Abstract
It has been proposed that the areas of marginally trapped or anti-trapped surfaces (also known as leaves of holographic screens) may encode some notion of entropy. To connect this to AdS/CFT, we study the case of marginally trapped surfaces anchored to an AdS boundary. We establish that such boundary-anchored leaves lie between the causal and extremal surfaces defined by the anchor and that they have area bounded below by that of the minimal extremal surface. This suggests that the area of any leaf represents a coarse-grained von Neumann entropy for the associated region of the dual CFT. We further demonstrate that the leading area-divergence of a boundary-anchored marginally trapped surface agrees with that for the associated extremal surface, though subleading divergences generally differ. Finally, we generalize an argument of Bousso and Engelhardt to show that holographic screens with all leaves anchored to the same boundary set have leaf-areas that increase monotonically along the screen, and we describe a construction through which this monotonicity can take the more standard form of requiring entropy to increase with boundary time. This construction is related to what one might call future causal holographic information, which in such cases also provides an upper bound on the area of the associated leaves.
Highlights
Despite this thermodynamic property, the nature of the entropy described by holographic screens has remained unclear.1 In contrast, the Ryu-Takayanagi formula computes the von Neumann entropy, tr(ρ log ρ), of the dual CFT density matrix ρ
We generalize an argument of Bousso and Engelhardt to show that holographic screens with all leaves anchored to the same boundary set have leaf-areas that increase monotonically along the screen, and we describe a construction through which this monotonicity can take the more standard form of requiring entropy to increase with boundary time
We explore the properties of such screens below when all leaves are anchored to the same boundary set ∂A, whereas for RT/HRT we take ∂A to be the boundary of a partial Cauchy surface A for the boundary spacetime
Summary
We provide some definitions and lemmas that will be used throughout the work below. A marginally trapped surface is a smooth, codimension-two achronal spacelike surface whose future directed orthogonal null congruences, k and , have expansions satisfying θk = 0, θ ≤ 0. We will focus in particular on future holographic screens where, for the boundaryanchored case, we define the k, null congruences as follows: consider a boundary region A and a marginally trapped surface σ homologous to A as above. If a smooth spacelike curve γ intersects the boundary of the future I+(S) of some set S at a point p, either i) γ enters the chronological future I+(S) or ii) all null generators of I+(S) through p intersect γ orthogonally. Unless λ has a conjugate point at p, at least in a neighborhood of p these generators define a smooth null surface N nowhere to the past of σ. Σ enters the future of some λ and enters I+(S)
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