Hello! Is it possible to convert from the time dilation effects caused by velocity to the time dilation effects caused by gravity? In other words, if one calculates the time dilation effect caused by a given velocity, how can one calculate the gravity needed to create that same effect using acceleration? I dont know if I am explaining this in a way that makes total sense, but I would like to know how to convert time dilation values due to velocity to time dilation values due to gravity. I know that acceleration is velocity changing over time so I assume any conversion would have to take that into consideration, I am just not sure how to due that using the time dilation formulas that I have (for velocity and gravity). Thank you for your help!

Gravitational time dilation and standard relativistic time dilation are actually measurably different. Nonetheless, it is possible to determine the velocity required to create a special relativistic time dilation equivalent to the time dilation due to a gravitational potential. This played a critical role in one of the early precision tests of General Relativity performed by Pound and Rebka.

The time dilation due to motion at a velocity v is given by

tSlow = Sqrt( 1 - v²/c² )tFast

while the time dilation due to being a distance r from an object of mass Mis given by

tSlow = Sqrt( 1 - 2GM/rc² ) tFast

In their experiment, Pound and Rebka balanced the gravitational blue shift of a photon falling down an elevator against the doppler redshift of a target moving away from the photon by adjusting the velocity of the target. By finding the velocity at which no shift in frequency occurred,they were able to measure the gravitational blue shift.

The gravitationally blue shifted frequency (after falling down an elevator of height h) is given by

fGravBlue = f0 Sqrt( 1 - 2GM/(r+h)c² ) / Sqrt( 1 - 2GM/rc² )

The doppler redshift due to moving at a velocity v is given by

fDopplerRed = f0 Sqrt( 1 - v/c ) / Sqrt( 1 + v/c )

Thus, Pound and Rebka needed to find the velocity v such that

1 = ( Sqrt( 1 - 2GM/(r+h)c² ) Sqrt( 1 - v/c ) ) / ( Sqrt( 1 - 2GM/rc² )Sqrt(1 + v/c ) )

They found that they needed v = gh/c, which confirmed the prediction of General Relativity.

Time dilation due to relative movement and time dilation due to being in gravity well are different things. If two observers are moving with respect to each-other, then they will both observe the other as experiencing time moving more slowly. The only way to reconcile this is to have one of them accelerate, which changes the rules of SR. Time dilation due to gravity however produces no apparent disagreement: time passes more slowly in a gravitational potential well than it does out of it, and both the observer on the satellite and the observer on the surface of the Earth would agree that this is the case, with no accelerations needed. Yes, you could calculate how fast the other observer deeper in gravity well would need to be moving, but not vice versa, because of this.