Thermalization of Causal Holographic Information

TL;DR

This work probes the causal holographic information χ_A and the causal information surface Ξ_A in time-dependent AdS spacetimes, using Vaidya-AdS as a simple model of quantum quench. By analyzing both three- and higher-dimensional cases, the authors map how χ_A evolves as a shell collapses, revealing regimes where χ_A tracks the entanglement entropy S_A and regimes where χ_A exhibits quasi-teleological behavior. A key finding is that in 3D, χ_A = S_A in Regimes 1, 2, and 4, while in Regime 3 χ_A > S_A with a calculable analytic form; near the regime boundaries, the growth is governed by distinct power laws (e.g., t_A^3 near t_A=0 and ε^2 near ε=φ_A). The results illuminate the relationship between causal bulk constructs and boundary information, offering both precise quantitative predictions and qualitative insights into when causal holographic information reflects or exceeds standard entanglement measures, with implications for identifying a dual field theory quantity to χ_A and Ξ_A.

Abstract

We study causal wedges associated with a given sub-region in the boundary of asymptotically AdS spacetimes. Part of our motivation is to better understand the recently proposed holographic observable, causal holographic information (CHI), which is given by the area of a bulk co-dimension two surface lying on the boundary of the causal wedge. It has been suggested that CHI captures the basic amount of information contained in the reduced density matrix about the bulk geometry. To explore its properties further we examine its behaviour in time-dependent situations. As a simple model we focus on null dust collapse in an asymptotically AdS spacetime, modeled by the Vaidya-AdS geometry. We argue that while CHI is generically quasi-telelogical in time-dependent backgrounds, for suitable choice of sub-regions in conformal field theories, the temporal evolution of CHI is entirely causal. We comment on the implications of this observation and more generally on features of causal constructions and contrast our results with the behaviour of holographic entanglement entropy. Along the way we also derive the rate of early time growth and late time saturation (to the thermal value) of both CHI and entanglement entropy in these backgrounds.

Figures (8)

• 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: 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: Radial profile of the causal wedge for fixed $t_{{\cal A}} = -1.5$ (left), $t_{{\cal A}} = 0$ (middle), and $t_{{\cal A}} = 1.5$ (right), for a set of $\cal A$, color-coded by size $a$. The thick black curve on right in each panel is the AdS boundary, the dashed black line on left is the origin, the dashed red curve the event horizon (whose final size is $r_h=2$ in AdS units), and the thin brown diagonal line the shell. The black dots denote the radial position of $\Xi_{\cal A}$ corresponding to the given ${\cal A}$ at time $t_{{\cal A}}$ and size $a$. Our coordinates are such that ingoing radial null geodesics are diagonal everywhere (i.e. parallel to the shell). The plots are made for Vaidya-AdS$_{3}$ spacetime.
• Figure 3: A plot of the causal information surface $\Xi_{\cal A}$ (thick blue curve) along with representative generators of $\partial_\pm(\blacklozenge_{{\cal A}})$ (thin null curves, color-coded by $r_{\rm x}$), in Regime 2 (left) and 3 (right), as discussed in text. For orientation we also show the boundary, imploding shell, corresponding event horizon whose final size is $r_h=2$, the region ${\cal A}$ (thick red curve) whose size is $\varphi_{{\cal A}}=\frac{2\pi}{5}$ and time $t_{{\cal A}}=-0.1$ (left) and $t_{{\cal A}}=0.6$ (right), the corresponding domain of dependence $\Diamond_{\cal A}$ (thin grey curves) with its future and past tips ${q^\wedge}$, ${q^\vee}$ as marked, as well as the extremal surface ${\mathfrak E}_{{\cal A}}$ (thick purple curve) for comparison.
• Figure 4: Top view of the same plot as in Fig. \ref{['f:CWAB']} (with the same color-coding scheme), i.e. all curves are projected onto the Poincare disk. For orientation, we also indicate the final black hole size $r_h$ (dashed red curve).
• Figure 5: (Left): The curves $\Xi_{\cal A}$ (color-coded by $t_{{\cal A}}$) for a range of $t_{{\cal A}}$ sampling across the 4 regimes (separated by the three transitions at $t_{{\cal A}}=-\varphi_{{\cal A}} , 0, \varphi_{{\cal A}}$ as labeled; the thin gray curves represent ${\cal A}$ at those times) in increments of $0.1 \varphi_{{\cal A}}$, for $\varphi_{{\cal A}}=2\pi/5$ and $r_h=2$. (Right): Same curves $\Xi_{\cal A}$ projected onto the Poincare disk, analogous to Fig. \ref{['f:PDAB']}.