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, partial derivative 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 partial derivative I is located on an equal-time Cauchy surface of the observer A when r(2) ((A)) >= 64G(N)kc.

Automated summary

This study applies the quantum extremal surface construction to the step-function Vaidya model of evaporating black holes, and finds that the location of the entanglement island boundary depends on the value of 8M - vA + vI when cutoff surface A is in Minkowski region III. Moreover, it is observed that the entanglement island still exists and partial derivative I is located on an equal-time Cauchy surface of observer A when cutoff surface A is in Minkowski region III after the black hole completely evaporates.