Seeing Page Curves and Islands with Blinders On

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.

💡 Research Summary

The paper “Seeing Page Curves and Islands with Blinders On” revisits the black‑hole information paradox, the Page curve, and the recent “island” prescription, responding specifically to the arguments presented in arXiv:2506.04311. The authors begin by reviewing a theorem that the algebra of observables at infinity is complete in quantum gravity, both in asymptotically AdS and asymptotically flat spacetimes. Because the Hamiltonian is a boundary term, it belongs to this asymptotic algebra and can be used by an exterior observer to evolve boundary operators back in time. Consequently, exterior observables already contain full information about the interior; the bulk Hilbert space cannot be factorized radially. This “principle of holography of information” invalidates the key assumption—used by Hawking and by Page—that interior and exterior operator algebras commute.

Armed with this principle, the authors argue that the usual Page curves found in the literature do not represent genuine information release from a black hole. A Page curve appears only when one artificially removes the Hamiltonian from the exterior algebra or restricts measurements to a sub‑region of the boundary (a “blind spot”). In AdS, dividing the boundary sphere into two parts and measuring only one yields an entanglement entropy that follows a Page‑like curve as a function of the solid angle, but this merely reflects redistribution of already‑available information between measured and unmeasured degrees of freedom. In asymptotically flat space, one can formally drop the Hamiltonian at future null infinity where the algebra becomes free, but this truncation is physically unnatural because realistic observables (e.g., the Riemann tensor) inevitably encode the Hamiltonian.

The paper then turns to the island prescription. The authors define islands as compact entanglement wedges that do not extend to the asymptotic region. Since exterior observables have access to the entire bulk, such compact wedges cannot have commuting algebras with their complements; any island that excludes part of the boundary necessarily omits the Hamiltonian and is therefore inconsistent in a standard theory of gravity. They acknowledge that “extended islands” which include a portion of the asymptotic boundary can be constructed, but these are essentially the same as the blind‑spot constructions that already require discarding the Hamiltonian.

When a black hole is coupled to a non‑gravitational bath, the combined system no longer obeys the standard constraints of pure‑gravity dynamics; the authors refer to this as a “non‑standard” theory of gravity. In this setting the Hamiltonian is shared with the bath, and information can be transferred from the gravitational sector to the bath in a way that reproduces the Page curve and island formulas. The paper shows that even here the underlying mechanism is the holographic redundancy of bulk information in the asymptotic observables; the bath merely provides a convenient external register for that information.

In summary, the authors claim that existing Page‑curve and island calculations are not mathematically wrong, but they obscure the essential gravitational physics. The Page curves obtained by removing the Hamiltonian or by imposing a blind spot describe only the flow of information between measured and unmeasured sectors, not a genuine recovery of information from the black hole interior. The principle of holography of information resolves Hawking’s paradox directly: the information is already present at infinity, and no additional “emergent” Page curve is needed. Only in non‑standard setups with a bath does a Page‑like behavior arise, and even then it reflects the same holographic redundancy rather than a new dynamical process.