2026/07/27 by Krishna Jalan
#hep-th #gr-qc
We study the island formula and Page curve for the asymptotically flat eternal and quasi-stationary evaporating black hole solutions within the dimensionally reduced Einstein-Hilbert (DREH) model. In this model, the four-dimensional Einstein-Hilbert action reduces to a two-dimensional dilaton gravity, on top of which the quantum corrections can be incorporated via the Polyakov-Liouville (PL) action for a two-dimensional conformal field theory with a large central charge c. This gravity model arises from the s-wave approximation of four-dimensional Einstein-Hilbert gravity and provides a fully gravitational setting in which we study the islands, i.e., we will not require a non-gravitational bath region to collect the radiation, instead it lives in the black hole spacetime itself. The fine-grained entropy of Hawking radiation in the eternal and evaporating black holes within the DREH model was derived using the island rule in \citedjordjevic2022eternal, djordevic2025evaporating. The island rule is based on the replica method using the Euclidean gravitational path integral. In this work, we complement the Euclidean approach by providing a Lorentzian prescription for the generalized entropy Sgen, in the island formula without invoking the replica trick to compute Sgen. The generalized entropy is shown to coincide with Wald-like Noether charge of the combined DREH-PL action. Using this generalized entropy, we determine the quantum extremal surface, the Page time, and the Page curve for the eternal and the evaporating black hole. We conclude with a discussion of the possible extension to the full four-dimensional geometry incorporating vacuum polarization corrections.