Spinning disk, length contraction, & equivalence principle

[Sorcerer]

Suppose we have a spinning disk with a very fast spin, an observer in the center, and an observer on the edge. Suppose that the observer on the edge measures the circumference of the spinning disk.

(1) Now, the observer on the edge at a given instant will be moving at a faster speed than the observer in the center. Therefore according to the Center observer the Edge's ruler will be length contracted at that instant, right? And should his clock run slow according to the Center observer over an infinitesimal instant? And for each subsequent instant?

(2) Suppose the Edge observer measures the circumference. If he calculates π based on his measurement, he should get a different answer than if the disk was at rest, correct?

(3) Equivalence principle: I know this is not the same situation as the elevator acceleration thought experiment, but would this indicate that in a gravitational field there should also be length contraction? (since the accelerating disk should have length contraction at instantaneous moments where the speed is calculated)
Or is this entirely invalid?Thanks for the replies.

 

[pervect]

The issue of the circumference of the spinning disk is Ehrenfest's paradox. There is so much written about it that I really wish I had some citation indices so I could be sure I was recommending the consensus resolution.

I'm afraid I can't vouch for it's popularity, but I find the arguments of Rizzi and Ruggerio to be my personal recommendation. See for instance https://arxiv.org/abs/gr-qc/0207104. You will find a broad agreement, though, that the ratio of the circumference to the diameter is not equal to pi in the rotating frame, though. Note that the value of pi itself is not frame dependent. Thus if you calculate pi, you'll get the same value you always did. It's just that the ratio of the circumference to the diameter will not be the number pi.

I don't understand your point 3 at all, sorry. I don't think there's any reason to believe that "a gravitatioanl field" causes "length contraction".

 

[A.T.]

(3) Equivalence principle: I know this is not the same situation as the elevator acceleration thought experiment, but would this indicate that in a gravitational field there should also be length contraction? (since the accelerating disk should have length contraction at instantaneous moments where the speed is calculated)

You should distinguish between "length contraction" (which usually means due to motion) and "distorted spatial geometry" (which is measured with rulers at rest in some common frame): The length contraction is observed by the non-rotating frame, which doesn't have a centrifugal acceleration field. In the rotating frame with the centrifugal acceleration field, the disc is static, so there is no length contraction but an distorted spatial geometry.

What the dics indicates via the Equivalence principle:
- that in a non-uniform gravitational field, the spatial geometry can be non Euclidean (cirumference / radius is not pi)
- that in a gravitational field there should be gravitational time dilation (resting clock rate depends on position)

 

[Sorcerer]

Thanks for the replies. Re: pervect, I was thinking in terms of the fact that an accelerating body has instantaneous velocities, and presumably at each one there would be length contraction, but it appears my speculation was wrong.

You should distinguish between "length contraction" (which usually means due to motion) and "distorted spatial geometry" (which is measured with rulers at rest in some common frame): The length contraction is observed by the non-rotating frame, which doesn't have a centrifugal acceleration field. In the rotating frame with the centrifugal acceleration field, the disc is static, so there is no length contraction but an distorted spatial geometry.

What the dics indicates via the Equivalence principle:
- that in a non-uniform gravitational field, the spatial geometry can be non Euclidean (cirumference / radius is not pi)
- that in a gravitational field there should be gravitational time dilation (resting clock rate depends on position)

But this time dilation cannot be experimentally indistinguishable as time dilation caused by instantaneous speed, correct? Because of the clock hypothesis?

This seems kind of paradoxical to me. An acceleration is a collection of changing speeds, so each speed should have time dilation. But if I understand the clock hypothesis, it is not acceleration that causes time dilation, and if that is so, then where does the equivalence principle come into play when it comes to gravitational time dilation?

Or is there a connection between instantaneous speed and the equivalence principle? (relating to gravitational time dilation)

 

[A.T.]

it is not acceleration that causes time dilation,

Time dilation between clocks at relative rest depends on the difference in potential between their positions, not on their proper accelerations.

and if that is so, then where does the equivalence principle come into play when it comes to gravitational time dilation?

The above applies to both: accelerated/rotating frames in outer space and gravitational fields near a big mass.

 

[jartsa]

Suppose we have a very large spinning disk, and some observers far away from the center. Suppose that the observers compare their clocks. The observers orbit the center at different speeds, so their clocks run at different rates.

If the aforementioned observers are closed in a small lab located on the disk, it is difficult for them to tell whether they are on a spinning disk, or inside an accelerating elevator, or on the surface of a planet. This is known as the equivalence principle.(Those clocks on the spinning disk are like the twin paradox twins. While clocks inside the elevator are not quite like the twins from the twin paradox.)

(Oh yes, we must put an additional restriction on the observers on the spinning disk: only short duration experiments are allowed. Gyroscopes must not detect the spinning, you see.)

 

[jartsa]

(Those clocks on the spinning disk are like the twin paradox twins. While clocks inside the elevator are not quite like the twins from the twin paradox.)

(Oh yes, we must put an additional restriction on the observers on the spinning disk: only short duration experiments are allowed. Gyroscopes must not detect the spinning, you see.)

Actually, if we consider only short periods of time, and a frame where the lab on the disk does not have a very high speed, a lab on a large spinning disk is like a lab on an accelerating elevator: Both labs accelerate to one direction.

What I said about twin paradox is only true if we consider long times, or if we are in the center of disk frame.

 

[Sorcerer]

Time dilation between clocks at relative rest depends on the difference in potential between their positions, not on their proper accelerations.

The above applies to both: accelerated/rotating frames in outer space and gravitational fields near a big mass.

So in the second part here, are you saying that, supposing you have an accelerating, rotating disk, that clocks will move slower on the edge than closer to the middle?

 

[A.T.]

supposing you have an accelerating, rotating disk, that clocks will move slower on the edge than closer to the middle?

On a rotating disk, the co-rotating clocks tick slower at the edge, than in the middle.