A hole-ographic spacetime

TL;DR

The paper embeds spherical Rindler space with a central hole into AdS and shows the hole’s gravitational entropy matches a boundary construction defined by a finite-time strip of local observers. It defines differential entropy as the residual uncertainty left by restricting to finite-time causal diamonds on the boundary and proves that, in AdS$_3$/CFT$_2$, this quantity equals the length of the hole’s boundary (and more generally the length of any closed curve) in AdS$_3$, saturating strong subadditivity. The work links UV/IR structure, entanglement, and causal holographic information to the emergence of bulk geometry, and discusses implications for higher dimensions, BTZ black holes, and RG-like flows that challenge naive tensor-factorization of the Hilbert space. It opens avenues to reinterpret holographic entanglement and the emergence of spacetime as governed by time-bounded observational data rather than strictly local tensor factorizations.

Abstract

We embed spherical Rindler space -- a geometry with a spherical hole in its center -- in asymptotically AdS spacetime and show that it carries a gravitational entropy proportional to the area of the hole. Spherical AdS-Rindler space is holographically dual to an ultraviolet sector of the boundary field theory given by restriction to a strip of finite duration in time. Because measurements have finite durations, local observers in the field theory can only access information about bounded spatial regions. We propose a notion of Residual Entropy that captures uncertainty about the state of a system left by the collection of local, finite-time observables. For two-dimensional conformal field theories we use holography and the strong subadditivity of entanglement to propose a formula for Residual Entropy and show that it precisely reproduces the areas of circular holes in AdS3. Extending the notion to field theories on strips with variable durations in time, we show more generally that Residual Entropy computes the areas of all closed, inhomogenous curves on a spatial slice of AdS3. We discuss the extension to higher dimensional field theories, the relation of Residual Entropy to entanglement between scales, and some implications for the emergence of space from the RG flow of entangled field theories.

Figures (5)

• Figure 1: Spherical Rindler-AdS space, with a hole inside it, is built up of regions visible to individual accelerating observers, each of whom observes physics that is holographically dual to the content of a single causal diamond. The union of the diamonds makes up a finite time strip in the boundary field theory.
• Figure 2: Combinations of boundary causal diamonds considered in the derivation in Sec. \ref{['formula']}.
• Figure 3: The spatial geodesics that extend across intervals of length $2\alpha_0$ and $2\alpha_0-\pi/K$. In the limit $K \to \infty$, the differences between their lengths arise only from the tips of the geodesics and make up a circle of radius $R_0$ in the center. The graphs show $2K = 16, 32, 64$. For arbitrary curves the cancellations are more subtle (see below.)
• Figure 4: Left: the notation of eqs. (\ref{['curve']}-\ref{['Deltatheta']}). Right: the geodesics, which make up eq. (\ref{['genbounddiscr']}).
• Figure 5: Left: The integrand in eq. (\ref{['genbound']}) is the length of the black, continuous geodesic minus the lengths of the red, dashed half-geodesics. The angle between the two straight red lines is $d\theta$. Center: The form (\ref{['thecure']}) adds the red, thickly dashed length and subtracts the green, finely dotted length. Right: The resulting integrand is that in eq. (\ref{['newintegrand']}), the length element along the curve $R(\tilde{\theta})$. The angle between the two straight black lines is $d\tilde{\theta}$.