Black Hole Interior and Quantum Error Correction with Dynamical Gravity

Abstract

According to the island formula, information in the code subspace defined in the black hole interior is embedded in the Hawking radiation after the Page time. At first sight, this embedding suggests that operations acting on the Hawking radiation could modify the information in the code subspace, potentially leading to an apparent violation of causality. Indeed, in previous studies based on the PSSY model, which incorporates only the topological degrees of freedom of gravity, it was shown that when the error is sufficiently large, a violation of causality can arise, as indicated by a nonvanishing mutual information. In this paper, we investigate the situation in which dynamical gravity also acts on the Hawking radiation. In this case, operations on the Hawking radiation induce nontrivial backreaction on the bulk spacetime appearing in the gravitational path integral for the mutual information -- an effect that is absent when the Hawking radiation is non-gravitating. We find that this backreaction renders the relevant mutual information vanishing. This result implies that, in theories with dynamical gravity, the apparent violation of causality is resolved.

The paper asks whether operations on Hawking radiation can disturb black-hole interior information encoded through an island, and whether gravitational backreaction changes the associated error-correction condition. It models a gravitating radiation bath as a second entangled black hole, with interior and exterior excitations represented by flavored code branes, and diagnoses decoupling using Rényi-2 mutual information between an interior reference and the exterior reference plus error environment. In the doubled topological PSSY/West Coast model, gravitational path integrals evaluated by dominant saddles give nonzero mutual information only in the post-Page window $d_{BH}<k<d_{BH}d_{in}$, for errors with sufficiently negative Rényi-2 coherent information. Here $k$ controls inter-system entanglement and $d_{in}$ is the interior code dimension. The nonzero contribution arises from a fully connected replica geometry. In dynamical JT gravity, errors are modeled as boundary scaling operators dual to massive branes of tension $\Delta$. Their backreaction introduces boundary cusps, adding approximately $2\Delta$ to the connected saddle’s entropy contribution. This shrinks the non-decoupling region and eliminates it for $\Delta>\tfrac12\log d_{in}$; pre-Page mutual information remains zero at leading saddle order. The result is conditional on this error model: sufficiently negative coherent information is not generally shown to imply large $\Delta$. A thermal holographic-CFT environment supplies an illustrative relation. Other limitations include prescription-dependent interior branes, subdominant-saddle corrections, neglected topology changes, and an unresolved microscopic protection mechanism. Microstate-dependent recovery operations and extensions beyond this two-dimensional setup remain open.