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Because all particles must follow their geodesics, is there any reason we make a differentiation between a geodesic and a "non-geodesic"? And is there any possible way to get a particle to follow a path outside its geodesic without exerting any sort of external force on it (including the 4 fundamental forces)?

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    There are of course paths through space(time) that are not geodesics; just because objects not influenced by forces will not follow those does not mean they are not there. – Marius Ladegård Meyer Feb 26 '24 at 16:56
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    In the absence of electromagnetic interaction and fermion degeneracy pressure, an initially stationary geodesic on Earth's surface is supposed to drop into Earth. At least, quantitatively, it is supposed to work like that. Standing on Earth and not dropping into the centre is itself a non-geodesic behaviour. – naturallyInconsistent Feb 26 '24 at 16:58
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    If every path had to be a geodesic you couldn't even stand on the floor. – Yukterez Feb 26 '24 at 17:34

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Two things:

Only free particles (as in, only affected by the curvature of spacetime) follow geodesics. Geodesics are paths with zero four-acceleration $a^\mu$; an external force will of course result in a nonzero acceleration and thus a non-geodesic trajectory. This is only true for point particles, though: extended bodies experience the so-called self-force which results in the loss of energy through gravitational radiation and trajectories that are not geodesics.

Additionally, even if it was true that every physical object follows a geodesic always, non-geodesics would still exist as mathematical curves. The definition of geodesic is mathematical, not physical, so the distinction is independent of what physical objects actually do.

Javier
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