Holographic Holes and Differential Entropy

TL;DR

The paper establishes a covariant equivalence between the differential entropy of a boundary family of intervals and the gravitational entropy of a corresponding bulk surface, generalizing the hole-ographic construction to time-varying holes and backgrounds with generalized planar symmetry. It provides a rigorous classical-mechanics proof that, under tangent-vector or null-vector alignment, the boundary differential entropy equals the bulk area-like functional, and extends the framework to Lovelock gravity with appropriate entropy functionals. It introduces a signed-area interpretation to accommodate cases where the alignment parameter changes sign along the curve, and develops both bulk-to-boundary and boundary-to-bulk constructions, including a concrete AdS$_3$ case study. The results broaden holographic diagnostics of bulk geometry, demonstrate the utility of entanglement wedges and their envelopes, and point to new directions in reconstructing bulk spacetimes from boundary data using differential observables.

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.

Figures (18)

• Figure 1: (Colour online) The causal diamonds for two neighbouring intervals are drawn above: $I_k$ with endpoints $\gamma_L(\lambda_k)$ and $\gamma_R(\lambda_k)$, and $I_{k+1}$ with $\gamma_L(\lambda_{k+1})$ and $\gamma_R(\lambda_{k+1})$. Red shading highlights the intersection region, which, of course, is the causal diamond for the interval $I_k\cap I_{k+1}$ with endpoints $\gamma_L(\lambda_{k+1})$ and $\gamma_R(\lambda_k)$.
• Figure 2: (Colour online) The bulk curve $\gamma_B(\lambda)$ is shown above in green, along with the tangent geodesics at each point. One such geodesic $\Gamma(s;\lambda^*)$ is highlighted in blue, along with a neighbouring geodesic at $\lambda^*-d\lambda$. The points $\gamma_L(\lambda^*)$ and $\gamma_R(\lambda^*)$ are explicitly drawn on the boundary at $z=0$.
• Figure 3: (Colour online) The causal diamonds for two neighbouring intervals, $I_k$ and $I_{k+1}$, in the case of a time varying bulk curve. Their respective endpoints $\gamma_L(\lambda_k)$ and $\gamma_R(\lambda_k)$, and $\gamma_L(\lambda_{k+1})$ and $\gamma_R(\lambda_{k+1})$. The intersection of these two causal diamonds (highlighted with red shading) is the causal diamond for the interval with endpoints $\gamma_L(\lambda_{k+1})$ and $\gamma_R(\lambda_k)$.
• Figure 4: (Colour online) For each point on the bulk curve $\gamma_B(\lambda)$, the intersection of the extremal curve $\Gamma(s;\lambda)$ with the boundary defines an interval between $\gamma_L(\lambda)$ and $\gamma_R(\lambda)$. We take the family of intervals as described by the curves $\gamma_L(\lambda),\gamma_R(\lambda)$ shown in yellow and orange respectively. The differential entropy of this family of intervals equals the gravitational entropy of the bulk curve.
• Figure 5: (Color online) A constant-time slice of AdS${}_3$, in the $(r,\theta)$ coordinate system described in the text. (a) A boundary interval $[\theta_L,\theta_R]$ with $\Delta<\pi/2$, and the corresponding bulk geodesic (black line). The yellow annulus is the union of the bulk regions corresponding to intervals with the same value of $\Delta$, and its boundary (the blue circle) is $\gamma_B$. (b) A boundary interval with $\Delta>\pi/2$, and the corresponding bulk geodesic. The yellow disc is the intersection of the bulk regions corresponding to intervals with that value of $\Delta$, and its boundary (the red circle) is $\gamma_B$.