Holographic entanglement entropy conjecture for general spacetimes

We present a natural generalization of holographic entanglement entropy proposals beyond the scope of $\mathrm{AdS}/\mathrm{CFT}$ 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 anchored 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.

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 $\mathrm{ext}(A)$ (orange) is anchored to its boundary. The causal region $D_{\sigma }$ is the green diamond (both interior and boundary). Note that $\mathrm{ext}A\subset D_{\sigma }$.

Figure 3

The proof of Lemma 1 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 2. 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 $\sigma$.

Figure 6

The domain of dependence $D_{\sigma (\tau )}$ for a late time leaf in the flat FRW universe (the small green triangle in the upper diagram) can be approximately mapped to a domain of dependence $D_{\tilde{\sigma }(\tau )}$ in empty de Sitter space (lower diagram). The mapping becomes increasingly accurate as $\tau$ becomes larger. The effect of increasing $\tau$ is to move the green triangle in the upper diagram into the top-left corner (along the blue curve), while the green triangle in the lower diagram moves to the right and approaches the entire left static wedge.

Figure 7

The upper hemisphere of a 3-sphere of radius $\alpha$ is half of a static slice in empty de Sitter space and serves as a good approximation for $D_{\sigma (\tau )}$ at large $\tau$. The blue 2-sphere (appearing as a circle here) lies at constant $z$ [equivalently, constant $r$, where $r$ is the radial coordinate in Eq. (4.9)]. This 2-sphere is an approximation for the leaf $\sigma (\tau )$. Green surfaces depict extremal spherical caps on $S^3$ that approximate $\mathrm{ext}{A}_{\psi }$ for various values of $\psi$. The many samples of extremal surfaces shown here have evenly spaced values of $\psi$. Figure 8 provides evidence that this static sphere approximation is accurate at late $\tau$.

Figure 8

Plots of $S(A_{\psi })$ and other quantities for two leaves at different times in a universe with dust and vacuum energy. In both plots, the red curve is the numerically computed holographic screen entanglement entropy of $A_{\psi }$. The dashed green curve is the static sphere approximation for $S(A\psi )$ which becomes more accurate at later $\tau$ (smaller $z$). The orange curve with a sharp peak is $S_{\text{Page}}(A_{\psi })$ as defined by Eq. (4.6) and the black curve is $S_{\text{extensive}}(A_{\psi })$. The horizontal line, provided for scale, marks the value of $\pi {\alpha }^2/2$ which is precisely one-fourth of the extensive entropy of the de Sitter horizon.

Figure 9

Both Penrose diagrams here are for the same spacetime: a closed FRW universe with dust. The red lines denote a null foliation and the black diagonals are the past and future holographic screens corresponding to the foliation. The two figures demonstrate that different foliations give rise to different screens. In both figures, the lower half of the diagonal is a past holographic screen and the upper half is a future holographic screen. Arrows show the direction of increasing area.