Global properties of causal wedges in asymptotically AdS spacetimes

TL;DR

The paper address how causal structures in asymptotically AdS spacetimes, via the causal wedge $\mathbb{\blacklozenge}_A$ and causal information surface ${\Xi}_A$, constrain bulk and boundary observables. It develops general, geometry-independent properties of these constructs, demonstrates that ${\Xi}_A$ can have non-trivial topology even for simple boundary regions, and shows that extremal surfaces ${\mathfrak E}_A$ generally lie outside the causal wedge, with important consequences for holographic entanglement entropy, including possible entanglement plateaus and saturation of Araki-Lieb-type relations in black hole backgrounds. The results are illustrated with explicit analyses in Schwarzschild-AdS$_{d+1}$ and BTZ, including boosted black holes and star geometries, revealing conditions under which ${\Xi}_A$ becomes disconnected and how that impacts ${\mathfrak E}_A$ and $S_A$. Overall, the work reveals deep links between bulk causal structure and boundary entanglement data, suggesting robust CFT interpretations of causal wedges and guiding future exploration of their duals.

Abstract

We examine general features of causal wedges in asymptotically AdS spacetimes and show that in a wide variety of cases they have non-trivial topology. We also prove some general results regarding minimal area surfaces on the causal wedge boundary and thereby derive constraints on the causal holographic information. We go on to demonstrate that certain properties of the causal wedge impact significantly on features of extremal surfaces which are relevant for computation of holographic entanglement entropy.

Figures (7)

• Figure 1: A sketch of the causal wedge $\blacklozenge_{{\cal A}}$ and associated quantities in planar AdS (left) and global AdS (right) in 3 dimensions taken from Hubeny:2013hz: in each panel, the region ${\cal A}$ is represented by the red curve on right, and the corresponding surface $\Xi_{\cal A}$ by blue curve on left; the causal wedge $\blacklozenge_{{\cal A}}$ lies between the AdS boundary and the null surfaces $\partial_+(\blacklozenge_{{\cal A}})$ (red surface) and $\partial_-(\blacklozenge_{{\cal A}})$ (blue surface).
• Figure 2: A plot of the intersection points of the future and past congruence, ${\cal X}_{t=0}$, plotted on the Poincaré disk. In each panel, the outer circle represents the AdS boundary (with the region ${\cal A}$ highlighted in red; $\varphi_{{\cal A}}=2.5$ in both panels). The black hole size is $r_h = 0.5$ (left) and $r_h = 0.2$ (right), denoted by red dashed curve (but obscured in the latter case). ${\cal X}_{t=0}$ is composed of the individual intersection points, color-coded by $\ell$ (from red at $\ell =0$ to purple at $\ell = 1$). For large enough black hole (left), $\varphi_{t=0}(\ell) <\pi$ for all $\ell$, and therefore ${\cal X}_{t=0} = \Xi_{\cal A}$. For small black hole (right) ${\cal X}_{t=0}$ self-intersects and therefore $\Xi_{\cal A}$ has two components as indicated.
• Figure 3: Causal wedge for the case $r_h=0.2$ and $\varphi_{{\cal A}}=2.5$, as in right panel of Fig. \ref{['f:Xi_PD']}. Same color-coding (by $\ell$) is applied to the null geodesic generators of $\partial_\pm \blacklozenge_{{\cal A}}$. In addition to the AdS boundary and horizon, the plot exhibits the region ${\cal A}$ (indicated by the thick red curve), the two components of $\Xi_{\cal A}$ (indicated by the thick blue curves), and the curves of caustics ${\cal C}^\pm$ (indicated by thick brown curves) which connect up the two components of $\Xi_{\cal A}$. The causal wedge $\blacklozenge_{{\cal A}}$ bounded by the null generators clearly exhibits a hole.
• Figure 4: The projection of the causal information surfaces $\Xi_{\cal A}$ for various $\varphi_{{\cal A}}$ onto the Poincaré disk of Schwarzschild-AdS$_5$ with fixed black hole size $r_h=0.2$, color-coded by $\varphi_{{\cal A}}$ which varies from $0$ (red) to $\pi$ (purple) in increments of $0.1$. (For example, the blue curve with $\varphi_{{\cal A}}=2.5$ corresponds to the projection of $\Xi_{\cal A}$ in Fig. \ref{['f:CWplot']}.) We can clearly see that $\Xi_{\cal A}$ pinches off; for larger $\varphi_{{\cal A}}$, the disconnected component of $\Xi_{\cal A}$ is located very near the horizon.
• Figure 5: The critical curves on $(\varphi_{{\cal A}},\rho_h)$ plane indicating where $\Xi_{\cal A}$ pinches off for Schwarzschild-AdS$_{d+1}$. $\Xi_{\cal A}$ has two components above the curve and only a single component below. To guide the eye, we also indicate the $\varphi_{{\cal A}}=\pi/2$ and $\varphi_{{\cal A}}=\pi$ (dashed lines); the latter gives the upper bound in $\varphi_{{\cal A}}$, while the former indicates the lower bound below which $\Xi_{\cal A}$ is connected for Schwarzschild-AdS black hole of any size. The topmost (red) curve corresponds to Schwarzschild-AdS$_4$ geometry where the effect of gravity is strongest, while the next (orange) curve is for Schwarzschild-AdS$_5$ which is our prime exhibit. Increasing the dimension results in slower growth of $\varphi_{{\cal A}}^\ast(\rho_h)$, as exemplified by $d=5$ (green), $d=6$ (blue), $d=7$ (purple), $d=19$ (purple dotted) and $d = 49$ (blue dashed). The effects of the weaker gravitational potential are clearly visible with the increasing dimension and the bottommost curve is close to the limiting behaviour for large $d$.