Abstract
We consider black holes in 2d de Sitter JT gravity coupled to a CFT, and entangled with matter in a disjoint non-gravitating universe. Tracing out the entangling matter leaves the CFT in a density matrix whose stress tensor backreacts on the de Sitter geometry, lengthening the wormhole behind the black hole horizon. Naively, the entropy of the entangling matter increases without bound as the strength of the entanglement increases, but the monogamy property predicts that this growth must level off. We compute the entropy via the replica trick, including wormholes between the replica copies of the de Sitter geometry, and find a competition between conventional field theory entanglement entropy and the surface area of extremal “islands” in the de Sitter geometry. The black hole and cosmological horizons both play a role in generating such islands in the backreacted geometry, and have the effect of stabilizing the entropy growth as required by monogamy. We first show this in a scenario in which the de Sitter spatial section has been decompactified to an interval. Then we consider the compact geometry, and argue for a novel interpretation of the island formula in the context of closed universes that recovers the Page curve. Finally, we comment on the application of our construction to the cosmological horizon in empty de Sitter space.
Highlights
Trick [9, 10]
Tracing out the entangling matter leaves the CFT in a density matrix whose stress tensor backreacts on the de Sitter geometry, lengthening the wormhole behind the black hole horizon
Our result suggests that the island formula, which is derived within the framework of semiclassical gravity coupled to effective field theory, can detect the finite dimensionality of the de Sitter quantum gravity Hilbert space
Summary
The island formula instructs us to minimize and extremize the generalized entropy, i.e. the surface area of the island plus effective field theory entropy of the island plus the auxiliary system, over all possible subregions of the gravitating universe. There would have to be subtle gravitational constraints which modify the axioms we have used in this argument in an unknown way which only appears in the discussion of closed universes Both of these options are clearly in tension with general beliefs, for example, that the Hilbert space of de Sitter quantum gravity has dimension of order e1/GN [29, 35].4. In the weak entanglement regime, the island formula interpreted this way would lead us to conclude that the entanglement of the auxiliary system with the closed universe is just the naive field theory entropy. When the entanglement becomes sufficiently large, the area of the small sphere dominates the entropy calculation In this way, it is possible for a auxiliary system to be entangled with a closed, gravitating universe while respecting the Page behavior. We re-compactify the extended geometry by introducing identifications on the spatial slice, recovering our proposed prescription for islands on compact universes
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