The Einstein Equation from an Informational-Geometrical Equivalence

Abstract: While any observer perceives its immediate neighborhood as locally flat, the metric field deviates from Minkowski space to relate infinitesimal distances assigned by local observers at distinct spacetime points. In a quantum description, these events should emerge from interactions (and thus correlations) between quantum systems, with one of them acting as a reference frame that defines relational localization. In this context, the metric field connects distances between interactions (events) \textit{from the perspective of} distinct physical local reference frames (LRFs). Building on this idea -- together with the connection between entanglement entropy and area (which, in turn, may be linked to the metric itself), and the fact that the Einstein equation does not require the explicit presence of these material frames -- we propose an \emph{informational-geometrical equivalence} (IGE): for a sufficiently small spacelike region $ B $, let $\rho_B $ and $ \sigma_B$ denote the reduced states of the quantum fields and the LRF within $ B $, respectively. In this picture, the relational content of the entropy of the quantum fields as seen by the LRF -- quantified by the conditional entropy $ \delta_\rho S(\rho_B | \sigma_B) $ -- is encoded in the variation $ \delta_{g,\rho} S(\rho_B) $ induced by a smooth geometric perturbation. When the reference frame has complete information about the infrared sector, this IGE recovers the semiclassical Einstein equation. Furthermore, considering the presence of quantum correlations between the system and the LRF reveals a positive cosmological constant related to the density of quantum correlation within $B$.