Seeing Page Curves and Islands with Blinders On

TL;DR

The paper argues that gravity’s holography of information makes asymptotic observables complete, so the interior of a black hole is in principle accessible from the exterior and the bulk Hilbert space does not factorize as $H_{ ext{full}} = H_{ ext{in}} \,\otimes\, H_{ ext{out}}$. Consequently, Page curves do not reflect fundamental information recovery in standard gravity; they arise only when one artificially restricts the exterior algebra (or couples to a nongravitational bath). Islands, defined as entanglement wedges not touching infinity, are inconsistent in standard gravity but can appear in nonstandard gravity or behind double horizons; relational observables fail to salvage a nonperturbative Page-curve and fine-grained entropy. The paper also discusses how information transfer into a bath can be understood through holography of information and outlines open questions about small AdS black holes and natural algebras. Overall, the results clarify the correct gravitational framework for black hole information and outline when Page curves and islands may appear, pointing to the boundary as the true repository of information in standard gravity.

Abstract

This paper summarizes recent discussions of the Page curve and the information paradox, and responds to the reasoning and examples from arXiv:2506.04311. We review arguments demonstrating that in quantum gravity the algebra of observables at infinity is complete, both in AdS and in asymptotically flat space. This completeness implies that the bulk Hilbert space in quantum gravity does not factorize along the radial direction, undermining a key common assumption in Hawking's argument for information loss and in initial derivations of the Page curve. As a consequence, in a standard theory of gravity, information does not ``emerge'' from a black hole in the manner suggested by the Page curve; rather, it is already encoded in asymptotic observables. Relatedly, the full black hole interior, and not just an ``island'', can be reconstructed from exterior data. Page curves and islands can be obtained by removing the Hamiltonian from the exterior algebra. This may be implemented operationally by restricting access to part of the asymptotic region (a detector with a ``blind spot'') or, in the special case of null infinity in asymptotically flat spacetimes, by formally discarding the Hamiltonian from the set of observables despite its physical accessibility. Such Page curves describe only the redistribution of information between measured and unmeasured degrees of freedom, rather than fundamental information recovery. Finally, Page curves and islands also arise when a black hole is coupled to a nongravitational bath, a setup that yields a nonstandard theory of gravity. We show how, even in this setting, the unusual localization of information in gravity provides a concrete physical mechanism for information transfer from the gravitational system into the bath.

Figures (22)

• Figure 1: Entanglement entropy of a boundary region plotted against its angular size for a single-sided black hole in AdS dual to a pure state. See section \ref{['subsectrivialpage']} for details of the simple computation.
• Figure 2: Flowchart for section \ref{['sechawkingpage']}.
• Figure 3: A nice slice in a black hole geometry. Hawking assumed that the Hilbert space would factorize into a part associated with the interior ${\cal H}_{\text{in}}$ and another part associated with the exterior ${\cal H}_{\text{out}}$.
• Figure 4: The Page curve Page:1993dfPage:1993wv was obtained by modeling the black hole and its radiation as a bipartite system $\mathcal{H} \simeq \mathcal{H}_m \otimes \mathcal{H}_{n}$, with $\operatorname{dim} \mathcal{H}_m=m$ and $\operatorname{dim} \mathcal{H}_{n}=n$ for the black hole and radiation subsystems, respectively. Page expected the subsystems to be nearly maximally mixed, so that the fine-grained entropy nearly saturates the bound set by the smaller Hilbert space $\log(\text{min}(m,n))$. By assuming the joint system is in a Haar random pure state, Page modeled the fine-grained entropy: for $n \ge m$, $S_{m,n} \approx \log m-\frac{m}{2 n}$; for $m \ge n$, $S_{m,n} \approx \log n-\frac{n}{2 m}$. The coarse-grained entropy of the radiation $s_r \approx \log m$ increases with time, while the coarse-grained entropy of the black hole $s_{h} \approx \log n$ decreases; the fine-grained entropy $S_{m,n}$ begins to decrease around when the coarse-grained entropies become comparable at the Page time.
• Figure 5: The left subfigure shows an asymptotic time band on the AdS boundary. In a gravitational theory, the algebra of a time band has no commutant. The right subfigure is the familiar figure in a nongravitational system in which the algebra of a time band corresponds to the bulk algebra on the $t=0$ slice for $r \in (\tan{\pi - T \over 2}, \infty)$ but has a nonzero commutant corresponding to the algebra of a diamond near the middle of AdS.

Theorems & Definitions (1)

• Definition