What happens to the geometry of a spinning disk as it rotates about its center?

[lovetruth]

If a circular disk rotates about its centre, what will happen to its geometry. Since a spinning disk has velocity gradients, different regions of the disk must contract by different proportions.
For example, a uniformly moving body undergoes length contraction and its new geometry is easily calculated.

 

[Dale]

Take a look at section 3.4.4 here:
http://www.lightandmatter.com/html_books/genrel/ch03/ch03.html#Section3.4

Also:
http://www.phys.uu.nl/igg/dieks/rotation.pdf

 

[yuiop]

If a circular disk rotates about its centre, what will happen to its geometry. Since a spinning disk has velocity gradients, different regions of the disk must contract by different proportions.
For example, a uniformly moving body undergoes length contraction and its new geometry is easily calculated.

If you had a thin circular disk, then yes (in principle) the outer parts would have greater relative motion than the inner parts and would length contract to a greater extent, deforming the disk into a dished shape. Unfortunately you would actually observe this in practice because of the enormous centripetal forces involved to keep the disk intact at relativistic speeds and the disk would be torn apart.

You could get an idea of what is happening by spinning a hoop at relativistic speeds and applying centripetal forces such that the proper length of the hoop remained constant according to an observer on the hoop and the hoop remained unstrained. Under such conditions, the radius of the hoop would be smaller for greater rotational speeds.

P.S. The above will confuse some people that think length contraction is a mathematical rather than physical effect.

 

[ctxyz]

P.S. The above will confuse some people that think length contraction is a mathematical rather than physical effect.

It is a measurement artifact. The result of measuring objects in motion wrt the measuring device is perceived as length contraction.

 

[yuiop]

It is a measurement artifact. The result of measuring objects in motion wrt the measuring device is perceived as length contraction.

How do you explain the Ehrenfest paradox http://en.wikipedia.org/wiki/Ehrenfest_paradox then?

For example let us say you have a 10m track, with 5X1m train carriages on it connected by 5x1m spring links. When the train is moving around the track at relativistic speeds, the springs have a measurable tension increase and the observers on the train also measure the spring lengths between consecutive carriages to increase, even though they are at rest with the springs. Clearly physical effects. With a 10m train on a 10m track, where does the tension come from without something changing its physical length?

 

[bcrowell]

How do you explain the Ehrenfest paradox http://en.wikipedia.org/wiki/Ehrenfest_paradox then?

A related but simpler way to get at the same issues is Bell's spaceship paradox: http://en.wikipedia.org/wiki/Bell's_spaceship_paradox

 

[ctxyz]

How do you explain the Ehrenfest paradox http://en.wikipedia.org/wiki/Ehrenfest_paradox then?

What I told you is applicable to inertial frames, I assume that you are familiar with SR length contraction.

In non-inertial frames, like for example in rotating frames, the situation is much more complicated since the mere act of spinning the disc up to speed introduces strain in it. If you read the reference you cited, you would notice that the disc shatters due to strain well before it reaches any relativistic speed.

For example let us say you have a 10m track, with 5X1m train carriages on it connected by 5x1m spring links. When the train is moving around the track at relativistic speeds, the springs have a measurable tension increase and the observers on the train also measure the spring lengths between consecutive carriages to increase, even though they are at rest with the springs. Clearly physical effects. With a 10m train on a 10m track, where does the tension come from without something changing its physical length?

See above, nothing to do with relativity , per se.

 

[bcrowell]

If you read the reference you cited, you would notice that the disc shatters due to strain well before it reaches any relativistic speed.

Do you mean the Wikipedia article? The one with the prominent warning at the top that says, "This article needs attention from an expert on the subject"?

There is no lower limit on the speed needed in order to produce relativistic effects. The effects are simply smaller when the speeds are smaller relative to c.

 

[ctxyz]

Do you mean the Wikipedia article? The one with the prominent warning at the top that says, "This article needs attention from an expert on the subject"?

There are some excellent references , I go by the references , not by the wiki page.

There is no lower limit on the speed needed in order to produce relativistic effects. The effects are simply smaller when the speeds are smaller relative to c.

The disc shatters at $v=\sqrt{GrS/m}<<c$.

 

[bcrowell]

There are some excellent references , I go by the references , not by the wiki page.

OK. But when you wrote "If you read the reference you cited," you were replying to yuiop, who posted the WP link, not DaleSpam, who posted the other two links.

The disc shatters at $v=\sqrt{GrS/m}<<c$.

Yes, but that doesn't mean that relativistic effects are zero, just that they're small. Also, Ehrenfest's paradox is equally paradoxical in the case where the rotating disk is held together by external forces. In fact, when people talk about rigid bodies in relativity, they normally understand that to mean Born rigidity http://en.wikipedia.org/wiki/Born_rigidity , and one of the standard warnings you'll often see when the concept of Born rigidity is introduced is that it should not be understood as a property of a material operating solely under its own cohesive forces. In general, Born rigidity can only be realized when external forces act throughout the body, and those external forces have to be coordinated in advance according to some plan. This is one of reasons that I agree with the warning banner at the top of the WP article on the Ehrenfest paradox; the discussion of the strengths of materials is irrelevant.

The following from p. 14 of the Dieks paper may also be relevant:

This makes it clear that the Lorentz contraction can be responsible for clearly dynamical effects—the contractions are not just a matter of “perspective” (see [9] and [10]). (Of course, this whole discussion is rather academical because centrifugal forces will tear the cylinder apart before the relativistic effects become noticeable.)

The first sentence supports yuiop's point about the Lorentz contraction's physical reality, and Bell's spaceship paradox is also widely viewed as supporting this. Bell famously got physicists in the CERN cafeteria to agree unanimously on the incorrect answer to his question, because they were all so sure that Lorentz contraction was only a matter of measurement, rather than a real physical effect.

 

[ctxyz]

The following from p. 14 of the Dieks paper may also be relevant:

From Dieks:
"Of course, this whole discussion is rather academical because centrifugal forces will tear the cylinder apart before the relativistic effects become noticeable.)"

Exactly my point.

The first sentence supports yuiop's point about the Lorentz contraction's physical reality, and Bell's spaceship paradox is also widely viewed as supporting this. Bell famously got physicists in the CERN cafeteria to agree unanimously on the incorrect answer to his question, because they were all so sure that Lorentz contraction was only a matter of measurement, rather than a real physical effect.

I do not care much for Bell's paradox as a "physical" proof of length contraction. Firstly, it is only a thought experiment, secondly it has alternative explanations that do not resort to length contraction, I wrote a few on this subject.

 

[bcrowell]

From Dieks:
"Of course, this whole discussion is rather academical because centrifugal forces will tear the cylinder apart before the relativistic effects become noticeable.)"

Exactly my point.

Certainly the whole thing is just a thought experiment. Nobody has suggested that it could actually be realized in a laboratory in the year 2011 in such a way that the relativistic effects could be measured using present-day measuring devices. Dieks hasn't suggested that it could, yuiop hasn't, and I haven't.

I do not care much for Bell's paradox as a "physical" proof of length contraction. Firstly, it is only a thought experiment, secondly it has alternative explanations that do not resort to length contraction, I wrote a few on this subject.

OK, can you point us to them? Of course these discussions about what is "real" or "physical" always get bogged down because nobody can ever define what they mean by "real" or "physical."

 

[ctxyz]

OK, can you point us to them?

Gladly

 

[bcrowell]

OK, can you point us to them? Of course these discussions about what is "real" or "physical" always get bogged down because nobody can ever define what they mean by "real" or "physical."

Gladly

Thanks. I guess this is section 2 excerpted from this paper http://fizika.hfd.hr/fizika_a/av10/a19p119.htm ?

 

[ctxyz]

Thanks. I guess this is section 2 excerpted from this paper http://fizika.hfd.hr/fizika_a/av10/a19p119.htm ?

yes, it is

 

[yuiop]

What I told you is applicable to inertial frames, I assume that you are familiar with SR length contraction.

I am familiar with SR length contraction and I am also familiar (as you should be) with the fact that SR can handle acceleration and non-inertial situations fine. The only thing that SR does not cover is gravity which is addressed by GR.

In non-inertial frames, like for example in rotating frames, the situation is much more complicated since the mere act of spinning the disc up to speed introduces strain in it. If you read the reference you cited, you would notice that the disc shatters due to strain well before it reaches any relativistic speed.

As a disk is spun to very high angular velocities it tends to expand due to the fact that the cohesive forces of the material become insufficient to provide the centripetal forces to hold the disk at constant size (as measured in the inertial frame). In the Ehrenfest paradox it is assumed that external forces are applied such that the material of the disk/train does not expand and maintains constant spatial dimensions as measured in the inertial frame. Since strain is defined as an increase in length there can be no strain on the circumference of the disk if we do not invoke relativistic length contraction. The reason the disk/train comes under stress and eventually breaks in this situation is that the length of the disk/train circumference is increasing as measured by an observer at rest with the rotating disk/train and this strain (that is direct result of length contraction) results in real stress that eventually breaks the disk perimeter of train connections.

I do not care much for Bell's paradox as a "physical" proof of length contraction. Firstly, it is only a thought experiment, secondly it has alternative explanations that do not resort to length contraction, I wrote a few on this subject.

The explanation you linked to does not mention length contraction but that does not mean it is not involved. In fact the article states "Thus, we can conclude that the distance between the rockets in the co-moving frame is larger than their distance L as measured in the launcher frame" which is in fact due to length contraction. In the inertial frame the rulers of the co-moving observers are length contracting and this causes the co-moving observers to measure the distance between the rockets as larger than the constant distance measured by the inertial observers. The string snaps due to increased strain causing increased physical stress in the string. Although we can have long philosophical debates about what is real and what is not, I think most people would agree that a string breaking is indisputably a physical effect. If the increased distance between the rockets as measured by the co-moving observers is just an artefact of measurements methods then the string would not actually break. Although the Bell rocket experiment has not actually been carried out, the mainstream and logical conclusion is that SR predicts the string will break. You can disagree, but basically that means you do not understand relativity or you do not believe that relativity is a correct theory.

Let me try an analogy. Let us say we have a rod made of a metallic material that expands with increasing temperature. We heat the rod up to a high temperature and clamp it in a very robust rigid frame at room temperature. Along the heated rod we place some small rulers that are at the same temperature as the rod. We allow the rod (and small rulers) to cool down to room temperature. The length of the rod appears to have increased according to the small cooled down rulers, but the length of the rod in the clamp frame appears to be constant. Is this a measurement artefact? No, it can be shown that the rod is now physically stressed as it had been stretched even though its length appears to remain constant in the clamp frame. In fact with sufficient heating and cooling the metal rod will break and this stressing of the rod due to temperature changes is just as physical as the stressing of the string in Bell's rocket paradox due to relativistic length contraction.