Islands outside the horizon

TL;DR

The paper shows that for nearly AdS$_2$ black holes in equilibrium with a bath, quantum extremal surfaces can lie outside the horizon, creating islands whose entanglement wedges include far-distant bath regions. Using JT gravity coupled to a CFT bath and the quantum focusing conjecture, the authors demonstrate causality-preserving scenarios where information localized near the horizon can be encoded in the bath, and they formulate an eternal-black-hole version of the information paradox in the Hartle–Hawking state. They provide a teleportation-based mechanism to recover island information, analyze a two-sided (thermofield-double) setup to obtain a Page-like entropy saturation with islands, and discuss phenomenological extensions to four-dimensional near-extremal black holes and transplanckian/cosmological contexts. Overall, islands outside horizons emerge as a consistent feature, offering a unitarity-respecting resolution to information flow in black holes coupled to baths.

Abstract

We consider an AdS$_2$ black hole in equilibrium with a bath, which we take to have a dual description as (0+1)-dimensional quantum mechanical system coupled to a (1+1)-dimensional field theory serving as the bath. We compute the entropies of both the quantum mechanical degrees of freedom and of the bath separately, while allowing contributions from entanglement wedge "islands". We find situations where the island extends {\it outside} the black hole horizon. This suggests possible causality paradoxes which we show are avoided because of the quantum focusing conjecture. Finally, we formulate a version of the information paradox for a black hole in contact with a bath in the Hartle-Hawking state, and demonstrate the role of islands in resolving this paradox.

Figures (9)

• Figure 1: On the left, we depict the Penrose diagram for a two-dimensional extremal black hole, coupled to a bath at zero temperature. The red part covers a region of AdS$_2$. The blue triangle is the Penrose diagram of half of Minkowski space. On the right, we have the dual quantum mechanical setup, with the black hole replaced by a quantum system that is coupled to a CFT$_2$ on a half-line (only the spatial dimension is represented).
• Figure 2: On the left, in green, we see the entanglement wedge of the region corresponding to the interval ${\pdfliteral direct { 2 Tr 0.6\space w } [0,b]\pdfliteral direct { 0 Tr 0 w }}$. It is also shaded in green on the right. Shaded in blue we see the interval ${\pdfliteral direct { 2 Tr 0.6\space w } [b,+\infty]\pdfliteral direct { 0 Tr 0 w }}$ and its entanglement wedge which also includes the island $[-\infty,-a]$, which is outside the horizon.
• Figure 3: Decoupling the black hole from the bath. (a) Naively, a signal from the island can reach the AdS boundary. This is wrong. (b) The decoupling process produces some energy that moves the horizon outwards. This means that the trajectory of the boundary particle (shown in solid black) reaches a maximum Poincaré time $x_m$. As long as $x_m < a$, we will have no contradiction since the signal will not be able to reach the physical boundary. This is guaranteed by the quantum focusing conjecture. (c) After we decouple, we can evolve the system backwards in time. Then we should also find that $x'_m >-a$ to avoid seeing the signal $s_2$.
• Figure 4: A black hole in thermal equilibrium with a bath. The full Hartle-Hawking state is dual to the thermofield double of the two quantum mechanical plus bath systems. We have two $y$-planes, one on the right and one on the left. Half of each $y$-plane belongs to the bath and the other half to the black hole exterior. The two $y$-planes fit into a $w$-plane where the state is the Minkowski vacuum for the quantum fields of the CFT. The $y$-planes are like Rindler wedges and the state looks thermal in the $y$-coordinates.
• Figure 5: A message inserted into the right boundary quantum mechanical system will exit its entanglement wedge after a scrambling time.