Casting Shadows on Holographic Reconstruction

TL;DR

The paper formalizes the notion of holographic shadows—bulk regions inaccessible to any extremal geometric probes—and develops a generalized framework that compares minimal-area surfaces, Wilson loops, and causal information surfaces as bulk probes. It proves coverage theorems linking boundary-region geometry to bulk coverage and applies these ideas to both stellar (regular) spacetimes and Schwarzschild–AdS geometries, quantifying shadow sizes across dimensions. Key findings include entanglement shadows surrounding dense stars in AdS and exponentially small shadows for large AdS black holes, with causal information surfaces often penetrating deepest, sometimes approaching the horizon exponentially in large or small BH limits. The results illuminate fundamental obstructions to bulk reconstruction from boundary data and hint at dual CFT interpretations ranging from nonlocal encodings to secret-sharing, motivating further exploration of alternative probes such as entwinement. The analysis provides a unified view across dimensions and probes, highlighting how phase-transition (switchover) behavior governs the reach of holographic information.

Abstract

In the context of the AdS/CFT correspondence, we study several holographic probes that relate information about the bulk spacetime to CFT data. The best-known example is the relation between minimal surfaces in the bulk and entanglement entropy of a subregion in the CFT. Building on earlier work, we identify "shadows" in the bulk: regions that are not illuminated by any of the bulk probes we consider, in the sense that the bulk surfaces do not pass through these regions. We quantify the size of the shadow in the near horizon region of a black hole and in the vicinity of a sufficiently dense star. The existence of shadows motivates further study of the bulk-boundary dictionary in order to identify CFT quantities that encode information about the shadow regions in the bulk. We speculate on the interpretation of our results from a dual field theory perspective.

Figures (28)

• Figure 1: The left figure shows a disconnected boundary region $a=\bigsqcup_i a_i$ (blue) and the corresponding disjoint minimal surface $b=\bigsqcup_jb_j$ in the bulk (red). As the boundary region is continuously increased, the bulk surfaces $b_j$ are pushed towards the dashed curve, at which point $b$ discontinuously switches to the new global minimum $b=\bigsqcup_jb_j'$ shown in the right figure. The region inside the dashed curves cannot be probed with this particular choice of bulk dual.
• Figure 2: The left figure shows a continuous foliation of minimal $n$-dimensional surfaces (red) on an $(n+1)$-dimensional equatorial slice of the bulk. The right figure shows how the angle between an $n$-sphere (blue circle) in the bulk and the foliation surfaces changes continuously from $0$ to $\pi/2$. Note that although the rightmost red surface is tangent to the blue circle at precisely $r_{*}$ in this plot, the proof does not rely on this.
• Figure 3: A minimal surface (red) with its symmetric point sitting at a finite radius $r_{*}$ cannot have other points approach arbitrarily close to $r=0$. Otherwise, a pinched-off version (blue) will have even smaller area.
• Figure 4: $\theta_{\infty}(r_*)$ for $GM = 2$ and stellar radii $R = 1.01l_{\rm AdS}$ (blue), $1.05l_{\rm AdS}$ (red), $1.1l_{\rm AdS}$ (black), $1.15l_{\rm AdS}$ (green), and $1.2l_{\rm AdS}$ (magenta). The case $R=1.2l_{\rm AdS}$ is insufficiently dense, and hence exhibits a monotonic function with no shadows. But the other cases, with $R < \sqrt{4/3}~l_{\rm AdS}$ (cf. \ref{['range']}), have a single maximum at finite radius $r_{\rm min}$, within which an entanglement shadow exists.
• Figure 5: Plots of extremal surfaces (blue) for stars of varying density. The solid black circle is the stellar radius $R$; the smaller, dotted black circle is the would-be horizon radius $r_H$. Note that in the first case, which is outside the range \ref{['range']}, there is no restriction against covering the entire bulk.