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

TL;DR

This paper explores how spacetime geometry and Einstein's equations can emerge from quantum entanglement in Hilbert space, using Radon transforms and entanglement equilibrium principles.

Contribution

It introduces a framework linking quantum entanglement to spacetime geometry without boundary conditions, utilizing Radon transforms and a generalized entanglement equilibrium approach.

Findings

Spacetime geometry can be reconstructed from entanglement entropy data.

Radon transforms enable conversion of entanglement data into a spatial metric.

Under certain assumptions, the emergent geometry obeys Einstein's equations in the weak-field limit.

Abstract

We consider the emergence from quantum entanglement of spacetime geometry in a bulk region. For certain classes of quantum states in an appropriately factorized Hilbert space, a spatial geometry can be defined by associating areas along codimension-one surfaces with the entanglement entropy between either side. We show how Radon transforms can be used to convert this data into a spatial metric. Under a particular set of assumptions, the time evolution of such a state traces out a four-dimensional spacetime geometry, and we argue using a modified version of Jacobson's "entanglement equilibrium" that the geometry should obey Einstein's equation in the weak-field limit. We also discuss how entanglement equilibrium is related to a generalization of the Ryu-Takayanagi formula in more general settings, and how quantum error correction can help specify the emergence map between the full quantum-gravity Hilbert space and the semiclassical limit of quantum fields propagating on a classical spacetime.