A Holographic Approach to Spacetime Entanglement

TL;DR

This work investigates how spacetime entanglement entropy can be captured holographically through differential entropy, providing a concrete boundary construction for encoding a bulk co-dimension-two hole. By proving, in Einstein gravity with planar symmetry, that the differential entropy of carefully chosen families of boundary intervals equals the Bekenstein–Hawking entropy of a bulk curve, the paper unifies boundary data with bulk geometry via tangent and null-vector alignment. It presents explicit AdS$_3$ realizations, a holographic lemma underpinning the boundary-to-bulk link, and a generalized boundary-to-bulk procedure using entanglement wedges, with extensions to higher dimensions and remarks on higher-curvature theories. The results deepen the understanding of spacetime entanglement and offer a practical framework for mapping arbitrary bulk holes to boundary entanglement data, with implications for interpreting holographic BH entropy as a leading term in spacetime entanglement. $S_{ m grav}=2\pi \mathcal{A}/\ell_p^{d-1}$ and $S(A)=\mathcal{A}(\gamma_A)/(4G_N^{(d+1)})$ serve as foundational equations linking area and entanglement in the holographic context, while the differential entropy captures a directional derivative of entanglement across a family of boundary intervals.

Abstract

Recently it has been proposed that the Bekenstein-Hawking formula for the entropy of spacetime horizons has a larger significance as the leading contribution to the entanglement entropy of general spacetime regions, in the underlying quantum theory [2]. This `spacetime entanglement conjecture' has a holographic realization that equates the entropy formula evaluated on an arbitrary space-like co-dimension two surface with the differential entropy of a particular family of co-dimension two regions on the boundary. The differential entropy can be thought of as a directional derivative of entanglement entropy along a family of surfaces. This holographic relation was first studied in [3] and extended in [4], and it has been proven to hold in Einstein gravity for bulk surfaces with planar symmetry (as well as for certain higher curvature theories) in [4]. In this essay, we review this proof and provide explicit examples of how to build the appropriate family of boundary intervals for a given bulk curve. Conversely, given a family of boundary intervals, we provide a method for constructing the corresponding bulk curve in terms of intersections of entanglement wedge boundaries. We work mainly in three dimensions, and comment on how the constructions extend to higher dimensions.

Figures (16)

• Figure 1: The extremal curve $\gamma_A$ is shown above for an interval $A$ on the boundary of AdS$_3$, drawn on a constant time slice.
• Figure 2: The causal diamond for the interval $I_k \cap I_{k+1}$ for two intervals on a constant time slice is shaded in red above in (a). When the intervals do not lie on the same time slice as in (b), the appropriate interval has endpoints $\gamma_L(\lambda_{k+1})$ and $\gamma_R(\lambda_k)$.
• Figure 3: 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 neighboring geodesic at $\lambda^*-d\lambda$. The points $\gamma_L(\lambda^*)$ and $\gamma_R(\lambda^*)$ are explicitly drawn on the boundary at $z=0$.
• Figure 4: 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: We consider evaluating the action only on a portion of the classical trajectory from the boundary to the bulk surface of interest. In the case of tangent vector alignment we can apply the lemma (\ref{['almost']}) to equate the differential entropy of the family of boundary intervals with the Bekenstein-Hawking of the bulk curve.