Page Curve for an Evaporating Schwarzschild Black Hole in Dimensionally-Reduced Model of Dilaton Gravity

We study entanglement entropy of quantum fields in a (1+1)-dimensional model of dilaton gravity derived from the four-dimensional Einstein-Hilbert action by dimensional reduction. Scalar matter is coupled to gravity, while the back-reaction is included via the Polyakov-Liouville action. This theory exhibits both eternal and evaporating Schwarzschild black hole solutions. The fine-grained entropy in a collapse scenario is investigated by applying the "island rule" via the quantum extremal surface approach. We demonstrate that the fine-grained entropy of the Hawking radiation follows the Page curve, and therefore the evolution of the Hawking radiation is unitary.