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.
Highlights
One of the most inspiring ideas in the recent development of string theory is the suggestion [1,2] that the classical spacetime is a consequence of the quantum entanglement without which two nearby regions of spacetime would come apart [1,2] and, that the Einstein equation itself is coming from a relation of entanglement entropy at least at the linearized level [3]
The difference in the recent activity [3,6] is that the entanglement first law and its gravity dual are derived from the conformal field theory (CFT), it gives only a linearized equation
Recent activities aim to derive the Einstein equation of the dual gravity of a CFT assuming the presence of holography
Summary
One of the most inspiring ideas in the recent development of string theory is the suggestion [1,2] that the classical spacetime is a consequence of the quantum entanglement without which two nearby regions of spacetime would come apart [1,2] and, that the Einstein equation itself is coming from a relation of entanglement entropy at least at the linearized level [3] The latter is a consequence of connecting two different descriptions of entanglement entropy (EE): one as the area of the Ryu-Takayanagi surface [4] and the other as the expectation value of the modular Hamiltonian [5]. The flux line, once the total flux is quantized, is analogous to the microscopic wormhole and is concentrated along the boundary of the entangled regions
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