Abstract
Identifying an entanglement island requires exquisite control over the entropy of quantum fields, which is available only in toy models. Here we present a set of sufficient conditions that guarantee the existence of an island and place an upper bound on the entropy computed by the island rule. This is enough to derive the main features of the Page curve for an evaporating black hole in any spacetime dimension. Our argument makes use of Wall's maximin formulation and the quantum focusing conjecture. As a corollary, we derive a novel entropy bound.
Highlights
An ad hoc proposal, it follows under certain assumptions from a Euclidean gravitational path integral [4,5,6,7,8]
This derivation implies that the RyuTakayanagi [3] (RT) prescription is not tied to the anti–de Sitter (AdS)=conformal field theory (CFT) correspondence but can be evaluated in any spacetime M
RT yields the Page curve [9] for the entropy of the bulk radiation emitted by a black hole [10,11]
Summary
The quantum-corrected [1], covariant [2] RyuTakayanagi [3] (RT) prescription computes the conformal field theory (CFT) entropy of a boundary region in terms of a dual asymptotically anti–de Sitter (AdS) bulk spacetime. An ad hoc proposal, it follows under certain assumptions from a Euclidean gravitational path integral [4,5,6,7,8] This derivation implies that the RT prescription is not tied to the AdS=CFT correspondence but can be evaluated in any spacetime M. RT yields the Page curve [9] for the entropy of the bulk radiation emitted by a black hole [10,11]. Before the Page time, this is greater than SðRÞ by definition, so Iis not a viable island candidate; one finds that IðtÞ 1⁄4 ∅, EðRÞ 1⁄4 R, and SðRÞ 1⁄4 SðRÞ. Entanglement islands can appear in cosmology, where their significance is less obvious [23,24]
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