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We consider the emergence from quantum entanglement of spacetime geometry ina bulk region. For certain classes of quantum states in an appropriatelyfactorized Hilbert space, a spatial geometry can be defined by associatingareas along codimension-one surfaces with the entanglement entropy betweeneither side. We show how Radon transforms can be used to convert this data intoa spatial metric. Under a particular set of assumptions, the time evolution ofsuch a state traces out a four-dimensional spacetime geometry, and we argueusing a modified version of Jacobson's "entanglement equilibrium" that thegeometry should obey Einstein's equation in the weak-field limit. We alsodiscuss how entanglement equilibrium is related to a generalization of theRyu-Takayanagi formula in more general settings, and how quantum errorcorrection can help specify the emergence map between the full quantum-gravityHilbert space and the semiclassical limit of quantum fields propagating on aclassical spacetime.

Bulk Entanglement Gravity without a Boundary: Towards Finding Einstein's Equation in Hilbert Space