Page curves for a family of exactly solvable evaporating black holes

We study the entanglement entropy of a one-parameter family of exactly solvable gravities in the two-dimensional asymptotically flat space. The islands and Page curves of eternal, evaporating, and bath-removed black holes are investigated. The different theories in this parameter class are identified through field redefinitions, which leave the island invariant. The Page transition is found to occur at the first and third of the black hole lifetimes in the evaporating case for this family of solutions. In addition, we consider gluing the equilibrium black hole and the evaporating one along a null trajectory and study the effect of gluing on the islands and Page curves. In the glued space, the island jumps across two different geometries at a certain retarded time. As a result, the Page transition is stretched and split into two separate ones---the first transition happens when the net entropy generation stops, and the second one occurs as the early radiation effectively starts to become purified. Finally, we discuss the issues concerning the inconsistent rates of purification and the paradox related to the state of the radiation.