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

TL;DR

This work analyzes the semiclassical thermodynamics of quantum extremal surfaces within Jackiw–Teitelboim gravity by solving backreaction exactly via the Polyakov action. It shows that the semiclassical Wald entropy reproduces the generalized entropy, including the time-dependent von Neumann contribution, and that extremizing this entropy yields a QES just outside the black hole horizon, defining a nested AdS2–Rindler wedge. Using covariant phase space methods, the authors derive Smarr relations and first laws for the nested wedge, including semiclassical corrections and coupling-variation extensions, and show that in the microcanonical ensemble the generalized entropy is stationary at fixed energy, making the thermodynamics of the nested wedge equivalent to QES thermodynamics. The analysis supports interpreting the QES entanglement wedge as a thermodynamic system and suggests that, when semiclassical effects are present, the microcanonical ensemble provides the natural framework for QES thermodynamics and islands in the information problem. The results offer a concrete, solvable model for entanglement island physics and lay groundwork for extensions to higher dimensions and dynamical settings.

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.

Figures (3)

• Figure 1: The relevant portion of a Penrose diagram of an eternal non-extremal AdS$_2$ black hole is presented. The diagonal lines denote the bifurcate Killing horizons and the vertical lines denote the conformal boundary. The dashed edges indicate that this is only a portion of the full Penrose diagram. The curved arrows indicate the directions of the Killing flow.
• Figure 2: The black circles denote a black hole and the wavy arrows represent Hawking radiation. In the Hartle-Hawking state (left Figure) a stationary observer at the AdS boundary measures $T=T_{H}$, whereas a stationary observer at the boundary measures $T=0$ in the Boulware state (right Figure). In the Hartle-Hawking state the black hole is in thermal equilibrium with all of its surroundings whereas the Boulware state can be interpreted as the black hole being in thermal equilibrium only with a membrane wrapped tight around the event horizon.
• Figure 3: Nested Rindler wedges. The larger AdS-Rindler wedge (shaded in blue) envelopes a smaller Rindler wedge (shaded in green). The bifurcation points of the larger and smaller (right) Rindler patches generally do not coincide, but they do in the limit when the boundary time interval goes to infinity, $\alpha\to\infty$. In this limit the boost Killing vector $\zeta$ of the nested Rindler wedge becomes proportional to the time-translation Killing vector $\partial_{t}$ of the black hole metric. We have illustrated a nested Rindler wedge whose extremal slice $\Sigma$ is located at $t=0$, but we also consider nested wedges centered at $t=t_0 \neq0.$