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,
nonrotating 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(1v^{2}/c^{2}) 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 r^{2} /
c^{2}) T_0 = sqrt(1 + 2
phi/c^{2}) T_0
where phi =  w^{2} r^{2} / 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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