Gravitational Dynamics From Entanglement "Thermodynamics"

TL;DR

The paper shows that in holographic CFTs, the entanglement-thermodynamics relation $\delta S_A = \delta E^{hyp}_A$ for ball-shaped regions constrains the dual bulk geometry to satisfy the linearized Einstein equations about AdS. By relating $\delta S$ to the area variation of extremal surfaces via the Ryu-Takayanagi prescription and $\delta E^{hyp}_A$ to boundary metric perturbations through Fefferman–Graham holography, the authors prove both directions: Einstein solutions imply $\delta S = \delta E$, and $\delta S = \delta E$ for all regions and frames implies the linearized equations. The results hold in general dimensions and emphasize that the universal sector of linearized bulk gravity is encoded in entanglement structure of the boundary CFT, with potential extensions to higher-derivative gravity and nonlinear regimes to be explored.

Abstract

In a general conformal field theory, perturbations to the vacuum state obey the relation δS = δE, where δS is the change in entanglement entropy of an arbitrary ball-shaped region, and δE is the change in "hyperbolic" energy of this region. In this note, we show that for holographic conformal field theories, this relation, together with the holographic connection between entanglement entropies and areas of extremal surfaces and the standard connection between the field theory stress tensor and the boundary behavior of the metric, implies that geometry dual to the perturbed state satisfies Einstein's equations expanded to linear order about pure AdS.