Holographic Holes in Higher Dimensions

TL;DR

This work extends the holographic hole program from AdS$_3$ to higher dimensions and general backgrounds, showing that the Bekenstein-Hawking entropy of bulk surfaces with planar symmetry can be obtained from a boundary differential entropy built from a family of entanglement entropies, with the outer envelope playing a central geometric role. The authors derive the higher-dimensional generalization by analyzing constant-$z$ and varying-$z(x)$ bulk surfaces, extend the construction to Lovelock gravity via the Jacobson-Myers entropy functional, and demonstrate that the differential entropy matches the bulk gravitational entropy in the continuum limit. They also analyze causal holographic information and find that using $ Chi$ in place of $S$ generally fails to reproduce the bulk entropy in higher dimensions due to nonlocal divergences. The results hold in general holographic backgrounds and with generalized entropy functionals, indicating a robust boundary–bulk link for gravitational entropy and providing a framework for understanding holographic reconstruction from boundary entanglement data. The work opens several avenues, including extending beyond planar symmetry, exploring boundary tilings, and clarifying the boundary interpretation of residual entropy.

Abstract

We extend the holographic construction from AdS3 to higher dimensions. In particular, we show that the Bekenstein-Hawking entropy of codimension-two surfaces in the bulk with planar symmetry can be evaluated in terms of the 'differential entropy' in the boundary theory. The differential entropy is a certain quantity constructed from the entanglement entropies associated with a family of regions covering a Cauchy surface in the boundary geometry. We demonstrate that a similar construction based on causal holographic information fails in higher dimensions, as it typically yields divergent results. We also show that our construction extends to holographic backgrounds other than AdS spacetime and can accommodate Lovelock theories of higher curvature gravity.

Figures (13)

• Figure 1: (Color online) Proof of strong subadditivity in a holographic framework: (a) Two intervals on the boundary of AdS$_3$ in global coordinates. The blue arcs indicate the geodesics used to evaluate $S(I_{1})$ and $S(I_2)$, while the green arcs are those which determine $S(I_{1}\cup I_2)$ and $S(I_{1}\cap I_2)$. (b) Rearranging the interconnection of the blue arcs at their intersection produces two new curves in the same homology classes as the green arcs. However, the lengths of the red and yellow curves must be longer than that of the homologous green arcs.
• Figure 2: (Color online) (a) This figure illustrates that the outer envelope depends on the details of the individual boundary intervals, not just their union. (b) An example with three boundary intervals and the corresponding outer envelope (in red).
• Figure 3: (Color online) Eight intervals and their outer envelope, which forms a closed curve in the bulk, (a) in empty AdS space and (b) in an AdS black hole spacetime. In case (a), the entanglement entropy of the global boundary state vanishes while it is non-vanishing in case (b).
• Figure 4: (Color online) (a) In the continuum limit of many identical intervals, the outer envelope becomes a circle of a fixed radius. (b) The continuum limit of many intervals whose length varies continuously produces a smooth outer envelope with a profile that varies in the bulk.
• Figure 5: (Color online) A bulk surface with a constant profile $z=z_*$. The two intersecting semi-circles of radius $r=z_*$ in AdS$_3$ are the extremal bulk surfaces determining the holographic entanglement entropy for two overlapping boundary intervals of length $\Delta x=2z_*$.