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A seemingly interesting question.

Newton's 1st law states that objects continue in straight lines, unless acted upon by external forces. Now consider a frictionless manifold. Since it is locally euclidian, the object will travel in a straight line. In other words, if we build a frictionless track surrounding the earth, neglect all other forces and air, will the object be able to circle the earth unless acted upon by external forces?

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Yes, "straight" lines on a manifold are geodesics, which obey the geodesic equation $$\frac{\text{d}^2x^\rho}{\text{d}\tau^2} = -\Gamma^\rho_{\mu\nu}\frac{\text{d}x^\mu}{\text{d}\tau}\frac{\text{d}x^\nu}{\text{d}\tau}$$

If we take the surface of a sphere as our manifold, the geodesics are great circles. A frictionless track will only be able to exert a perpendicular force on the object, so any object traveling along it will always remain at constant speed and will circle the Earth forever.

The track exerts two forces on the object. The first is the normal force which points radially. If the track is not a great circle, it will exert an additional force of constraint on the object which is tangential and also perpendicular to the track.

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  • $\begingroup$ Free fall needs a gravity force, so Newton's 1st law doesn't apply $\endgroup$
    – FGSUZ
    Jun 30 '21 at 11:11
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    $\begingroup$ @FGSUZ: The point of the equivalence principle is that gravity is freely falling frames are actually the closest analog to inertial frames in a curved spacetime. Any frames in which you experience a “gravitational force” is, in the context of GR, a non-inertial frame. $\endgroup$ Jun 30 '21 at 13:24

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