Entropic Interpretation of Einstein Equation in dS/CFT

TL;DR

This paper shows that the first law of holographic pseudo-entropy in a 2D CFT is equivalent to Einstein's equations in 3D de Sitter space, revealing a connection between pseudo-entropy, complex geodesics, and emergent time.

Contribution

It establishes a novel link between holographic pseudo-entropy and Einstein equations in dS space, highlighting the emergence of time from Euclidean CFT in dS/CFT.

Findings

Pseudo-entropy obeys a Klein-Gordon equation in dS space.

Geodesics for pseudo-entropy involve complex and timelike curves.

First law of pseudo-entropy corresponds to Einstein's equations in dS.

Abstract

In this paper, we demonstrate that the first law of holographic pseudo-entropy, which is a non-Hermitian generalization of entanglement entropy in a two-dimensional conformal field theory (CFT), is equivalent to the perturbative Einstein equation in three-dimensional de Sitter (dS) space, assuming the dS/CFT correspondence. Our analysis reveals that the geodesic that accurately satisfies the first law of holographic pseudo-entropy consists of a timelike curve and a curve whose coordinates are complex. We also demonstrate that infinitesimal changes to the pseudo entropy satisfy a Klein-Gordon equation in two-dimensional de Sitter space. These imply the emergence of a time coordinate from a Euclidean CFT in dS/CFT.