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Failure of the split property in gravity and the information paradox
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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.
Forward citations
Cited by 5 Pith papers
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Hollow-grams: Generalized Entanglement Wedges from the Gravitational Path Integral
The entropy of a bulk region in holographic states equals the generalized entropy of the smallest wedge containing it, derived from a replica path integral via a hollow-graphic construction.
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Cosmological correlators in gravitationally-constrained de Sitter states
Cosmological correlators in gravitationally constrained de Sitter states are conformally invariant and differ from QFT vacuum correlators, but relational observables with a heavy background state can reproduce QFT results.
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An apologia for islands
Entanglement islands and Page curves can arise in massless gravity without an external bath, and compactly supported gauge-invariant operators exist in islands around generic symmetry-breaking black hole backgrounds.
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Wheeler-DeWitt equation and Bondi-Metzner-Sachs (BMS) symmetry
BRST-invariant extensions of the BMS generators are constructed and used to define matrix elements between Wheeler-DeWitt states.
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The Mechanism behind the Information Encoding for Islands
Island information is encoded in the bath because island operators are gravitationally dressed to the bath via the massive-graviton Stückelberg field, realized as Wilson lines in the Karch-Randall braneworld.
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