# A bench-top test of special relativity ?

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## Main Question or Discussion Point

The line element in a rotating plane, with angular velocity \omega is given,
in polar coordinates (t, r, \theta) by ( ref 1)

$$ds^2 = -(c^2 - r^2\omega^2)dt^2 + dr^2 + 2r\omega^2dtd\theta + r^2d\theta^2$$

So two clocks placed at r=0 and r=r_1 will have relative rate

$$f = \sqrt{\frac{c^2 - r_1^2\omega^2}{c^2}}$$.

This means also that a light from a source at r=r_1 shining towards the center will be frequency shifted by the same factor.

Looking at available equipment, I find a centrifuge with a radius of about .14 m ( 14cm) and a top speed of 6500 rpm. This translates to \omega = 2*pi*6500/60 = 680 rad/sec. Which gives $$\omega^2r^2 = 9063 m^2s^{-2}$$ and f=0.99998 which is 2 parts in 100,000. Using high precision gratings, is it possible to detect such a small effect today ?

It may also be difficult to get a laser that works at about 1500g acceleration.

refs:
1. "THE THEORY OF RELATIVITY", C. M0ELLER (1958), OUP.

Related Special and General Relativity News on Phys.org
Gamma and the ultracentrifuge

Hello:

It looks like the frequency f is an inverted gamma:

$$\gamma = \frac{1}{\sqrt{1 - \beta^2}}=1/f\quad eq ~1$$

Moeller was good, I just needed to translate it into a term I was more familiar with.

In my days as a biologist, I worked with ultracentrifuges. These are used to separate different biological materials, to purify a circle of DNA (called a plasmid) from DNA that was not a circle. Those can get up to a million g for a table top machine. I recall having to make sure the tubes that go into the rotors are very well balanced!

I would imagine (meaning I have no experience) that one would mount a solid state "laser on a chip" sort of device on the outside edge of the rotor, and have it fire not toward the center, but perpendicular to that, in the direction of motion and away from the direction of motion. Have a second solid state laser, same make, firing away, as a reference. There are several reasons why not to put it at the center. Most important is that the length of the laser is not zero. As the centrifuge speeds up, that should be observable, a speed effect. This is not linear, it goes as eq. 1 predicts. Now you could compare the reference laser to the pointing in the direction of motion laser, as well as the one opposite the direction of motion, while changing speeds of the rotor.

Can one get a small enough solid state laser with high enough frequency stability to mount in a spinning rotor? I have no idea.

Just for fun, you could turn this into a test of GR by tipping the centrifuge over on its side. At the top of the rotation, the laser would be under a smaller gravitational field than when it is at the bottom. I bet if you did a calculation on it, the necessary precision would not make it practical.

Doug

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Hi Doug,

after some more thought I concluded that the experiment in this form is not practicable. It is possible to get up to 20000 rpm, but only with a 25cm diameter apparatus. Anything on the rim would be crushed, so even if it brings us into the precision range, the high speed will destroy the light source.

So I'm now thinking about a low-speed experiment using twin clocks. It is not difficult to build a small crystal controlled ticker that is accurate to 1 part in a million. If we had two near identical clocks they could be compared after spinning one for some time. I haven't done the calculation yet.

So it comes down to very accurate clocks again, like the circumnavigation experiments.

Try finding some info on solid state gyroscopes where they use fiber optic loops at right angles. By comparing phase changes in the light singles traveling around the loops they calculate changes in three dimensional orientations just as can be done with physical gyroscopes.

I suspect the math required to do that depends on and demonstrates most of the same relativistic principles you seek to show.

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Randall, thanks for the feedback. I looked up fibre-optic gyroscopes and the Sagnac effect. Very interesting. In fact it may well be 'bench-top relativity' in itself. I haven't checked the maths.

The idea that the same clock will run at different rates in different frames has always seemed mysterious and fascinating to me, hence my desire to witness it.

Ich
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Ich, you're a hero. Thanks.

This experiment was conducted in 1963 and it must be counted as one of the best confirmations of SR. From the abstract

"... a relative frequency shift the magnitude of which agreed, to within an accuracy of about 2 per cent, with that predicted by the relativistic expression (Δν/ν) = (va2 - vs2)/2c2, where va and vs are the velocities of the absorber and source. This expression may be obtained either in terms of the time dilatation of special relativity or in terms of the pseudo-gravitational potential difference between source and absorber."
(my emphasis)
It's a good thing someone did it.

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Thanks to everyone who responded. Unfortunately I made a dumb error in my original calculations and dropped a factor of c. This means instead of a precision of 1 part in 10^5, we actually need 1 part in 10^10.

Something I thought would take a few hours, would actually take many years.

NateTG
Homework Helper
I'd be curious to see if you can get measurable relativistic doppler shift by bouncing laser light off of a piezo vibrator and into a spectrometer. Since the laser light is basically monochromatic (you could also use a helium lamp) you could compare the measured light frequency range as a function of the frequency... Actually, it's probably better to use a high-speed spindle with a bit that has a mirrored tip and an encoder section for use with an optical tachometer.

More generally, it seems like you should be able to put mirrors on your spinner, and move the light source off of it, or, at least, move the laser to the center of the spinner.

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NateTG,
I'd be curious to see if you can get measurable relativistic doppler shift by bouncing laser light off of a piezo vibrator and into a spectrometer. Since the laser light is basically monochromatic (you could also use a helium lamp) you could compare the measured light frequency range as a function of the frequency.
Again, the effect would be of the order 1/c and very hard to spot. Measuring the broadening of the spectrum is good idea.

So far as I now know, the phase shift in light of the Sagnac effect is the only relativistic effect that is observable with kitchen-sink physics ( if the kitchen is equipped with lasers and fibre optic cable).

Ich