# Time dilation in a centrifuge: effect of velocity and acceleration?

An object in a centrifuge has a tangential velocity of $$v = \omega r$$ and a radial acceleration of $$g = \omega^2 r$$. By itself, the velocity would cause a time dilation of $$1/\sqrt{1-(v/c)^2}$$ = $$1/\sqrt{1-(\omega r/c)^2}$$.

The potential associated with the radial acceleration is $$U = -(\omega r)^2/2$$, so the time dilation due to the potential alone would be $$1/\sqrt{1+2U/c^2}$$ = $$1/\sqrt{1-(\omega r/c)^2}$$, which interestingly has the same value.

I feel like I ought to be able to multiply these together to get $$1/(1-(\omega r/c)^2)$$ for the combined time dilation, but I doubt it's that simple.

I see a comprehensive answer from Crowell to a related question but I don't really understand the answer. It seems like a well-defined question with just two parameters, $$\omega$$ and $$r$$, so I assume the answer is a relatively simple expression?

• This is discussed in Can a ultracentrifuge be used to test general relativity? though I'm reluctant to close this as a duplicate because I think Dale's answer here is spot on! Apr 19 at 6:09
• @JohnRennie thanks for pointing that out. You have a good answer there too. Apr 19 at 17:26

Acceleration, in and of itself, does not cause time dilation. This is the clock postulate and is well confirmed by experiment e.g. in particle accelerators. So only the speed (relative to the observer) causes time dilation. If that speed is constant (even though the velocity is changing) then you can just use the ordinary SR time dilation formula. In the case of a centrifuge, you could instead use coordinates which are co-rotating, in which case the speed is 0 but you'd get the same value for time dilation by introducing a "gravitational" potential. But you shouldn't do both at the same time, because that would be mixing coordinate systems.

• that link is very helpful, thanks! Jun 1, 2021 at 20:32

which interestingly has the same value.

I feel like I ought to be able to multipy these together

It is not a coincidence that they have the same value. They are both the same time dilation from different perspectives. The velocity formula is the time dilation from the perspective of the inertial frame. The potential formula is the time dilation from the perspective of the rotating frame.

You do not multiply them. The potential doesn’t exist in the inertial frame. The velocity doesn’t exist in the rotating frame. So they are different ways of calculating the same thing, and cannot be combined