Is it true that a freely-falling body accelerates because, as time slows toward the center of the earth, each second is longer than the preceding one, giving the body further to travel? A simple yes or no will do it for me.
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No.
If you want an answer beyond a simple yes or no have a look at my answer to How does "curved space" explain gravitational attraction?. The acceleration is caused by the curvature of spacetime. This curvature does also cause time to run more slowly as you approach a spherical mass, but the change in the rate taht time runs does not cause the acceleration.

John Rennie
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John, how to differ in a better way between an acceleration caused by a force like circular motion in a particle generator and a free fall which is not an acceleration? – HolgerFiedler Feb 06 '16 at 16:41
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is it a duplicate of "How does "curved space" explain gravitational attraction?" ? – Feb 06 '16 at 18:24