Entanglement islands in higher dimensions (1911.09666v3)

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.

• The paper presents a numerical study of entanglement islands in 5D AdS gravity, extending the Page curve analysis to higher dimensions.
• It employs a Randall-Sundrum brane and Neumann boundary conditions to compute quantum extremal surfaces as proxies for entanglement entropy.
• The findings reveal an early-time horizon-penetrating entropy growth that transitions to a stable island-dominated saturation, in line with unitarity.

Analyzing "Entanglement Islands in Higher Dimensions"

The paper "Entanglement islands in higher dimensions" by Almheiri, Mahajan, and Santos provides a numerical investigation of entanglement islands within the context of higher-dimensional asymptotically AdS gravity. Previous studies on the Page curve of Hawking radiation have established that entanglement islands in the gravity region are crucial to addressing the information paradox in black hole physics, particularly within two-dimensional Jackiw-Teitelboim (JT) gravity. The authors aim to extend these concepts to higher-dimensional setups.

Summary and Computational Approach

The central focus of this work is on a five-dimensional (5D) asymptotically Anti-de Sitter (AdS) geometry that can model a four-dimensional (4D) Hartle-Hawking state of an eternal AdS black hole coupled with a bath. By considering 4D AdS black holes in equilibrium with a bath, this paper extends the investigation of extremal surfaces related to entanglement entropy beyond two dimensions. Specifically, the approach involves solving Einstein’s equations with specific boundary conditions through numerical implementation using the DeTurck trick.

The authors employ a Randall-Sundrum type brane within the 5D setup to facilitate the computations of quantum extremal surfaces as RT surfaces in 5D. This reduces the complexity of the problem, allowing for the construction of a static geometry that represents a higher-dimensional model for studying quantum entanglement properties. The use of Neumann boundary conditions set on the brane allows the numerical solutions to represent an equilibrium state of a black hole geometry intersected by the brane.

Numerical Results

The paper successfully constructs a geometry where quantum extremal surfaces manifest within the entanglement wedge. The authors identify two distinct classes of extremal surfaces: one penetrating the horizon, increasing linearly with time due to the horizon's internal geometry, and the other terminating on the brane, remaining outside the horizon. The variety and switching of these surfaces as a function of time illustrates the existence of entanglement islands—an effect that incorporates higher-dimensional interactions into traditional notions of black hole information theory.

One key finding is that at early times, the entropy derived from extremal surfaces penetrating the horizon grows due to spacetime stretching within the black hole. However, in later times, the entropy saturates, consistent with unitarity, as the surfaces ending on the brane dominate the dynamics. In the terminology of black hole thermodynamics, this behavior provides a compelling description of the transition phases of the entanglement entropy, aligning with the principles of holography.

Implications and Future Directions

This paper makes significant strides toward generalizing lower-dimensional insights (from 2D gravity) into more physically realistic higher-dimensional scenarios, confronting limitations posed by previous studies confined to lower dimensions. Thus, it alleviates concerns about the applicability of JT gravity results to broader spacetimes and sets a precedent for future explorations of quantum gravity phenomena, including information recovery from evaporating black holes.

Further research could involve analyzing time-dependent extensions to explore real-time evolution and potential deviations introduced by quantum corrections, such as greybody factors. Furthermore, the approach lays a foundation for newly examining quantum information-theoretic frameworks and their emergent holographic dualities with implications for unifying quantum mechanics with general relativity. The methodology could also extend to include string-theoretical backgrounds where fully quantum gravitational effects may reveal deeper corporate structures of spacetime geometry.

In conclusion, this research elucidates critical insights into how higher-dimensional environments can host analogous features found in lower-dimensional toy models, specifically in resolving the information paradox with entanglement islands—a key advance in marrying holographic principles with practical computational techniques.