In work on Galaxy dynamics, from stellar kinematics, we measure relative velocities and relative positions, of nearby stars relative to the Sun (or really the Solar System barycenter). These relative positions and velocities are coordinate free, in the sense that they don't imply a rest frame for anything (and indeed, the SS barycenter is not anywhere near the rest-frame position or rest-frame velocity of the Milky Way or Local Group or anything else).
In addition to this, any measurements we make are insensitive to any overall or external acceleration: If the Milky way is in free-fall, accelerating towards some external “great attractor” or anything else, none of these observables are affected in any way by that acceleration. So what is it that stellar kinematics can really be used to measure? I think somehow the answer has to be Galilean covariant (covariant to boosts and translations), but even better it should be generally covariant (in the Newtonian sense, which is well defined, apparently).
I did some research on this subject, and the literature is all about Newton–Cartan theory, but this theory is a Newtonian limit of general relativity. That isn't quite what we care about in stellar kinematics, since in stellar kinematics, we don't get to see any orbits as a function of time (we don't observe geodesics or geodesic deviation). What, exactly do we observe? I think what we observe is something about gradients of accelerations, but I don't know yet. Great project for this summer.
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