Is Periodicity Preserved in Relativistic Phenomena?

In summary, the question of whether a phenomenon that is periodic in one frame of reference is also periodic in another frame moving at a constant speed can be answered by general relativity, which suggests that it is not always the case. Special relativity also has some implications, such as the need for clock synchronization, which can be approximated by using small enough clocks.
  • #1
Eynstone
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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.
 
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  • #2
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.

Eynstone said:
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 (see p. 9)
http://math.ucr.edu/home/baez/physics/Relativity/SR/clock.html
 
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1. What is periodicity in relation to relativity?

Periodicity refers to the repetition of a particular event or phenomenon at regular intervals. In the context of relativity, periodicity refers to the repetition of events in a recurring pattern in different reference frames.

2. How does relativistic time dilation affect periodicity?

Relativistic time dilation is the phenomenon where time appears to slow down for an observer moving at high speeds. This can affect the perceived periodicity of events, as time intervals may appear longer or shorter for different observers depending on their relative motion.

3. Can periodicity be observed in different reference frames?

Yes, periodicity can be observed in different reference frames. However, the perceived frequency or time intervals of the periodic event may differ for observers in different frames due to the effects of relativity.

4. How does the speed of light play a role in periodicity?

The speed of light is the ultimate speed limit in the universe, and it plays a crucial role in determining the perceived periodicity of events. As an object approaches the speed of light, time dilation and length contraction effects become more significant, altering the observed periodicity.

5. Can periodicity be used to test the principles of relativity?

Yes, periodicity can be used to test the principles of relativity. The predicted effects of time dilation and length contraction on the observed periodicity of events have been confirmed through various experiments, providing evidence for the validity of relativity.

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