Complete Einstein equation 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 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.

5 pages, 4 figures, v2: title shortened, v4: proof improved. result enhanced v5: result restated as "gravity dual expression of relative entropy is equivalent to Einstein equation"

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• Emergent gravity in superplastic crystals
• Off-diagonal cosmological solutions in emergent gravity theories and Grigory Perelman entropy for geometric flows
• Bit Threads: From Entanglement to Geometric Entropies