Can a Centrifuge Experiment Demonstrate Reciprocity in Special Relativity?

[Jeff273]

Hi all,

In another forum, I've been exploring the idea of an experiment to demonstrate what has been referred to as 'reciprocity' in SR. That is, the mutual observation of time dilation by two observers with relative velocity(Bob says Mary's clock is dilated while Mary says Bob's clock is dilated).

The motivation for this is the claim by some that it is physically impossible for this to occur. I of course disagree. However, on looking for evidence of this, including questions submitted to NASA, Cornell, et al, to my amazement, there appears to be no experiment that demonstrates this.

A key feature of the proposed experiment is the use of a long-radius centrifuge. The timing apparatus will experience approx. 25g constant acceleration.

The question is, from the frame of the rotating clock (and the clock at the center) can I expect time dilation based solely on the clocks relative speed (gamma) or must I incorporate GR time dilation effects since acceleration is involved?

Thanks.

 

[JesseM]

The question is, from the frame of the rotating clock (and the clock at the center) can I expect time dilation based solely on the clocks relative speed (gamma) or must I incorporate GR time dilation effects since acceleration is involved?

You don't really need to use GR, you can analyze the problem in the frame of the non-rotating clock at the center to figure out what the rotating clock sees--just imagine the non-rotating clock sends out signals every time it ticks, and figure out what the rotating clock reads at the moments it receives successive signals. Of course, this only tells you what the rotating clock sees using light-signals, not what coordinates the event of each tick would be assigned in its "frame", but I don't know if there's any canonical way to define the coordinate system of a non-inertial observer, there should be different possible ways of coming up with a non-inertial coordinate system where the rotating clock's position coordinates don't change over time. But since the rotating clock maintains a constant distance from the non-rotating one, I'd think that in any "reasonable" coordinate system the time between successive ticks of the non-rotating clock would be identical to the time between the events of the rotating clock receiving successive signals generated by the ticks, as measured in terms of the rotating clock's proper time. And if you work this out, you see that the situation is not reciprocal--both the rotating clock and the non-rotating clock agree that the non-rotating clock ticks faster, by the same factor.

 

[Jeff273]

Well, JesseM, that's the problem. If this system is non-reciprocal (which I suspected it was) then the experiment, in it's current configuration, is pointless.

Thanks.

 

[kleinwolf]

For this problem : how about deriving Lorentz transformations from the invariance of space-time interval in polar coordinates ?

 

[pervect]

Well, JesseM, that's the problem. If this system is non-reciprocal (which I suspected it was) then the experiment, in it's current configuration, is pointless.

Thanks.

I'm not sure what the question is here. I thought you were asking about what relativity predicted. The answer to that question is that you can either use a gravitational time dilation formula (if you use a rotating frame of reference), or you can use a velocity-dependent gamma formula (if you use a non-rotating frame of reference). The former approach requires GR, the later approach requires only SR. The result from either approach is the same - the clock that's not accelerating ticks faster than the clock that is accelerating, and the magnitude of the effect can be given by 1/sqrt(1-(w*r)^2/c^2).

The formulas that apply are either

t/tau = 1/sqrt(1-v^2/c^2) (the SR approach)
t/tau = 1/sqrt(1-2*U/c^2), a weak-field GR approach where U = -Phi is the negative of the Newtonian potential energy.

Either approach alone is correct - applying a "double whammy" would give the wrong answer (there is no approach where it makes sense to have both a velocity red-shift and a gravitational one).

As far as whether or not the specific centrifuge experiment tests some specific non-SR theory, that would depend on the specific non-SR theory. The centrifuge type tests I've read about are designed to look for long-range vector fields that might hypothetically exist. They have been sometimes called "ether-drift" experiments, because these sorts of long-range vector forces would allow one to determine one's absolute velocity if they existed. MTW's gravitation talks a bit about these tests on pg 1063. Results have been (should I say of course?) negative - the results have been in accordance with SR. Such a field would show up as a dependence of the doppler shift with angle, rather than a constant doppler shift.

 

[yogi]

Try putting one clock on one arm of your long radius centrifuge and another clock on a second long radius arm of the same length and rotate the arms in opposite directions (better space their plane of rotation apart a little or your results will be broken clocks which won't reveal much about SR). The two clocks are now in relative motion - what would you expect to find as to how each clock aged relative to the other? How about with respect to a third clock located at the center axis?

 

[JesseM]

Try putting one clock on one arm of your long radius centrifuge and another clock on a second long radius arm of the same length and rotate the arms in opposite directions (better space their plane of rotation apart a little or your results will be broken clocks which won't reveal much about SR). The two clocks are now in relative motion - what would you expect to find as to how each clock aged relative to the other? How about with respect to a third clock located at the center axis?

Each clock will age at the same rate. Once again, the rules of SR only apply to inertial frames, and if you analyze this problem from the point of view of an inertial frame (such as that of the clock at the center) it is easy to see this is true.

 

[Ich]

The third clock would of course age differently.

 

[JesseM]

Yeah, I just meant the two clocks on the spinning arms would age at the same rate, not the one at the center. And by "age at the same rate" I mean that each successive time they pass each other they will have aged the same total amount...depending on what frame you're using, they can be ticking at different rates at different points along the circle.

 

[Jeff273]

So, any thoughts on how to demonstrate mutual time dilation on a budget?

I.E. each measure the other to be dilated by:

$$\gamma \equiv \frac{1}{\sqrt{1 - v^2/c^2}}$$

while in uniform motion.

 

[JesseM]

So, any thoughts on how to demonstrate mutual time dilation on a budget?

Well, if you can demonstrate that clocks moving relative to you slow down by the predicted amount, and that rulers moving relative to you shrink by the predicted amount, and that light moves at the same speed in all directions in your frame, and you assume that all observers synchronize their clocks using the procedure Einstein laid out in his paper (where each observer assumes that light travels at the same speed in all directions relative to himself, so that if a flash is set off at the midpoint of two clocks, each clock should be set to read the same time at the moment the light reaches it), then it's not too hard to prove that mutual time dilation follows necessarily from these things. The main problem is that it's a lot harder to measure length contraction than time dilation, and also some people might argue that Einstein's synchronization procedure is not really the correct one, although they cannot suggest an experimental procedure that will allow all observers to synchronize their clocks in such a way that there is no disagreement about simultaneity.

 

[Jeff273]

JesseM,

I completely agree that mutual dilation must follow, as you say, from existing observations of dilation in satellite and aircraft experiments. I've been trying to devise a way of actually showing it happen. I have access to high speed electronics and instrumentation able to measure accurately to 1ns. The whole problem is getting enough uniform speed for long enough to accrue a few nanoseconds worth of dilation.

 

[pervect]

Existing centrifuge experiments use the Mossbauer effect, the same effect was used in the Erza&Pound gravitational redshift experiment.

This yields a very sensitive detector

http://hyperphysics.phy-astr.gsu.edu/hbase/nuclear/mossfe.html

notes that doppler shift from a velocity of only .2 mm/sec towards or away from the detector will detune the resonance.

 

[Jeff273]

Interesting. Now I just need a couple of Mossbauer Spectrometers.

 

[pervect]

I saw some being advertised for only about 20 grand - minus the radiation source :-) :-). (The radiation source is probably not terribly expensive, but will probably have a lot of regulatory hurdles).