Decay criteria for asymptotic freedom in plane gravitational waves

Abstract

We investigate when plane-wave memory admits standard outgoing free data beyond the idealized sandwich-wave approximation. For a Brinkmann plane wave with profile $A(U)$, the commonly used condition $A(U)|_{U\to\infty}=0$ is not sufficient to guarantee ordinary asymptotically free motion. From the integral form of the transverse geodesic equation, we derive weighted decay criteria which divide the asymptotic dynamics into strongly asymptotically free, weakly asymptotically free, and non-asymptotically free motions. These motions are realized explicitly by the new analytical solutions of three typical examples: a Scarf profile, an inverse-cubic profile, and an inverse-square profile. A surprising feature is that the drift correction in the weakly asymptotically free motion affects only trajectories with nonzero outgoing velocity and therefore does not obstruct displacement memory. We further express the classification in terms of the accumulated tidal matrix, showing that it is an intrinsic curvature effect rather than a coordinate artifact.