Islands and Page Curves for Evaporating Black Holes in JT Gravity

TL;DR

The paper analyzes how a CFT-induced shockwave interacting with a JT gravity eternal black hole in a thermal bath reshapes the entanglement structure. By solving the backreaction exactly in the semiclassical high-temperature regime and applying the generalized entropy (including quantum extremal surfaces) framework, it maps out the evolution of Page curves and the formation/location of islands under non-equilibrium conditions. The results show that shock energy can delay island formation and Page transitions, while shock entropy can hasten them, and that the QES can migrate from inside to outside the horizon as the system relaxes back to equilibrium. The extremal-black-hole case is treated analogously, illustrating robust island dynamics even at zero bath temperature and highlighting the broad relevance of replica-wormhole-inspired entropy prescriptions for dynamical black-hole information questions.

Abstract

The effect of a CFT shockwave on the entanglement structure of an eternal black hole in Jackiw-Teitelboim gravity, that is in thermal equilibrium with a thermal bath, is considered. The shockwave carries energy and entropy into the black hole and heats the black hole up leading to evaporation and the eventual recovery of equilibrium. We find an analytical description of the entire relaxational process within the semiclassical high temperature regime. If the shockwave is inserted around the Page time then several scenarios are possible depending on the parameters. The Page time can be delayed or hastened and there can be more than one transition. The final entropy saddle has a quantum extremal surface that generically starts inside the horizon but at some later time moves outside. In general, increased shockwave energy and slow evaporation rate favour the extremal surface to be inside the horizon. The shockwave also disrupts the scrambling properties of the black hole. The same analysis is then applied to a shockwave inserted into the extremal black hole with similar conclusions.

Figures (11)

• Figure 1: The coordinates of the eternal black hole pair along with their half-Minkowski space bath regions. The pink region is part of the AdS geometry outside the right black hole. The yellow region is the right bath region. The right Schwarzschild coordinates $y^\pm$ cover the pink and yellow regions. The global coordinates $w^\pm$ cover all regions on both the left and the right.
• Figure 2: The calculation of the entropy of the baths region to the AdS region involves calculating a CFT entanglement entropy for an interval $D$ across the AdS region between boundary points at a given time $t$.
• Figure 3: The generalized entropy for points $p_2$ and $p_4$ at the boundaries with QES's at points $p_1$ and $p_3$ in the bulk involves calculating the entropy of disjoint intervals as shown.
• Figure 4: The time evolution of the entanglement structure. For early boundary times times (left), the entanglement wedge of the green boundary points consists of the entire AdS region. For late times, past the Page time, (right) the minimal entropy is captured by a configuration with 2 QES's outside the horizons. The entanglement wedges of the red boundary points are now much smaller and an island forms between the QES's (the pink region). Also shown is a Hawking mode and its partner behind the horizon. Before the Page time, the both modes are in the entanglement wedge of the boundary point whereas after the Page time the partner mode is now in the island.
• Figure 5: The set-up consists of the two-sided black hole in thermal equilibrium with the left and right flat space with the CFT acting as a thermal bath. Operator insertions in the baths create shockwaves that enter the AdS region and the black hole. The symmetry between the left and right is chosen to simplify the analysis.