When an object accelerates, the leading edge accelerates slower than the tailing edge.
Why?
Relativistic effects inherant in acceleration itself. Or were you looking for a more specific answer.
Please.
When an object, whether it is an elevator, rocket, flat earth, what have you, accelerates, time is dilated more the further from the source of the acceleration you go. So, a clock at a higher elevation relative to the source will run slow compared to one at a lower elevation. This, along with length contraction, causes the dv/dt to be skewed based on the gamma factor. I have a good shortcut somewhere you might want to take a look at, I'll try to find it later.
And? If FET is correct, the entire surface is the leading edge. So, one would weigh the least at the peak of Everest and the most at the bottom of the Mariana Trench is what you are saying, and that does not coincide with observed phenomena.
Are you saying that you do not weigh less on Everest than at a lower elevation?
Unless you are near the event horizon of a black hole, would this be a measurable effect?
We're talking about acceleration, not gravitation. So what you mean to ask is "Unless you are near the speed of light, would this be a measurable effect" in which case I say, yes, that we would be going fast enough by now to take relativistic effects into consideration.