Failure of the split property in gravity and the information paradox

TL;DR

This work argues that the split property, central to many quantum information intuitions in nongravitational QFT, fails in gravity due to holography of information, meaning boundary observables determine the full state on a Cauchy slice. By identifying the precise point at which Hawking’s information-loss reasoning relies on a factorization of the Hilbert space, the authors show that the paradox is avoided once gravitational non-factorization is acknowledged, without invoking firewalls. They also explain why Page-curve calculations to date rely on nongravitational baths or massive gravity and do not apply to standard gravity, where the exterior entropy of an evaporating black hole does not generically follow a Page curve for fine-grained entropy. The discussion highlights that gravity localizes information in a way fundamentally different from nongravitational theories and that any Page-curve-like behavior requires coarse-graining or nonstandard gravitational dynamics, not a resolution within standard gravity alone.

Abstract

In an ordinary quantum field theory, the "split property" implies that the state of the system can be specified independently on a bounded subregion of a Cauchy slice and its complement. This property does not hold for theories of gravity, where observables near the boundary of the Cauchy slice uniquely fix the state on the entire slice. The original formulation of the information paradox explicitly assumed the split property and we follow this assumption to isolate the precise error in Hawking's argument. A similar assumption also underpins the monogamy paradox of Mathur and AMPS. Finally the same assumption is used to support the common idea that the entanglement entropy of the region outside a black hole should follow a Page curve. It is for this reason that computations of the Page curve have been performed only in nonstandard theories of gravity, which include a nongravitational bath and massive gravitons. The fine-grained entropy at ${\cal I}^{+}$ does not obey a Page curve for an evaporating black hole in standard theories of gravity but we discuss possibilities for coarse graining that might lead to a Page curve in such cases.

Figures (3)

• Figure 1: The configuration of interest. We study a region $R$ (blue), surrounded by a collar region $\epsilon$ (white) and the complement of $R \cup \epsilon$, which is denoted by $\overline{R}_{\epsilon}$ (purple). The region, $\overline{R}_{\epsilon}$ extends to infinity. In a nongravitational theory, the state on $\overline{R}_{\epsilon}$ and $R$ can be specified entirely independently but in a theory of gravity the state on $\overline{R}_{\epsilon}$ completely fixes the state on $R$.
• Figure 2: The extended Penrose diagram of an evaporating black hole. A common incorrect assumption is that the Hilbert space factorizes into a part $H_2$ associated with the interior and a part $H_3$ associated with the exterior, up to constraints imposed by mass and global charges. But the Hilbert space does not factorize on a nice slice or even on ${\cal I}^{+}$. Information available in the algebra of all operators on ${\cal I}^{+}$ (marked in blue) is available in the algebra ${\cal {\cal A}}_{\text{bdry}}$ of operators from the red region near ${\cal I}^{+}_{-}$.
• Figure 3: A two step process clarifying the nature of information transfer in computations of the Page curve. First, we prepare a black hole in AdS (left subfigure). According to the principle of holography of information, information about the microstate resides near the boundary of the spacetime. Second, we couple the system to a nongravitational bath (pink shaded region in the right subfigure). Information flows across a nongravitational interface (dashed line) and this information transfer is described by the Page curve.