Island and Page curve for one-sided asymptotically flat black hole

TL;DR

This paper extends the island rule analysis to one-sided asymptotically flat black holes formed by gravitational collapse, revealing how the island boundary depends on the cutoff surface and affects entropy bounds and Page time.

Contribution

It investigates the island rule for a collapsing black hole in asymptotically flat spacetime, showing novel boundary behaviors and state-dependent entropy effects.

Findings

Island emerges at late times, preserving entropy bounds.

Boundary of island depends on cutoff surface position.

Different states influence entanglement entropy and Page time.

Abstract

Great breakthrough in solving black hole information paradox took place when semiclassical island rule for entanglement entropy of Hawking radiation was proposed in recent years. Up to now, most papers which discussed island rule of asymptotic flat black hole with $D \ge 4$ focus on eternal black hole. In this paper, we take one more step further by discussing island of "in" vacuum state which describes one-sided asymptotically flat black hole formed by gravitational collapse in $D \ge 4$. We find that island $I$ emerges at late time and saves entropy bound. And boundary of island $\partial I$ depends on the position of cutoff surface. When cutoff surface is far from horizon, $\partial I$ is inside and near horizon. When cutoff surface is set to be near horizon, $\partial I$ is outside and near horizon. This is different from the case of eternal black hole in which $\partial I$ is always outside horizon no matter cutoff surface is far from or near horizon. We will see that different states will manifestly affect $S_{\text{ent}}$ in island formula when cutoff surface is far from horizon and thus have different result for Page time.