Entanglement islands in higher dimensions

TL;DR

The paper demonstrates that entanglement islands persist in higher dimensions by constructing a static five-dimensional geometry with a Planck brane that realizes a four-dimensional doubly holographic setup. By solving Einstein equations via the DeTurck trick and computing five-dimensional RT surfaces, the authors identify two competing extremal surfaces: a horizon-penetrating one that grows entropy over time and a brane-ending one that saturates, signaling an island and restoring unitarity. This provides a concrete higher-dimensional generalization of the island paradigm, showing that the Page-curve-like saturation of Hawking radiation is robust beyond the 2d AdS/JT framework. The results strengthen the view that islands are a general feature of gravitational entanglement wedges and support unitarity in black-hole information processing across dimensions.

Abstract

It has been suggested in recent work that the Page curve of Hawking radiation can be recovered using computations in semi-classical gravity provided one allows for "islands" in the gravity region of quantum systems coupled to gravity. The explicit computations so far have been restricted to black holes in two-dimensional Jackiw-Teitelboim gravity. In this note, we numerically construct a five-dimensional asymptotically AdS geometry whose boundary realizes a four-dimensional Hartle-Hawking state on an eternal AdS black hole in equilibrium with a bath. We also numerically find two types of extremal surfaces: ones that correspond to having or not having an island. The version of the information paradox involving the eternal black hole exists in this setup, and it is avoided by the presence of islands. Thus, recent computations exhibiting islands in two-dimensional gravity generalize to higher dimensions as well.

Figures (7)

• Figure 1: A simple geometry with a RS or a Planck brane, discussed in Takayanagi:2011zk. The RS or the Planck brane along lies along the locus $z=-w \tan \theta$. The induced geometry on the brane is AdS$_4$ with length scale (\ref{['eq:L4']}). The angle $\theta$ is fixed by the tension parameter $\alpha$ in the action (\ref{['eq:action']}) via the relationship (\ref{['eq:alphathetarelation']}).
• Figure 2: On the left, we show the Penrose diagram of the 4d geometry. We have a two-sided AdS black hole, with each side coupled to a bath. On the right, we show the integration domain used in the numerics $x\in (0,1)$ and $y \in (-1,1)$. The objective is to solve for five metric functions $Q_1, \ldots, Q_5$ of two variables each (\ref{['eq:ansatz']}), in this domain. We numerically solve only in the region $y>0$, the rest is obtained simply by symmetry. On the left edge of this diagram, at $x=0$, we have the RS or the Planck brane where the 4d gravity region lives and the boundary condition (\ref{['eq:bc']}) is imposed. On the top and bottom edges we have the two baths. As $x \to 1$, the metric approaches that of a 5d planar AdS-Schwarzschild black hole. The reader might find it useful to note the points $ABHCD$ on both diagrams. The precise induced geometry on the segment $BC$ is determined by the numerical solution, and the left picture is just a cartoon.
• Figure 3: Plots of $-g_{tt}/L_4^2$ and $g_{w_1\,w_1}/L_4^2$ on the RS brane (located at $x=0$) as a function of the proper distance from the horizon $\mathcal{P}_{4D}$. In the top row, $\theta \approx 1.47113$, and in the bottom row, $\theta \approx 0.343024$. The blue disks correspond to the numerical data, and the solid blue lines are obtained from the 4d planar AdS black hole geometry. It is clear that as $\theta$ becomes smaller, the induced geometry on the brane gets closer to that of a 4d planar AdS black hole
• Figure 4: Shown here is a two-sided 4d black hole (with two of the spatial dimensions suppressed) coupled to two baths. See also figure \ref{['fig:intdom']}. We want to compute the entanglement entropy of the union of the two blue regions shown. This diagram lives on the boundary of a static 5d spacetime whose exterior region was computed numerically in section \ref{['sec-numerics']}.
• Figure 5: The two types of extremal surfaces, computed numerically at $t=0$ in the background geometry found numerically in section \ref{['sec-numerics']}. In this figure, we have taken $\theta = \pi/4$ and $x_{\partial}=1/2$. The horizontal black dotted lines $\partial$ at the top and bottom are the left and right baths. The dashed black line $\mathcal{B}$ along the left edge is the location of the brane, which contains the 4d black hole. The horizontal red dashed-dotted line in the middle is the 5d bifurcate horizon, which meets the brane at the 4d horizon. Compare with figure \ref{['fig:intdom']}. The orange curve corresponds to an extremal surface ending on the brane with $y_{\mathcal{B}}\approx0.31602(1)$, while the blue curve correspond to an extremal surface that penetrates the bifurcating Killing surface smoothly. There is, in fact, a continuous family of orange extremal surfaces and there is a unique one amongst them with the smallest area, see figure \ref{['fig:mar']}.