Complete Einstein equations from the generalized First Law of Entanglement

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.

Figure 1

(a) Entanglement wedge and flow of vector field $\xi$ and $V$. (b) Flow of vector field $V$ within $\mathrm{\Sigma }$. The red circle is the Ryu-Takayanagi surface.

Figure 2

(a) Flow of the entanglement vector field $V_E$. (b) Cartoon of three-dimensional version of left figure where it is rotated around the $z$ axis.