Persistent gravitational wave observables: Curve deviation in asymptotically flat spacetimes

In the first paper in this series, a class of observables that generalized the gravitational wave memory effect were introduced and given the name ``persistent gravitational wave observables.'' These observables are all nonlocal in time, nonzero in spacetimes with gravitational radiation, and have an observable effect that persists after the gravitational waves have passed. In this paper, we focus on the persistent observable known as ``curve deviation,'' and we compute the observable using the Bondi-Sachs approach to asymptotically flat spacetimes at the leading, nontrivial order in inverse Bondi radius. The curve deviation is related to the final separation of two observers who have an initial separation, initial relative velocity, and relative acceleration. The displacement gravitational wave memory effect is the part of the curve deviation that depends on the initial separation and is the entire contribution for initially comoving, inertial observers at large Bondi radius. The spin and center-of-mass memory effects are contained within the dependence of the curve deviation on the initial relative velocity, and the dependence of the curve deviation on relative acceleration contains observables distinct from these known memory effects. We find that the full curve deviation observable can be written in terms of differences in nonradiative data before and after the radiation (which we call the ``charge'' contribution), along with a nonlinear ``flux'' contribution that vanishes in the absence of gravitational radiation. This splitting generalizes the notion of ``ordinary'' and ``null'' memory that exists for the displacement, spin, and center-of-mass gravitational wave memory effects to the full curve deviation observable.