Complete Einstein equation from the generalized First Law of Entanglement

TL;DR

This paper demonstrates that the full non-linear Einstein equation can be derived from the gravity dual of relative entropy, extending previous linearized results, and introduces an entanglement vector field with flux equal to entanglement entropy.

Contribution

It shows that the gravity dual of relative entropy encodes the complete Einstein equation, and constructs an entanglement vector field representing entanglement entropy flux.

Findings

Full non-linear Einstein equation derived from entanglement principles.

Introduction of an entanglement vector field with flux equal to entanglement entropy.

Connection between entanglement flow and spacetime geometry sewing.

Abstract

Recently it was observed that the first law of Entanglement leads to the linearized Einstein equation. In this paper, we point out that the gravity dual of an relative entropy expression is equivalent to the full non-linear Einstein equation. We also construct an entanglement vector field $V_{E}$ whose flux is the entanglement entropy. The flow of the vector field looks like sewing two space regions along the interface.