Inconsistency of Islands in Theories with Long-Range Gravity

TL;DR

The paper identifies a fundamental inconsistency for islands—entanglement wedges disconnected from the asymptotic boundary—in theories with long-range gravity due to the Gauss-law constraint, which would let island excitations be detected outside the island. It argues that this puzzle is resolved when the graviton is effectively massive, as realized on AdS branes (Karch-Randall setups), where dimensional reduction yields a lower-dimensional massive gravity that lacks a Gauss law, allowing island degrees of freedom to be described by the radiation region without conflicting with the exterior. The authors develop a detailed analysis of the gravitational constraints in flat space and in AdS spacetimes with branes, demonstrating how the KK mass spectrum $m_n^2>0$ arises and how the reduced constraints permit consistent island physics. They also formulate a consistency condition for islands in decoupled systems, which disfavors islands as interior degrees of freedom for evaporating black holes in higher dimensions. Overall, the work suggests islands are not generic features of massless gravity and that their existence hinges on graviton mass or bath-induced effects, with significant implications for the applicability of the island rule beyond JT-like models.

Abstract

In ordinary gravitational theories, any local bulk operator in an entanglement wedge is accompanied by a long-range gravitational dressing that extends to the asymptotic part of the wedge. Islands are the only known examples of entanglement wedges that are disconnected from the asymptotic region of spacetime. In this paper, we show that the lack of an asymptotic region in islands creates a potential puzzle that involves the gravitational Gauss law, independently of whether or not there is a non-gravitational bath. In a theory with long-range gravity, the energy of an excitation localized to the island can be detected from outside the island, in contradiction with the principle that operators in an entanglement wedge should commute with operators from its complement. In several known examples, we show that this tension is resolved because islands appear in conjunction with a massive graviton. We also derive some additional consistency conditions that must be obeyed by islands in decoupled systems. Our arguments suggest that islands might not constitute consistent entanglement wedges in standard theories of massless gravity where the Gauss law applies.

Figures (11)

• Figure 1: A cartoon of a constant-time slice of a black hole with a brane embedded. $R$ is the union of regions on two asymptotic boundaries and $\overline{R}$ is its complementary region. The horizons in the bulk are marked by $H$. The separation between horizons is meant to convey that the Cauchy slice under examination is a late-time slice on which the wormhole is of a finite length. The dominant RT surface for the region $R$ is shown in purple. The region on the brane marked $I$ becomes the "island" in the lower-dimensional picture of Figure \ref{['figlowerd']}. In this figure, both the horizontal and the vertical directions are spatial.
• Figure 2: A spacetime diagram of the same system of branes in a black hole in the $d$ dimensional description. The entanglement wedge for the region $R$ is now an "island". In this figure, the horizontal direction is spatial and time runs along the vertical direction.
• Figure 3: A point $P$ that is part of multiple entanglement wedges. The quasilocal bulk operator must be dressed to $R_1$ on the left and to $R_2$ on the right. The figure shows a time slice of global AdS.
• Figure 4: An excitation at the point $P$ inside the island (pink shaded disk) can be detected using a two-point correlator outside the island. The two-point correlator involves an integral of the asymptotic metric (indicated by the dashed line) and another operator obtained by taking the limit of point $P'$ to $P_{B}$ on the boundary of the gravitational region. The bath is not shown in this figure, which shows a time slice of AdS.
• Figure 5: In the higher-dimensional setup, an operator at point $P$ can be dressed to the boundary through the higher-dimension. The setup is the same as that of Figure \ref{['fighigherd']} and this Figure illustrates how an operator at $P$ can be dressed to the boundary while bypassing the entanglement wedge of $\overline{R}$. In this figure both the horizontal and vertical directions are spatial.

Theorems & Definitions (1)

• Definition