entropy S BH and von Neumann entropy S vN of quantum matter, S gen = S BH + S vN .…”

“…In this section we derive both of these relations for de Sitter JT gravity using the Noether charge formalism [63,77] (see [30] or Appendix C for a summary).…”