Entropic Interpretation of Einstein Equation in dS/CFT

TL;DR

This work extends the holographic relation between entanglement and spacetime dynamics to de Sitter holography by studying the first law of holographic pseudo entropy in $dS_3$$/CFT_2$. Through analytic continuation from AdS$_3$ to $dS_3$ and by allowing complexified geodesics, the authors show that the perturbative Einstein equation in $dS_3$ is equivalent to a first-law condition for pseudo entropy, with infinitesimal changes $\Delta S_A$ obeying a Klein-Gordon equation on $dS_2$, signaling an emergent time from the Euclidean CFT. Real-space geodesic ansätze fail to reproduce the first law, but complex geometry resolves this, linking holographic pseudo entropy to a complexified bulk geometry and reinforcing the role of non-unitary CFTs in dS/CFT. The analysis highlights a consistent framework where the holographic pseudo entropy on the $dS_3$ boundary encodes the bulk dynamics, and where time may emerge from Euclidean data via a $dS_2$ scalar equation, suggesting deep connections between complexified bulk geodesics, emergent time, and quantum gravity in de Sitter space.

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.

Figures (6)

• Figure 1: Contours of $\theta$ integral of the entropy calculation in the AdS$_3$ (left) and dS$_3$ (right). In the above picture we took $\theta_1+\theta_2=\pi$ and thus $\psi_1=0$ and $\psi_2=\pi$.
• Figure 2: dS geodesics in global patch and Poincaré patch. To get the whole length of half Lorentzian region, we have to calculate (I)$+$(IV).
• Figure 3: dS$_3$ geodesic in Poincaré patch.
• Figure 4: Complex contour of integration $\frac{1}{\sinh u}$. White poles are corresponding to the poles whose residue are $+2i \pi$ while black ones are $-2i\pi$.
• Figure 5: The contour in Fig \ref{['fig:complex _contour_of_u']} can be interpreted as a geodesics in complexified dS$_3$.