Page curves and entanglement islands for the step-function Vaidya model of evaporating black holes.

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 uA< uH at very late times, we find that the location of the boundary of island ∂I depends on the value of 8M − vA + vI. Specifically, ∂I is inside, at or outside the horizon when 8M − vA + vI is less than, equal to or larger than zero respectively. Moreover, when cutoff surface A is located in Minkowski region III with uA> uH after the black hole evaporates completely, we find that entanglement island still exists and ∂I is located on an equal-time Cauchy surface of the observer A when r A 2 ≥ 64GNκc. [ABSTRACT FROM AUTHOR]