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- Is there a relationship between gravitational time dilation and falling bodies? Does the fact that time is relative with respect to altitude have anything to do with why a body will fall? Is so, what is the relationship? I have no conceptual explanation for why this should happen.

**Motivation for why I'm asking this:**

I'm trying to better understand why an object that is initially at rest with respect to a massive body will fall simply by virtue of the curvature of spacetime. If it were moving through space with respect to the Earth initially, I could see the curved path warping it to the Earth, but if it's at rest in space, what is the impetus to begin the motion through space?

To remedy this ignorance, I began looking at some math that is a bit above my head. I found a derivation that starts at, after some approxiations, the Schwartzchild metric for a weak gravitational field. This is that source:

https://aapt.scitation.org/doi/10.1119/1.4972045

There is one part in this derivation I did not understand at all (the killing vector). However, after the article got passed that step, I was able to follow along (my understanding was decent as of section (6.) ), which, by considering some obvious approximations (including slow speed), led to this:

##\frac{dx}{dt} = (2ax)^{1/2}##

And from there it's easy to see how you'd get an object falling with the familiar x = (1/2)at^2 equation.

It thus seems to me that simply having that curved spacetime metric automatically results in a body falling.

My major conceptual problem:

My major conceptual problem:

BUT, this is a mathematical explanation and doesn't help my intuition that is demanding the at rest apple have some sort of push to begin motion through space. So I started thinking about time dilation and wondering if its a transformation between coordinate system of the apple and the Earth wherein hides the conversion of motion only through time to motion through space and time. Unfortunately, it then occurred to me that one of the assumptions made in the above derivation was that dx/dt is approximately dx/dτ (because of the non-relativistic speed of the object). If that's true, the gravitational time dilation shouldn't have much of an effect, I wouldn't think.

So I'm thinking that conceptual explanation for why the object at rest begins to fall is faulty.

If you have some insight on what would cause the initially at rest apple to fall to Earth, please post it. I listed this as intermediate, but all levels of math are welcomed as long as an explanation comes with them. Thanks, as always.