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Is a radially falling rod in free fall?

Which frequency shift of light emitted at the ends of the rod would observers at rest at the respective other ends measure?

Here are some assumptions. Please correct accordingly.

What can one tell about the velocity of the ends of the rod, measured locally relative to freely falling test particles? Presumably the upper end falls faster and the lower end slower than the resp. particle. If true, there should be a point on the rod which is in rest with the particle at a moment.

In my opinion this would mean that no point on the rod is in free fall, means that no point follows a geodesic.

What can one say about the position of this very point? The gravitational acceleration is larger at the lower end of the rod. Can one conclude from this that this point is in the lower half and moves towards the lower end during the fall? Given the radial coordinates of the ends of the rod how would one calculate (in principle) the position of this point?

Two observers in radial free fall see themselves redshifted. So, I expect that the ends of the rod are less redshifted, maybe even blueshifted. I found this calculation

https://www.physicsforums.com/threads/black-hole.104577/#post-861282 #5

very interesting, but suspect that to calculate of the frequency shift along the rod is much more difficult.

Any help is very appreciated.