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In general relativity, if an object is free-falling in one reference frame, does that mean it's free-falling in all reference frames? A particular case that I have in mind is in the absence of gravity, where one frame is inertial and therefore has a flat spacetime and the other frame is accelerating or rotating relative to the inertial frame and therefore has a curved spacetime. If the world line of the object is a straight line in the inertial frame, must it necessarily be a geodesic in the non-inertial frame?

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An object is called "free-falling" if its motion obeys the geodesic equation. This equation is covariant, so if it's obeyed in one reference frame than it's obeyed in any other reference frame as well. So yes, the notion of "free-falling" is indeed frame-independent.

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