Covariant Constraints on Hole-ography

TL;DR

The paper establishes covariant constraints on hole-ography by analyzing extremal surfaces in spacetimes with holographic screens. Under a null curvature condition and suitable symmetry, extremal surfaces inside the interior of screens cannot be fully reconstructed from boundary-anchored data, indicating a coarse-grained limitation to hole-ography. The authors formalize this through a set of theorems (traffic, main, and deformability) and illustrate them with explicit examples in AdS, dS, and AdS-Vaidya collapse, then discuss implications for bulk reconstruction and potential quantum extensions via generalized entropy and quantum extremal surfaces. These results refine our understanding of how boundary data encode bulk geometry and highlight intrinsic information barriers associated with holographic screens.

Abstract

Hole-ography is a prescription relating the areas of surfaces in an AdS bulk to the differential entropy of a family of intervals in the dual CFT. In (2+1) bulk dimensions, or in higher dimensions when the bulk features a sufficient degree of symmetry, we prove that there are surfaces in the bulk that cannot be completely reconstructed using known hole-ographic approaches, even if extremal surfaces reach them. Such surfaces lie in easily identifiable regions: the interiors of holographic screens. These screens admit a holographic interpretation in terms of the Bousso bound. We speculate that this incompleteness of the reconstruction is a form of coarse-graining, with the missing information associated to the holographic screen. We comment on perturbative quantum extensions of our classical results.

Figures (16)

• Figure 1: An arbitrary closed curve $\gamma$ on a static time slice of global AdS$_3$. The set of all geodesics tangent to $\gamma$ define a family of regions on the boundary parametrized by a (possibly multi-valued) function $\alpha(\theta)$. The differential entropy of these regions gives the length of $\gamma$.
• Figure 2: \ref{['subfig:holepoint']}: reconstruction of bulk points via hole-ography. The curve $\gamma$ is shrunk to be arbitrarily small and centered at $p$, so that $p$ is identified by the common intersection of all the geodesics generated by $\alpha(\theta)$. \ref{['subfig:holedist']}: reconstruction of geodesic distances via hole-ography. The curve $\gamma$ is shrunk to be an arbitrarily thin convex curve (thick red line) encircling two points $p$ and $q$. The geodesic distance between $p$ and $q$ is then given by the differential entropy of the resulting boundary intervals.
• Figure 3: The two null congruences of an extremal surface $X$ of codimension two anchored to a timelike boundary $\partial M$.
• Figure 4: An illustration for Lemma \ref{['lem:aron']}. In Minkowski space, an extremal surface $X$ is just a plane (drawn here as a straight line). If $X$ is tangent to an expanding light cone, it lies nowhere to the cone's future, and the cone has positive expansion. If $X$ is tangent to a shrinking light cone, it lies nowhere to the cone's past, and thus the shrinking light cone has negative expansion.
• Figure 5: Constructing a preferred holographic screen from a null foliation of a spacetime. The dashed diagonal lines are the leaves of the foliation; the dot on each leaf marks the leaflet $\sigma_s$ where the expansion of the leaf changes sign. The union of all the leaflets is a preferred holographic screen.

Theorems & Definitions (16)

• Lemma 1
• proof
• Lemma 2
• proof
• Definition 1
• Definition 2
• Definition 3
• Theorem 1
• Definition 4
• Theorem 2