Imagine a 2D uniform circular motion of constant magnitude but changing direction in an area of zero g. The forces will be equal all the way round - it will be a perfect circle.
Now imagine the same circular motion on earth with the 2D circle 'upended'. If the circle described were to still be a perfect circle, considering just the effect of gravity and ignoring atmosphere and materials, more energy would have to be used for the upwards portion than the downwards. So from the perspective of physics or geometry of spacetime, would it no longer be a 'perfect' circle? Is the upwards portion geometrically 'longer' (needing more force to maintain same velocity) and the downwards portion 'shorter'?
What I'm trying to get at is, would the forces created by uniform circular motions of a certain mass in different regions tell us something about - or even help us identify - the local shape of spacetime?