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My question regards the limitation of the principle to point masses. Does universality of free-fall not hold for extended objects? It seems to me that in Newtonian gravity if one focuses on the center of mass of a body universality of free-fall is satisfied.The trajectory of a point mass in a gravitational field depends only on its initial position and velocity, and is independent of its composition and structure.

What about GR? If the particle is point-like, it is well-known that universality of free-fall is embodied in the fact that the particle trajectories are the geodesics of the spacetime manifold. Do extended objects move geodetically?

I assume the object is extended but still has a small mass, so that one can neglect the gravitational influence of the object.