Holographic Holes and Differential Entropy

TL;DR

This paper extends the holographic interpretation of gravitational entropy as differential entropy to time-varying surfaces and provides a general proof applicable to various holographic backgrounds and higher-curvature gravity theories.

Contribution

It generalizes the differential entropy construction to dynamic bulk surfaces and proves the equality with gravitational entropy for a broad class of theories.

Findings

Extended the differential entropy framework to time-dependent surfaces.

Provided a general proof for the equality of gravitational and differential entropy.

Applicable to higher-curvature gravity theories and backgrounds with planar symmetry.

Abstract

Recently, it has been shown by Balasubramanian et al. and Myers et al. that the Bekenstein-Hawking entropy formula evaluated on certain closed surfaces in the bulk of a holographic spacetime has an interpretation as the differential entropy of a particular family of intervals (or strips) in the boundary theory. We first extend this construction to bulk surfaces which vary in time. We then give a general proof of the equality between the gravitational entropy and the differential entropy. This proof applies to a broad class of holographic backgrounds possessing a generalized planar symmetry and to certain classes of higher-curvature theories of gravity. To apply this theorem, one can begin with a bulk surface and determine the appropriate family of boundary intervals by considering extremal surfaces tangent to the given surface in the bulk. Alternatively, one can begin with a family of boundary intervals; as we show, the differential entropy then equals the gravitational entropy of a bulk surface that emerges from the intersection of the neighboring entanglement wedges, in a continuum limit.