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I'm searching for a book or journal article that supports what seems to be a very elegant proof of how a central stationary clock "measures" the proper time $T$ of a revolving clock, and most interestingly, vice versa. See

http://www.physicsinsights.org/revolving_clock.html

Essentially, the conclusion is that when $c=1$

and

$\gamma = \dfrac{1}{\sqrt{1 - {v^2}}}$

the stationary clock measures the proper time of the revolving clock as

$\frac{1}{\gamma}{*T}$

while the revolving clock measures the stationary clock's proper time as

$\gamma*T$

Therefore, each clock agrees about the proper time of the other clock.

Has anyone come across a publication that shows this? Although I've seen discussions in online forums I think any textbook that covers this might be worth having.

Qmechanic
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argonaut
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1 Answers1

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Clock in the center always measures dilation of rotating clocks. Rotating clock measures acceleration of clock in the center. Clocks on opposite sides of the circumference tick "at the same rate", i.e. rotating observer will see, that a clock on the opposite side of circumference neither dilates nor accelerates.

Accelerates I mean ticks faster.

Confirmed by experiments: Champeney and Moon time dilation test, Kholmetskii at all time dilation test.

It is most convenient to consider this case through Transverse Doppler Effect, when source is in the center and absorber rotates and vice versa.

Very simple animation, last episode in particular: https://www.youtube.com/watch?v=hnphFr2Iai4

Links: https://www.researchgate.net/publication/304781760_Specific_Features_of_Time_Dilation_During_Circular_Movement

Also the last paragraph at Mathpages http://mathpages.com/home/kmath587/kmath587.htm

It is also interesting to consider a rotating mirror. Source in the center emits photon. Photon comes to mirror at oblique angle. Mirror gains energy of photon, i.e. photon is blueshifted at the mirror. Mirror spits photon out and loses the same amount of energy. Photon leaves mirror at oblique angle again and travels back to the source. It comes to the source at right angle and redshifts, i.e. source receives the same frequency (i.e. the same energy back) as it was released. Mirror was a "moving clock", i.e. dilates. If photon went through a center and went further, it blueshifts again. i.e. if photon was "green" (it's proper frequency) when it started from a mirror on certain point of circumference it will be "green" again on the opposite side of circumference. Sure, we consider perfect mirrors. It is interesting to note that rotating clocks do not move relatively to each other.