Causality Criteria for Island Models

TL;DR

The paper investigates how micro-causality emerges in island constructions and clarifies when effective descriptions inherit bulk locality. It distinguishes bulk-first from boundary-first double holography and argues that nonlocal operator reconstruction does not create new propagation channels. A three-part criterion, $L$, $D$, and $M$, is proposed and shown to be sufficient (and, in practice, necessary) for ensuring effective micro-causality from bulk micro-causality; brane-world DES satisfy all three and remain causally consistent, even in time-dependent processes like island formation and evaporation. This structural perspective provides a robust framework for understanding causal consistency in emergent spacetime and guides future explorations in more general and dynamical holographic settings.

Abstract

Island models offer a compelling resolution of the black hole information paradox, but they also raise persistent questions about causal consistency in effective descriptions. In particular, effective theories arising in double holography can exhibit apparent violations of micro-causality, despite the underlying bulk dynamics being local and causal. The aim of this work is to clarify the physical origin of this phenomenon and to identify the structural features that control causal consistency in island models. We argue that the apparent non-causality in double holography is neither intrinsic to island physics nor a consequence of nonlocal operator reconstruction. Rather, it reflects a mismatch between effective spacetime separation and bulk causal accessibility, a feature already implicit in earlier analyses. Nonlocal reconstruction instead encodes quantum error correction within a restricted code subspace and does not introduce independent propagation channels. Motivated by this perspective, we formulate a structural criterion for micro-causality in effective island descriptions. The criterion consists of three conditions: the absence of independent propagation channels beyond those of the bulk theory, a local bulk-supported operator dictionary, and a consistent matching between effective spacelike separation and dynamically accessible bulk causal curves. When these conditions are satisfied, effective micro-causality follows directly from bulk micro-causality. We apply the criterion to brane world realizations of island models, including the defect-extremal-surface construction, and show that they preserve causal consistency, in contrast to double holography. We further demonstrate that the criterion remains robust in time-dependent processes such as island formation and evaporation.

Figures (5)

• Figure 1: Three interrelated descriptions in double holography: the one-dimensional boundary description, the two-dimensional effective boundary description with gravity, and the three-dimensional bulk description with a Planck brane.
• Figure 2: Causal wedges of a bath interval and its island can either intersect (left) or remain disjoint (right) in bulk-first double holography, despite the regions being disconnected in the effective boundary description.
• Figure 3: Distinct notions of spacetime separation for operators localized on the Planck brane $Q$ and the asymptotic boundary $\Sigma$, viewed from the bulk perspective.
• Figure 4: Effective spacetime separation between an operator on $Q$ and an operator on $\Sigma$, defined intrinsically in the effective boundary description.
• Figure 5: In brane world constructions with localized field-theory degrees of freedom, the causal wedge of an island region on the brane can never intersect the causal wedge of a bath interval on the asymptotic boundary. This geometric property follows directly from the absence of independent bulk propagation channels from the brane to the bath.