Islands and stretched horizon

TL;DR

The work extends the island paradigm to asymptotically flat Schwarzschild black holes, showing that black-hole information can be read off from a stretched horizon at a critical radius $b_c$ where the quantum extremal surface becomes unstable. Through replica-trick calculations of entanglement entropy for the Hawking radiation region $R$ and its complements, the authors demonstrate that the island $I$ and a hidden island $I'$ reorganize the degrees of freedom in a way that preserves unitarity and yields a Page-time transition. The results draw a parallel with AdS setups containing an auxiliary flat space, where information localization occurs at interfaces, and connect the island mechanism to the membrane paradigm via a leading-order semiclassical picture. Limitations are acknowledged, including the dependence on vacua (Unruh vs Hartle–Hawking) and the restriction to leading order, motivating future work on higher-order corrections and dynamical collapse scenarios.

Abstract

Recently it was proposed that the entanglement entropy of the Hawking radiation contains the information of a region including the interior of the event horizon, which is called "island." In studies of the entanglement entropy of the Hawking radiation, the total system in the black hole geometry is separated into the Hawking radiation and black hole. In this paper, we study the entanglement entropy of the black hole in the asymptotically flat Schwarzschild spacetime. Consistency with the island rule for the Hawking radiation implies that the information of the black hole is located in a different region than the island. We found an instability of the island in the calculation of the entanglement entropy of the region outside a surface near the horizon. This implies that the region contains all information of the total system and the information of the black hole is localized on the surface. Thus the surface would be interpreted as the stretched horizon. This structure also resembles black holes in the AdS spacetime with an auxiliary flat spacetime, where the information of the black hole is localized at the interface between the AdS spacetime and the flat spacetime.

Figures (10)

• Figure 1: The entanglement entropy of the Hawking radiation is identified with that in the region $R = R_+\cup R_-$ (left). After the Page time, the island $I$ appears as a consequence of the replica trick in gravitational theories. The region of the black hole $B = B_+\cup B_-$ is naively the complement of $R\cup I$ (right), which includes a region which is added by a similar mechanism to that for the island (see Fig. \ref{['fig:BH-crit']}(right)). The boundaries between $R$ and $B$, $b_\pm$ are introduced by hand, while those of the island, $a_\pm$ are determined by the prescription of the quantum extremal surface.
• Figure 2: In the case of black holes in AdS spacetime, an auxiliary system is introduced outside the boundary of AdS spacetime. The region of the black hole is put in the auxiliary flat spacetime (left). It is extended into the AdS spacetime due to gravitational effects, and ends at the quantum extremal surface after the Page time (right). Before the Page time, it continues to the region in the other side of the event horizon. For the entanglement entropy of a region only in one of the two auxiliary flat spacetime, the region in the AdS spacetime always ends at the extremal surface.
• Figure 3: The island can be maximally extended by putting $b_\pm$ as close to $a_\pm$ as possible (left). The entanglement entropy becomes zero when $b_\pm$ is identical to $a_\pm$, while the quantum extremal surface becomes unstable when the radius at $b_\pm$ becomes $b_c$, implying that the quantum extremal surface rolls down to a global minimum at $b_\pm$. A part of $B$ overlaps with the maximally extended island. This part $I'$ in $B$ is interpreted as a consequence of the replica trick, and referred to as "hidden island" in this paper. Thus, the region $B$ consists of the essential region of the black hole $B'$ and the hidden island $I'$ (right). The information of the black hole is localized at the stretched horizon at $b_c$.
• Figure 4: The statement that the island appears after the Page time naively sounds as if it is absent before the Page time. Then, the region of the black hole subsystem $B$ covers all the other region than $R$, namely, $B = B_0$ (left). However, the black hole subsystem would be able to separated into that in the left wedge $B_-$ and that in the right wedge $B_+$. Information of the island $I$ cannot be reproduced by either of $B_+$ or $B_-$, and the region $B$ contains the island $I$, namely, $B_0 = B \cup I$, where $B = B_+ \cup B_-$ (right).
• Figure 5: The effective region of the Hawking radiation only in the right wedge $R_+$ (left) and that of the black hole (right). The entanglement entropy of $R_+$ contains no contribution from the island even after the Page time. The black hole subsystem $B$ can also be separated into that in the right wedge $B_+$ and that in the left wedge $B_-$. The effective region of $B_+$ does not includes the island, implying that the black hole subsystem in each side has no information of the island.