Observer Agreement and Entanglement Decay From Stochastic Relational Time Correlations

Abstract

We show that gravitational coarse-graining renders relational time a spatially correlated Gaussian stochastic field, with consequences for observer agreement and entanglement decay in spatially separated systems. Working in the ADM formulation of linearized gravity with a massless scalar field, we integrate out the transverse–traceless (TT) graviton sector within an effective field theory regulated at a coarse-graining scale. The Gaussian structure of the TT sector renders the path integral exact within the quadratic truncation, and the resulting influence functional admits an exact Hubbard–Stratonovich representation in terms of a Gaussian stochastic field. The accumulated relational time shift is a diffusive stochastic process whose variance grows linearly with observation time and carries an overall $M_{\mathrm{Pl}}^{-4}$ suppression. The normalized spatial correlation between clock fluctuations exhibits an $r^{-3}$ decay and is independent of the overall gravitational coupling strength. These correlations enter two physical observables with opposite dependence: the visibility of observer agreement decays as a function of one minus the spatial correlation, while the concurrence of a spatially separated Bell pair decays as a function of one plus it. This asymmetry arises because observer agreement is governed by the difference of relational time shifts, whereas Bell-pair coherence is governed by their sum. Spatial correlations therefore improve observer agreement while accelerating entanglement decay.