Abstract
We use the replica method to compute the entanglement entropy of a universe without gravity entangled in a thermofield-double-like state with a disjoint gravitating universe. Including wormholes between replicas of the latter gives an entropy functional which includes an “island” on the gravitating universe. We solve the back-reaction equations when the cosmological constant is negative to show that this island coincides with a causal shadow region that is created by the entanglement in the gravitating geometry. At high entanglement temperatures, the island contribution to the entropy functional leads to a bound on entanglement entropy, analogous to the Page behavior of evaporating black holes. We demonstrate that the entanglement wedge of the non-gravitating universe grows with the entanglement temperature until, eventually, the gravitating universe can be entirely reconstructed from the non-gravitating one.
Highlights
If these ideas are fundamental to quantum gravity, they should apply generally, for any boundary conditions.1 To test this, we will set up the simplest possible scenario: two disjoint but entangled two-dimensional universes, both carrying quantum field theories, and one gravitating according to the Jackiw-Teitelboim (JT) model
We suppose that the entangled CFTs have a dual 3d gravitational description in terms of an eternal BTZ black hole whose boundaries coincide with the original 2d universes. (We want these boundaries to be segments, not circles, so we introduce end-of-the world branes to truncate the geometries.) In this setting, the CFT entanglement entropies can be computed via the Ryu-Takayanagi formula as the lengths of geodesics in the BTZ bulk
We find that the bulk entanglement wedge of the boundary non-gravitating universe is defined by geodesics that terminate at the two endpoints of the causal shadow region in the gravitating universe B
Summary
We consider two disconnected universes, A and B, both with Cauchy surfaces and both carrying quantum field theories. These two theories can be different in general, but for simplicity we will assume that they are two identical conformal field theories. When we search for combined saddle-points of the gravity-plus-field theory system, we will solve the gravitational equations of motion in B with a source given by the quantum expectation value of the stress tensor in some state that is entangled with the field theory A. Since the CFT degrees of freedom on B must be entangled with gravitational degrees of freedom, we can probe the unitarity of quantum gravity by forcing the entanglement between CFT fields on A and B to be large. We will see that monogamy of entanglement between the gravitational degrees of freedom on B, matter fields on B, and matter fields on A will lead to large modifications of the entanglement entropy of the field theory state (2.2) despite the fact that the gravitational microstates are not explicitly included
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