Microcanonical action of causal diamonds and the entropy of Hawking radiation

The island formula, an extremization prescription for generalized entropy, is known to result in a Page curve for the entropy of Hawking radiation, arguably resolving the information paradox. The formula has been derived for Jackiw-Teitelboim gravity coupled to conformal matter in two-dimensional anti-de Sitter space using the replica trick. Thus far a derivation of the island formula for other gravitational theories remains elusive. In this talk, motivated by the thermodynamics of quantum extremal surfaces, I offer an alternative first principles explanation. Specifically, I show how the extremization of generalized entropy directly follows from minimizing the gravitational microcanonical action of an entanglement wedge, demonstrating consistency with the island formula. Crucially, this argument holds for any two-dimensional theory of semi-classical gravity.