Island Formula from Wald-like Entropy with Backreaction

We propose a Lorentzian derivation of the generalized entropy associated with the island formula for black holes as a Wald-like entropy without reference to the exterior non-gravitating region or field-theoretic von Neumann entropy of Hawking radiation in a fixed curved spacetime background. We illustrate this idea by studying two-dimensional black holes in the Jackiw-Teitelboim gravity and the Russo-Susskind-Thorlacius model in which Hawking radiation is represented by conformal scalars. With some prescriptions assumed, we show that the generalized entropy for the island formula can be reproduced as the Wald-like entropy of the two-dimensional dilaton-gravity theories upon the inclusion of the backreaction from Hawking radiation described by conformal anomaly. We give a discussion on how a similar idea can be applied to higher-dimensional black holes. It is emphasized that the generalized entropy is obtained in a fully gravitational fashion, yet it yields the same Page curve as that of the half-gravitational set-up. We argue that the results in this paper exacerbate the issues raised in the work of massive islands and inconsistency of islands in theories of long-range gravity.