Failure of the split property in gravity and the information paradox

TL;DR

This paper demonstrates that the split property, crucial in quantum field theory, fails in gravity, affecting the assumptions behind the black hole information paradox and the expected Page curve of entanglement entropy.

Contribution

It clarifies the failure of the split property in gravitational theories and analyzes its implications for the information paradox and entanglement entropy computations.

Findings

Split property does not hold in gravity.

Standard assumptions in the information paradox are invalid.

Page curve computations require nonstandard gravity theories.

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.