Abstract
Behind certain marginally trapped surfaces one can construct a geometry containing an extremal surface of equal, but not larger area. This construction underlies the Engelhardt-Wall proposal for explaining the Bekenstein-Hawking entropy as a coarse-grained entropy. The construction can be proven to exist classically but fails if the null energy condition is violated. Here we extend the coarse-graining construction to semiclassical gravity. Its validity is conjectural, but we are able to extract an interesting nongravitational limit. Our proposal implies Wall's ant conjecture on the minimum energy of a completion of a quantum field theory state on a half-space. It further constrains the properties of the minimum energy state; for example, the minimum completion energy must be localized as a shock at the cut. We verify that the predicted properties hold in a recent explicit construction of Ceyhan and Faulkner, which proves our conjecture in the nongravitational limit.
Highlights
There is a remarkable interplay between testable lowenergy properties of quantum field theory (QFT), and certain conjectures about quantum gravity, in which the area of surfaces is associated to an entropy
A quantum focusing conjecture (QFC) was proposed to hold in the semiclassical regime; it implements a quantum correction to the classical statement by replacing the area with the area plus exterior entropy, i.e., the generalized entropy. This was a guess about quantum gravity, but it led to a new result in QFT
Boundary dual Within AdS=conformal field theory (CFT), it is natural to ask whether the coarsegraining prescription for Souter in Sec
Summary
There is a remarkable interplay between testable lowenergy properties of quantum field theory (QFT), and certain conjectures about quantum gravity, in which the area of surfaces is associated to an entropy. A quantum focusing conjecture (QFC) was proposed to hold in the semiclassical regime; it implements a quantum correction to the classical statement by replacing the area with the area plus exterior entropy, i.e., the generalized entropy This was a guess about quantum gravity, but it led to a new result in QFT. We apply our construction to states on a fixed background black hole spacetime with a complete Killing horizon In this limit, coarse graining requires the existence of QFT states with specific and somewhat surprising properties, which we list. The QNEC follows from this conjecture, but it has been directly proven.) Our proposal implies that a state that maximizes the generalized entropy minimizes the nongravitational energy inside of a cut of a Killing horizon, subject to holding fixed the state on the outside.
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