Page curves and Entanglement Islands for the Step-Function Vaidya Model of Evaporating Black Holes

TL;DR

This paper investigates the Page curve and entanglement islands in a four-dimensional step-function Vaidya black hole model, revealing how the island boundary depends on the cutoff surface location and persists after evaporation.

Contribution

It applies the quantum extremal surface method to a step-function Vaidya model, analyzing late-time entanglement islands and their relation to the black hole horizon.

Findings

The island boundary location depends on the cutoff surface parameters.

Entanglement islands exist even after the black hole has fully evaporated.

The position of the island boundary varies with the cutoff surface in Minkowski regions.

Abstract

It was proposed recently that the fine-grained entropy of the Hawking radiation can be expressed by the semiclassical island formula, which reproduces the unitary Page curve. In this paper, we choose the ``in'' vacuum state and apply the quantum extremal surface construction to study the Page curve for the step-function Vaidya model of evaporating black holes in four dimensions, which is produced by the spherical null shells. Metrics of the three regions of this spacetimes are obtained. In addition, the entanglement islands for the step-function Vaidya model of evaporating black holes at very late times are studied. When cutoff surface $A$ is located in Minkowski region III with $u_A < u_H$ at very late times, we find that the location of the boundary of island $\partial I$ depends on the value of $8M-v_A+v_I$. Specifically, $\partial I$ is inside, at or outside the horizon when $8M-v_A+v_I$ is less than, equal to or larger than zero respectively. Moreover, when cutoff surface $A$ is located in Minkowski region III with $u_A > u_H$ after the black hole evaporates completely, we find that entanglement island still exists and $\partial I$ is located on an equal-time Cauchy surface of the observer $A$ when $r_{(A)}^2\geq64G_N\kappa c $.