# Is periodicity relativistic?

Is a phenomenon which is periodic in a frame A of reference also periodic in another frame B moving at a constant speed v with respect to A ?
I think general relativity will answer this in the negative. How about special relativity?
Consider a world line in A with the equation x=f(t) ; with f(t)= f(t+T) , T being the period. This won't transforms into x' = g(t') with a periodic g() as x,t depend on both x',t' . What form must f have in order to preserve periodicity?
(For instance f(t) =ct transforms well , but f(t) = sin wt doesn't.)
Since we determine time by periodic phenomena, I'd also like to ask how the arguments involving 'clock synchronization' in special relativity are to hold valid.

bcrowell
Staff Emeritus
Gold Member
I think periodicity of a world-line can sort of be defined in a coordinate-invariant way in GR. At least, there are certain properties of a periodically varying world-line that we can verify are periodic, in a coordinate-invariant way. For example, if an observer moves along the world-line with an accelerometer, he can verify that the reading on the accelerometer is a periodic function of proper time. I think this is weaker than the SR notion, however. E.g., it won't tell you anything useful about whether a geodesic is periodic, since the accelerometer will read zero the whole time. Maybe you can sense higher covariant derivatives, though. You can also sense, e.g., whether the variation of a curvature scalar with respect to proper time is periodic.

Since we determine time by periodic phenomena, I'd also like to ask how the arguments involving 'clock synchronization' in special relativity are to hold valid.

This is much easier. The clocks just have to be small enough so that, by the equivalence principle, SR is a good approximation. This is related to what people sometimes refer to as the "clock postulate," although it isn't really a postulate because it can be proved from the ordinary postulates of SR. Some references:

http://www.phys.uu.nl/igg/dieks/rotation.pdf [Broken] (see p. 9)
http://math.ucr.edu/home/baez/physics/Relativity/SR/clock.html

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