Failure of the split property in gravity and the information paradox

In an ordinary quantum field theory, the "split property" implies that the state of a 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. It can be shown in specific examples that observables near the boundary of a 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 recent computations of the Page curve have been performed only in nonstandard theories of gravity, which include a nongravitational bath and massive gravitons. We discuss possibilities for coarse-graining that might lead to a Page curve in standard theories of gravity.