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So in regular Newtonian mechanics, orbits are generally stable so if there is a metal sphere orbiting the Earth assuming no tidal interactions, they should stay where they are.

But then gravitational wave radiation can cause the metal sphere to fall into the planet, so isn't this not following the geodesic? So after 10^20 years, the metal sphere would fall in. So is the object following geodesic and how is this different from the metal sphere around Earth where there is more atmospheric drag which causes the metal sphere to slowly fall into the planet.

So are they both following geodesics because they are doing the same thing except the air resistance version is faster version?

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While the gravitational wave (though tiny) is radiating energy away from the earth - satellite system it is also changing the shape of the geodesic. The geodesic is a function of all gravitational effects. However, the satellite is also being decelerated, so the satellite will not follow its geodesic path exactly.
In the case of atmospheric drag, an additional force is acting on the satellite acting in opposition to its velocity and causing it to decelerate. In this case the satellite does not follow the geodesic as that path is only followed by an object in free fall, not an object undergoing acceleration.

In effect as in both cases there is acceleration and not free fall, neither is following a geodesic.

(Of course, Newtonian mechanics doesn't have gravitational waves. So for that reason the satellite, without atmospheric drag should continue to orbit, in free fall, for eternity.)

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  • $\begingroup$ So doesn't that mean that free falling objects don't actually follow geodesics even without drag from atmosphere. So only objects moving in straight line with no gravity are truly following geodesics then? $\endgroup$
    – Roghan Arun
    Commented Apr 7 at 17:10
  • $\begingroup$ As long as the object is in free fall (no external forces) it will follow a geodesic. A geodesic is the equivalent of a straight line (the shortest distance between two points) when spacetime is curved , but it is definitely not straight. $\endgroup$
    – Rich
    Commented Apr 7 at 17:23
  • $\begingroup$ @Milton The system is radiating energy-momentum, so the geodesics are changing. $\endgroup$
    – PM 2Ring
    Commented Apr 8 at 3:44

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