Information Transfer with a Gravitating Bath

TL;DR

This paper investigates how entanglement islands and Page curves extend to gravitational baths using Karch-Randall braneworlds. It develops a doubly-holographic, quantum-extremal-surface framework to compute entanglement entropies for radiation and for left/right divisions in a gravitating setting. A key result is that the entropy of radiation on a gravitating bath is time-independent, while a left/right entanglement entropy can exhibit a Page curve only when both branes lie below a Page angle, with a universal critical angle theta_c controlling island existence. The findings reveal sharp, angle-dependent transitions and discontinuous entropy behavior that are robust across dimensions, shedding light on information transfer in gravity-augmented holographic systems.

Abstract

Late-time dominance of entanglement islands plays a critical role in addressing the information paradox for black holes in AdS coupled to an asymptotic non-gravitational bath. A natural question is how this observation can be extended to gravitational systems. To gain insight into this question, we explore how this story is modified within the context of Karch-Randall braneworlds when we allow the asymptotic bath to couple to dynamical gravity. We find that because of the inability to separate degrees of freedom by spatial location when defining the radiation region, the entanglement entropy of radiation emitted into the bath is a time-independent constant, consistent with recent work on black hole information in asymptotically flat space. If we instead consider an entanglement entropy between two sectors of a specific division of the Hilbert space, we then find non-trivial time-dependence, with the Page time a monotonically decreasing function of the brane angle -- provided both branes are below a particular angle. However, the properties of the entropy depend discontinuously on this angle, which is the first example of such discontinuous behavior for an AdS brane in AdS space.

Figures (10)

• Figure 1: (a) Embedding of a KR braneworld with two subcritical branes in anti-de Sitter space. We refer to the left brane as the "physical" brane, and the right brane as the "bath" brane. $\mathcal{R}$ denotes the radiation region whose entanglement entropy we wish to calculate, and the dashed green line connected to the boundary of $\mathcal{R}$ is the candidate RT surface. $\mathcal{I}$ is the entanglement island on the brane corresponding to the RT surface in green. (b) Embedding of a KR braneworld with the same tensions in the black string geometry. The dashed black line is the black string horizon separating the exterior and interior regions.
• Figure 2: A constant-$t$ slice of global AdS$_{d+1}$ with two branes present. The defect is shown in red, and various candidate extremal surfaces (in order of decreasing area from left to right) are in green. For each surface, $\mathcal{R}$ and $\mathcal{I}$ are respectively the radiation region and island, which end orthogonally on both branes. The minimal, zero-area entanglement surface $r(\mu) = 0$ is shown on the right as a limit, cutting through the middle of the space.
• Figure 3: The critical anchor on the brane, given here in four dimensions, is a monotonically decreasing function of the brane angle which vanishes at the critical angle, which in this case is approximately $\theta_c \approx .98687$. The critical anchor tends toward the horizon distance ($u_h = 1$) as $\theta \to 0$ and tends toward zero as $\theta \to \theta_c$.
• Figure 4: Overview for potential island surfaces within the black string geometry, each of which satisfies the boundary condition for some corresponding brane placed below the critical angle. Each surface ends at its critical anchor, which is where the island begins. One loses the island surfaces as the angle decreases, and the island fills the brane at the critical angle. The area difference between the Hartman-Maldacena surface and the island surface vanishes slightly below the critical angle. While one would need many digits of precision to show this, the island surfaces appear to fill the island commonwealth, labelled as region I, without crossing one another. They never make it into the region II, which we call independent territory.
• Figure 5: The $t = 0$ area difference \ref{['areaDiff']} between the Hartman-Maldacena surface and the island surface, given as a function of brane angle is in orange, given for the black string at $d=4$. This should be compared to the behavior of the brane's induced Newton's constant $G_N^{-1} \sim 1/\theta^2$ in blue. As we approach the critical angle the area of the island surface approaches the area of Hartman-Maldacena in empty AdS. This means the difference becomes negative in the region near the critical angle, which for $d=4$ is approximately $\theta_c \approx 0.98687$.