Complete Einstein equations from the generalized First Law of Entanglement

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 nonlinear 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.