Equivalent Equations of Motion for Gravity and Entropy

TL;DR

This work demonstrates an equivalence between the wave equation for entanglement-entropy perturbations in holographic CFTs and the linearized bulk Einstein equations in AdS, realized through the kinematic-space formalism. By relating OPE-blocks to Radon transforms and extending to tensor data via tensor Radon transforms, the authors show that the kinematic-space dynamics encode bulk gravitational dynamics through conformal Casimir intertwining. They derive a kinematic-space equation for the area perturbation $\delta A$ that matches the Einstein equations and recover the quantum-corrected Ryu-Takayanagi formula, $H_{mod}=\frac{\delta A}{4G_N}+H_{bulk}$, recovering the FLM correction in this framework. The results illuminate how entanglement structure controls gravity in AdS from a kinematic-space perspective, while also highlighting technical caveats (notably tensor-Radon invertibility) and paving the way for incorporating interactions and generalized gravity theories.

Abstract

We demonstrate an equivalence between the wave equation obeyed by the entanglement entropy of CFT subregions and the linearized bulk Einstein equation in Anti-de Sitter pace. In doing so, we make use of the formalism of kinematic space [arXiv:1505.05515] and fields on this space, introduced in [arXiv:1604.03110]. We show that the gravitational dynamics are equivalent to a gauge invariant wave-equation on kinematic space and that this equation arises in natural correspondence to the conformal Casimir equation in the CFT.

Figures (3)

• Figure 1: The kinematic space for spherical regions that lie on a single time slice is given by $d$-dimensional Lorentzian de Sitter space. The de Sitter space is a corresponding slice of the larger kinematic space for all (boosted) spherical regions, which is $2d$-dimensional with signature $(d,d)$.
• Figure 2: A point in kinematic space labels two timelike-separated boundary points, $x_1,x_2$, or equivalently a codimension-2 sphere formed by their causal domain. Operators $\sigma(x_1,x_2)$ localized on the sphere have an OPE expansion in terms of OPE blocks for primaries $\mathcal{O}_k$, where contributions can be identified as bulk surface operators $\phi_k$. Figure from Czech:2016xec.
• Figure 3: The retarded de Sitter propagator corresponds to an integral over all the geodesics that lie on the $t=0$ slice of the causal wedge. This integral reconstructs the bulk modular Hamiltonian, which is also an integral over the same spatial slice.