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When a rock falls from a ledge, why does it head to the surface and not up to where time runs faster?

If a rock, free from forces, follows a worldline of maximum aging, why would that rock approach Earth where the rate of time runs slower, and so would slow down the rocks aging? Shouldn’t the rock avoid earth?

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Related: physics.stackexchange.com/q/38348/2451 –  Qmechanic Jun 28 '13 at 19:22
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A rising trajectory is a geodesic. A falling trajectory is a geodesic. The principle of maximum aging doesn't select one geodesic over other geodesics. It tells us which world-lines are geodesics and which ones aren't. Given two spacetime events A and B, there is typically exactly one geodesic connecting them. In your example, you've only specified A, not B.

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