A Holographic Entanglement Entropy Conjecture for General Spacetimes

TL;DR

Addresses how to define holographic entanglement entropy for general spacetimes by anchoring extremal surfaces to holographic screens rather than a boundary. Introduces a maximin construction to obtain ext(A) within the causal region D_sigma and defines S(A) as area(ext(A))/4, conjecturing it equals von Neumann entropy in a putative holographic dual. Proves strong subadditivity and Page bound for S(A) and shows reduction to HRT in AdS scenarios. Applies the framework to FRW cosmologies, demonstrating late-time saturation to a random-entanglement limit and providing a tractable static-sphere approximation. Collectively, the work lays groundwork for entanglement structure and holography in general spacetimes, including cosmology.

Abstract

We present a natural generalization of holographic entanglement entropy proposals beyond the scope of AdS/CFT by anchoring extremal surfaces to holographic screens. Holographic screens are a natural extension of the AdS boundary to arbitrary spacetimes and are preferred codimension 1 surfaces from the viewpoint of the covariant entropy bound. A broad class of screens have a unique preferred foliation into codimension 2 surfaces called leaves. Our proposal is to find the areas of extremal surfaces achored to the boundaries of regions in leaves. We show that the properties of holographic screens are sufficient to prove, under generic conditions, that extremal surfaces anchored in this way always lie within a causal region associated with a given leaf. Within this causal region, a maximin construction similar to that of Wall proves that our proposed quantity satisfies standard properties of entanglement entropy like strong subadditivity. We conjecture that our prescription computes entanglement entropies in quantum states that holographically define arbitrary spacetimes, including those in a cosmological setting with no obvious boundary on which to anchor extremal surfaces.

Figures (9)

• Figure 1: An example of a past holographic screen $\mathcal{H}$. One particular leaf $\sigma$ is highlighted here along with its null orthogonal vector fields $k$ and $l$ satisfying $\theta^k=0$ and $\theta^l>0$. The causal region $D_\sigma$ plays a critical role in our generalization of holographic entanglement entropy.
• Figure 2: This figure depicts our construction of holographic entanglement entropy in general spacetimes. The horn-shaped surface is a past holographic screen $\mathcal{H}$. The black and red codimension 2 regions together form a single leaf $\sigma$. The black segment represents a region $A$ and the extremal surface $\textrm{ext} \: (A)$ (orange) is anchored to its boundary. The causal region $D_\sigma$ is the green diamond (both interior and boundary). Note that $\textrm{ext} \: A \subset D_\sigma$.
• Figure 3: The proof of lemma \ref{['inside_lem']} involves a continuous family of surfaces $A_s$ along with their extremal surfaces (dotted curves).
• Figure 4: The idea of a compact restriction is shown here. The restriction $R$ is the shaded region along with its boundary, the blue and orange lines. $\partial R$ consists of two parts: an extremal surface barrier $B$ (blue) and a portion of $\partial D_\sigma$ (orange). In this figure, the barrier $B$ protects extremal surfaces in $R$ from a singularity. Not shown are extremal surfaces in $R$, none of which contact $\partial R$ except at their anchor on the leaf $\sigma$.
• Figure 5: This figure depicts the argument of case 1 of the proof of theorem \ref{['mmext']}. Note that the surface $S$ is shown here for reference and that it does not play a critical role in the proof. The shaded region is $D_\sigma = D(S)$ and the green dot is (a cross-section of) the leaf $\sigma$.

Theorems & Definitions (11)

• Lemma 1
• proof
• Theorem 1
• Theorem 2
• proof
• Corollary 1
• proof
• Theorem 3
• proof : Sketch of Proof:
• Corollary 2