How can you prove that acceleration due to gravity is the same in a resting and moving frame?

Answer 1

Cristina, I have no idea what level you are, so I am going to assume you are not dealing with Lorenez factors.

But let me summarize on the force equation.

F=ma=m v/t' where t is the dilated time at relativistic speeds. Under constant acceleration, mass does not change, but because of the high speeds, time is dilated, so it takes a greater force to accelerate it. But with gravity, the greater force is provided by the dilated mass already present, so in fact, under the force of gravity, with dilation of time, the acceleration remains the same.

For a detailed argument,
http://www.mrelativity.net/TimeEnergyIG/TimeEnergyIG2.htm
And the only way to actually prove it is to test it, which has been done.

Answer 2

The apparent acceleration of a body that is only subjected to gravity is independent of the velocity of the coordinate system, as long that system is moving in a straight line at constant velocity. If the coordinate system is accelerating, there will be other apparent forces (Coriolis and pseudo-gravity) that lead to additional acceleration in that coordinate system. In a linearly accelerating coordinate system, there is no way to tell if the additional apparent force is due to gravity or the motion of the coordinate system. (A related statement is that there is no way to tell if you are really weightless in a free-falling elevator). This is one of the imporant principles that led Einstein to the General Theory of Relativity.

The short answer to your question is that the acceleration due to gravity alone IS independent of coordinate system motion, but the proof is complex and reduces to the fact that that experiment supports it.