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Does anyone know of attempts to re-prove such theorems on the basis of plausible constraints on the matter Lagrangian (or general forms of the Lagrangian) rather than the traditional energy conditions?

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*There are papers proving rigorously that if you carefully take the limit as a body shrinks in size and mass, that it must follow a geodesic of the background geometry. Such theorems as I've seen must assume an energy condition as part of the proof. I have also seen a paper that shows that the energy conditions is *necessary*. That is, if you do the limiting process without any constraint on T, not only is non-geodesic motion possible, but even spacelike paths are possible. This is not really surprising given the possible properties of exotic matter.