Abstract
In ordinary gravitational theories, any local bulk operator in an entanglement wedge is accompanied by a long-range gravitational dressing that extends to the asymptotic part of the wedge. Islands are the only known examples of entanglement wedges that are disconnected from the asymptotic region of spacetime. In this paper, we show that the lack of an asymptotic region in islands creates a potential puzzle that involves the gravitational Gauss law, independently of whether or not there is a non-gravitational bath. In a theory with long-range gravity, the energy of an excitation localized to the island can be detected from outside the island, in contradiction with the principle that operators in an entanglement wedge should commute with operators from its complement. In several known examples, we show that this tension is resolved because islands appear in conjunction with a massive graviton. We also derive some additional consistency conditions that must be obeyed by islands in decoupled systems. Our arguments suggest that islands might not constitute consistent entanglement wedges in standard theories of massless gravity where the Gauss law applies.
Highlights
Is a manifestation of a more general phenomenon: when a gravitational theory in AdS is coupled to a non-gravitational bath, the graviton in AdS picks up a mass
It is sometimes believed that the non-gravitational bath and the massive graviton that appear in such models are merely technicalities, and that the general lessons regarding islands and the Page curve should be applicable to other physical systems including realistic black holes in asymptotically flat space [41, 42]
When gravity is dynamical in the bath, as it should be in realistic models of black holes, it was found that the fine-grained entropy of radiation was constant, consistent with the previously obtained results [44] that the Page curve of radiation is trivial for black holes in asymptotically flat spaces
Summary
In this paper we will use the phrase “island” strictly in accordance with the following definition. When we refer to the Gauss law in this paper, we are referring to the relationship between the total energy on a Cauchy slice and the integral of an appropriate component of the asymptotic metric This relationship follows from the Hamiltonian constraint in standard theories of gravity. We adopt the perspective that a consistent entanglement wedge is one where it is possible to reconstruct approximately local bulk operators that, in the limit pl → 0, reduce to standard quantum-field operators. The conclusion of [44], which is consistent with the results of this paper, can be interpreted as the claim that information about a black hole microstate is always available outside for sufficiently detailed measurements in a standard theory of long-range gravity This could not possibly be true in a local quantum field theory, where spacelike-separated operators commute.
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