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Hello! How are you guys? I have a question about a gravitational potential formula I have been using for physics calculations, but I'm not sure if I'm using it for the right applications. I'd like to attach a document showing my work but I cant do it here. Can you please send me an email so I can get it to you guys? Thanks!
Answer 1:

Your question is very interesting! There's one mistake you made, however: the formula you're using for the time dilation in general relativity (also called the gravitational redshift) is only valid for the Schwarzschild spacetime; that is, it's only valid for the spacetime outside of a spherically symmetric, non-rotating mass distribution (like a star or a black hole). In particular, the notion of a gravitational potential does not exist in general relativity as it does in Newtonian gravity, so you can't simply take the Schwarzschild formula and apply it to any potential. That means it doesn't apply to the problem you're considering (a centrifugal potential). If you want to figure out the time dilation felt by a particle in a particle accelerator, the best approach is to simply use normal special relativity and not bother with effective gravitational potentials and gravitational redshift. To do that, you can write the trajectory of the particle in the laboratory frame (which is an intertial frame, to which special relativity applies), and then calculate the proper time along the trajectory of the particle as a function of the time elapsed; what you'll end up finding is the standard formula for time dilation from special relativity:

T = sqrt(1-v2/c2) T_0

(sorry for not being able to make the formulas prettier!). If you wanted to, you could write the speed as v = wr (I'm using w instead of omega) so that

T = sqrt(1 - w,sup>2 r2 / c2) T_0 = sqrt(1 + 2 phi/c2) T_0

where phi = - w2 r2 / 2 is the centrifugal potential you came up with. So, the correct formula for the time dilation of a rotating particle is the one above, not the one you gave. Great thinking, though!



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