Spinning disk and the speed of light

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I was wondering about something. Say you had a really large disk, impossibly huge, and you span it around. Presumably you could only spin it so fast that the edge of the disk cannot be spinning faster than the speed of light. But why? What stops you from spinning it just a little bit faster?
 

disregardthat

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I believe it's because the mass of the disc moving near the speed of the light becomes so heavy that to accelerate it further takes extremely much force. if the object span at c, it would be at infinite weight, which means you have to use infinite energy\force to accelerate it, which means it is impossible.
 

Chris Hillman

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andytoh just asked whether open questions remain in str, and I replied by mentioning various "relativistic paradoxes" as evidence that suprises may remain in store. So I just thought I'd point out that this spinning disk is one of the oldest, the Ehrenfest spinning disk paradox; see http://en.wikipedia.org/w/index.php?title=Born_coordinates&oldid=53957524
and http://www.math.ucr.edu/home/baez/physics/Relativity/SR/rigid_disk.html

Obligatory warning: arXiv eprints on relativistic "paradoxes" unfortunately have a greater than average chance, in my experience, of being badly written and even incorrect. I wish to avoid "debunking" in PF so I decline to name names :-/ but I should point out that quite a few recent eprints "rediscover" old errors which, unknown to the authors, were cleared up literally decades ago. The fact that arguments continue even in the arXiv does not imply that the mathematical facts are unknown (of course not, since these "controversies" mostly come down to making a computation, whose result is unambiguous). Rather, this should be taken as an embarrassing indication that the quality of "research" in physics varies widely, to put it as kindly as possible. Always remember: arXiv eprints are unrefereed, some "research journals" publish frequently inadequately unrefereed articles, and even the best journals sometimes publish nonsense (c.f. the notorious Bogdanov case).

Also, there is at least one long-running edit war in Wikipedia between someone who doesn't accept the mainstream view concerning at least one such paradox and the other editors, so it is particularly important to be very cautious in reading Wikipedia articles on relativistic "paradoxes".
 
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Demystifier

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Chris, at least you could indicate which explanations on these paradoxes you find correct.
 

Chris Hillman

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I take your point...

Chris, at least you could indicate which explanations on these paradoxes you find correct.
Well, there's a long list, but regarding the Bell's "paradox" and Ehrenfest's "paradox", see the cited versions of two Wikipedia articles (which are almost entirely due to myself) listed at http://en.wikipedia.org/wiki/User:Hillman/Archive, namely "Rindler coordinates", "Born coordinates", "Bell's spaceship paradox", "Ehrenfest's paradox" (four articles, all relevant to discussing these two "paradoxes"). See also the citations, especially the review paper by Oyvind Gron.

Part of the confusion over the "Ehrenfest paradox" involves confusion over what the "proper" statement of the alleged paradox should be, so I hesitate to try to give a short answer there, but Bell's paradox at least is easily disposed of: the string must eventually break, because the Bell congruence is nonrigid while the Rindler congruence is rigid.

Hope you like the articles! (I believe that pervect has worked on at least one of these more recently, by the way.)
 
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Wooaaahhh!! When I made this thread I had no idea this was some kind of famous paradox, in fact I know close to nothing about relativity - all I know is that nothing goes faster than light (I imagine an assymptotic situation where the energy required to get a mass up to the speed of light shoots off to infinity at c).

The only reason I asked this question is because in my fluid mech class we were looking at vector fields and one of them was basically a spinning disk with no boundary. One of the questions was, why is this situation unphysical? Because particles at the edge(?) would be moving faster than the speed of light!! It baffled me at first but I kind of get it now, it's just not possible to make a disk big enough without it collapsing under its own weight, or even if you did you just couldn't spin it very fast.
 
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disregardthat

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If you managed to build an unbreakable disc, to move it to the speed of light would be impossible. It would either need to break, or you would have to push infinite force on it.
 

Hans de Vries

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So I just thought I'd point out that this spinning disk is one of the oldest, the Ehrenfest spinning disk paradox
I once found an amusing numerical coincidence here in regard to semi-classical
microscopic systems:

A spinning wheel with its mass at a distance of half the Compton radius [itex]\hbar/(mc)[/itex]
has the following angular momenta for various gamma:

[tex]
\begin{array} {l}
\gamma\ =\ 2 \qquad \Rightarrow \qquad r \times p \ =\
\sqrt{\frac{1}{2}\left(\frac{1}{2}+1 \right)} \hbar \\ \gamma\ =\ 3
\qquad \Rightarrow \qquad r \times p \ =\ \sqrt{1\left(1+1 \right)}
\hbar
\\ \gamma\ =\ 4 \qquad \Rightarrow \qquad r \times p \ =\
\sqrt{\frac{3}{2}\left(\frac{3}{2}+1 \right)} \hbar \\ \gamma\ =\ 5
\qquad \Rightarrow \qquad r \times p \ =\ \sqrt{2\left(2+1 \right)}
\hbar
\end{array}
[/tex]

So for whole values of Lorentz contraction one gets the angular momenta
of spin 1/2, 1, 3/2, 2 .... A sort of "Lorentz contraction quantization" :smile:


Regards, Hans
 
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Well, there's a long list, but regarding the Bell's "paradox" and Ehrenfest's "paradox", see the cited versions of two Wikipedia articles (which are almost entirely due to myself) listed at http://en.wikipedia.org/wiki/User:Hillman/Archive, namely "Bell's spaceship paradox", "Rindler coordinates", "Born coordinates" (three articles, all relevant to discussing these two "paradoxes"). See also the citations, especially the review paper by Oyvind Gron.
I have some questions with regards to the Ehrenfest paradox as writen in the referenced article.

In particular with Einstein's referenced usage of [itex]2 \pi r /\sqrt{1 - v^2}[/itex].

While I understand and do not argue against the fact that the moving observer measures a longer circumference than the observer at rest I still have some questions:

It is entirely accurate to use the velocity v for the accelerated motion around the platform in calculating the Lorentz contractions? Afteral the trains not just move at v but in addition to that they constantly accelerate towards the center of the circle.

The assumption that r is similar for both the accelerating observer and the observer on the platform leaves me with some questions. While it is true that the center of the circle is orthogonal to the direction of motion and thus there is no Lorentz contraction the accelerating observer has a proper constant acceleration towards the center of the disk so it seems that that must influence the measurement of distance.

Where am I going wrong?
 
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Chris Hillman

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Which version of which article?

Hi, MeJennifer,

I have some questions with regards to the Ehrenfest paradox as writen in the referenced article.
Are you referring to http://en.wikipedia.org/w/index.php?title=Born_coordinates&oldid=53957524 ? (Note: I cited the version of 01:12, 19 May 2006, not the current version, which might be considerably better, or which might be much worse--- who knows? I won't take the time to check since I am no longer editing the WP, and there seems little point to reading the thing carefully if I am not intending to try to correct any mistakes I find.).

In particular with Einstein's referenced usage of [itex]2 \pi r /\sqrt{1 - v^2}[/itex].
I can't seem to find that right now--- which section does it occur in?

Why do you say "usage"? If you are discussing the "measured circumference" of the disk, this quantity is to be computed from first principles, not to be assumed.

While I understand and do not argue against the fact that the moving observer measures a longer circumference than the observer at rest I still have some questions:
Caution: the whole point of the article was that one must be careful to explain what measurement procedure is intended here! Failure to recognize this simple point is one cause of a good deal of the confusion exhibited in the bad papers on this topic.

It is entirely accurate to use the velocity v for the accelerated motion around the platform in calculating the Lorentz contractions? Afteral the trains not just move at v but in addition to that they constantly accelerate towards the center of the circle.
I don't understand what is troubling you, but it might help to recognize that the kinematical decomposition (acceleration vector, expansion tensor, vorticity tensor) of the timelike congruence consisting of the world lines of bits of matter in the "rigidly spinning" disk (the notion "rigidity" intended here--- vanishing expansion--- also requires careful explanation) is just what you need to treat the acceleration properly.

The assumption that r is similar for both the accelerating observer and the observer on the platform leaves me with some questions.
I'll stop here since I'm not even sure you are reading what I intended you to read.
 

Hans de Vries

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Where people often go wrong is that they consider the rotating frame as a
valid reference frame. It's not. An observer on a circular track a a speed
close to c sees himself at the end of a highly flattened ellipse as you might
expect and he is continuously changing from one reference frame to another.

Some points:

1) The remark of the Wikipedia article that Einstein said that the rotating
observer sees a longer path by gamma ([itex]2\pi r/\sqrt{1-v^2/c^2}[/itex] ) instead of
shorter is remarkable. I can't believe that to be correct. The total path he
sees is the sum of all the shorter paths he sees when rotating at the end
of the flat ellipse and therefor shorter as well (by a factor of gamma).


2) Another claim (often made in connection with Thomas Precession) is that
the rotating observer goes through an angle of [itex]2\pi /\sqrt{1-v^2/c^2}[/itex] instead of [itex]2\pi[/itex].
This is wrong. The argument is that the observer is always at the end of the
flat ellipse making a very sharp turn and that therefor the total angle he sees
is more than 360 degrees. It is true that the angle per unit of time he goes
through is greater by a factor gamma, but, the total time he observes as
needed to make a complete turn is also smaller by a factor gamma as a result
of time dilation. The two effects cancel and the total observed angle is 2 pi.


3) A real spinning disk would have to resist Lorentz contraction at the edge.
Say, if gamma is 1.1 then an observer laying on the rigid wheel along the
edge with hands and feet bound would be stretched by 10% :eek: in his
reference frame.


Regards, Hans
 
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Chris Hillman

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Yes, but what version?

Hi, MeJennifer,

It is essential that you read the same version as the one I am looking at. The version I am looking at is the one listed at http://en.wikipedia.org/wiki/User:Hillman/Archive, namely http://en.wikipedia.org/w/index.php?title=Ehrenfest_paradox&oldid=58681705 (the version from 00:59, 15 June 2006). You need to read that in conjunction with the supporting articles in the versions listed in my archive, http://en.wikipedia.org/w/index.php?title=Rindler_coordinates&oldid=51749949 and http://en.wikipedia.org/w/index.php?title=Born_coordinates&oldid=53957524

"Circumference": same comment I had for Hugo: you need to be clearly aware of an ambiguity in what you consider to be "one circuit around the track". BTW, I should have said that User:Pjacobi made significant contribs to the cited version of the article "Ehrenfest's paradox", in particular the discussion of the circular train track.
 
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Chris Hillman

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Einstein said what?!

Hi, Hans,

Where people often go wrong is that they consider the rotating frame as a
valid reference frame.
Please, please, do not dismiss my comments about the neccessity of clarifying any technical terms you use, bearing in mind that many here are not familiar with the relevant research literature.

I consistently use "frame field" to denote the concept discussed in http://en.wikipedia.org/w/index.php?title=Frame_fields_in_general_relativity&oldid=42117350, which is standard usage in the literature. It is simply not possible to understand many "paradoxes" without being comfortable with this fundamental notion. As I carefully explained in a post in another thread today (see "Any Research Left to do in Special Relativity?"), frame fields, congruences, kinematic decomposition, components of tensors wrt a frame, etc., are not "part of gtr", but are an essential tool in relavistic physics irrespective of gravitation or even curved spacetimes.

1) The remark of the Wikipedia article that Einstein said that the rotating
observer sees a longer path by gamma ([itex]2\pi r/\sqrt{1-v^2/c^2}[/itex] ) instead of
shorter is remarkable. I can't believe that to be correct. The total path he
sees is the sum of all the shorter paths he sees when rotating at the end
of the flat ellipse and therefor shorter as well (by a factor of gamma).
If you are not looking at the version cited above, I am not responsible for what you are examining. Assuming that you are looking at the same version that I am:

If you mean that you don't believe Einstein stated the opinion described, the article (in the version cited) cites the original paper. Your confusion may be based upon misunderstanding what one means by "sees" and "path". It is essential to recognize that the informational content of this and the other WP articles I have cited resides in the mathematics and its interpretation, not in an attempt to summarize the interpretation in natural language which is open to misunderstanding.

If you read these articles "actively", verifying the math as you go, you should be able to understand what the stated result describes geometrically and why it is true (in the unambiguous mathematical sense of resulting from a simple computation).

2) Another claim (often made in connection with Thomas Precession) is that
the rotating observer goes through an angle of [itex]2\pi /\sqrt{1-v^2/c^2}[/itex] instead of [itex]2\pi[/itex].
This is wrong.
No, it's ambiguous if you haven't specified what you mean by "one circuit" (a picture is worth a thousand words; please draw one and make the obvious guess from your drawing about what ambiguity I have in mind), or what is pretty much the same thing, what measurement procedure you have iin mind.

3) A real spinning disk would have to resist Lorentz contraction at the edge.
Please be careful to avoid glibly conflating kinematics with dynamics. The phrase "resist Lorentz contraction" is, I should think, obviously suspect.

There is no doubt about what str predicts, once we agree upon how to model the situation. An additional potential source of difficulty here is that we wish to avoid (if we are wise) trying to model the spin-up phase of an elastic disk (in which case we would need to use an appropriate relavistic version of Hooke's law, which is incompatible with str as stated in elementary physics).

I took care in writing the articles I cited to try to avoid misunderstanding, and I provided citations, especially to a useful review paper. I would naturally expect anyone who wishes to contradict my assertions to be familiar with the citations in that review, and to be familiar with the mathematical techniques I used in the analysis I presented. (See the book by Eric Poisson for a recent introduction to the kinematical decomposition wrt a congruence. See the old book by Harley Flanders, Differential Forms and their Application to Physics for a readable introduction to frame fields.) I know that puts a heavy burden on you and MeJennifer, but the techniques at least or worth learning for many other reasons.
 
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Chris Hillman

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Spinning up a disk?

If you managed to build an unbreakable disc, to move it to the speed of light would be impossible. It would either need to break, or you would have to push infinite force on it.
I was careful to remark in various of the cited versions of the four articles that there is something to explain here, regarding "rigid" rotation (constant angular velocity) versus spinning up a neccessarily non-rigid disk (neccessarily non-rigid because a torque applied at the axle cannot have an instantaneous effect throughout the disk). Some of the papers discussed by Gron do attempt to model spinup, which results in various degrees of catastrophic failure (in the papers, as well as, ultimately, in the disk).
 

pervect

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Hi, Hans,

I consistently use "frame field" to denote the concept discussed in http://en.wikipedia.org/w/index.php?title=Frame_fields_in_general_relativity&oldid=42117350, which is standard usage in the literature. It is simply not possible to understand many "paradoxes" without being comfortable with this fundamental notion. As I carefully explained in a post in another thread today (see "Any Research Left to do in Special Relativity?"), frame fields, congruences, kinematic decomposition, components of tensors wrt a frame, etc., are not "part of gtr", but are an essential tool in relavistic physics irrespective of gravitation or even curved spacetimes.
Unfortunately, many students of STR here at Physics Forums are not going to be comfortable or familiar with these tools, as they are quite advanced - while some of the basics of STR can be learned at the high school level.


If you are not looking at the version cited above, I am not responsible for what you are examining.
I notice that Rod Ball has been editing the Ehrenfest paradox article, looking at the history of the article. (But not recently, his last edit was in August).

Users with an interest in personalities might look at
http://en.wikipedia.org/wiki/Wikipedia:Requests_for_comment/Rod_Ball to see why I would think that fact is significant.
 

pervect

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Hi, Hans,

There is no doubt about what str predicts, once we agree upon how to model the situation. An additional potential source of difficulty here is that we wish to avoid (if we are wise) trying to model the spin-up phase of an elastic disk (in which case we would need to use an appropriate relavistic version of Hooke's law, which is incompatible with str as stated in elementary physics).
I think we got into this argument a little bit before, in another thread involving the relativistic disk.

While Hooke's law is not relativistic, I do not believe it has to be to be useful.

The idea is basically that we can use Dixon's formalism (as mentioned by Stingray in the previous thread) to provide a rigorous defintion of dynamic rigidity. This is the tricky part of the problem, IMO.

To be more specific, Proc Roy. Soc. Lond. A. 314, 499-527 (1970) talks about this defintion of dynamic rigidity.

Once we have this notion, this means that we can talk about "spinning up" a disk in an unambiguous fashion.

What this means, basically, is that we can consider ideas like taking a stress-free rotating disk with a stress energy tensor that looks like the one below in a cylindrical frame field [itex] ( \hat{t},\hat{r},\hat{\theta},\hat{\phi} )[/itex]

[tex]
\begin{pmatrix}
E(r) & 0 & P(r) & 0\\
0 & 0 & 0 & 0\\
P(r) & 0 & 0 & 0\\
0 & 0 & 0 & 0
\end{pmatrix}
[/tex]

and slow it down, reversibly, into a stressed non-rotating disk. (If we spin it up again, the disk will become non-stressed).

Or we can start with a non-stressed non-rotating disk, and spin it up into a stressed spinning disk.

The biggest thing to beware of is that at some point assuming Hooke's law will probably wind up violating one of the energy conditions, which would be a bit unphysical.
 
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Hans de Vries

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Hi, Chris.


A few remarks:


About 1) The circumference of a circle as measured by the rotating observer.

Imagine traveling at say gamma is 10 on a straight line on a multi year trip.
However, do you really know if you are on a straight and not on a very large
circle? If the "longer path" assumption is true then the path measured by
the traveling observer in the latter case would be 10x10 = 100 times longer
in the same period .... This difference should be instantaneously visible.

One could further simplify the situation by traveling on a square path instead
of a circle avoiding over complications. (The, by itself correct, statement
that more, shorter rulers fit on on the closed trajectory applies here as well).



About 2) The total angle observed while completing a circular trajectory.


I assume the standard SR measurement methodology here. I did some
extensive simulations two or three years ago when searching for equilibriums
in off axis retarded Lienard Wiechert potentials in the semi-classical case of
two opposite charges chasing each other on circular and other paths at
relativistic speeds.

Rotating on a circle, close to c, one observes the path as a flat ellipse.
The rotating observer finds himself always at the far end of the ellipse
continuously making a sharp turn. The angular speed by which he turns
depends on the curvature of the sharp end of the ellipse. piece by piece
he moves on a circle with the same curvature. This is a small circle which
can become arbitrary small depending on how close his speed is to c.
However, he will always observe the center at the same side (right or left)
at a 90 degree angle with his path at a distance which is potentially
much larger as the diameter of the circle determined by the curvature
of the sharp edge of the ellipse. After having observed a total angle of
360 degrees on this circle he will also have completed the round trip on
the circle as observed from a stationary position.

(I don't know if this agrees or disagrees with one of the (your?) Wikipedia
entries)



About 3) The rigid rotating disc


Taking into consideration that the solutions of both the wave-equations
of the potentials of the EM field and (for instance) the Klein-Gordon equation
for matter waves are automatically Lorentz contracted when moving, and do
so by them self without a single reference needed to SR. (I can do the
the derivations here if you want).

What this amounts to is that objects on the edge of the disc have to be
considered physically stretched if they are not contracted. That was my
point.



Regards, Hans.
 
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Chris Hillman

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Handing this over to pervect

Hi, pervect,

stress-free rotating disk
I agree that you wrote down a pressure-free stress tensor, and I agree that this is often appropriate in a weak-field situation (and in this problem we are of course neglecting gravitation entirely). On the other hand, those authors who wish to carefully consider whether or not internal stresses in the disk play a decisive role cannot assume these stresses are neglible (see some of the papers cited by Gron), so depending upon context, it may or may not be appropriate to try to evade (or to confront!) the issue of stresses.

While Hooke's law is not relativistic, I do not believe it has to be to be useful.
Again, I think that this depends on context, but no doubt there is more than one way to flay a paradox! I don't deny that the urge to seek an "elementary resolution" is natural, although I question whether it is, in this instance, the most efficient route.

Once we have this notion, this means that we can talk about "spinning up" a disk in an unambiguous fashion.
Not sure I entirely agree, but never mind, since:

Unfortunately, many students of STR here at Physics Forums are not going to be comfortable or familiar with these tools, as they are quite advanced - while some of the basics of STR can be learned at the high school level.
I think my point is that going beyond high school mathematics makes it, at the very least, much easier to efficiently resolve some of the trickier paradoxes, but I don't disagree that the techniques I used in my analysis might pass over the head of most members of PF (unless they make the effort to study them).

For this reason, and because of the difficulty I seem to be encountering in conveying the importance that everyone be studying the same version of the same WP article, and because I have no wish to repeat here my bad experience in Wikipedia, I'll happily hand this thread over to you now.
 

Hans de Vries

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I notice that Rod Ball has been editing the Ehrenfest paradox article, looking at the history of the article. (But not recently, his last edit was in August).

Users with an interest in personalities might look at
http://en.wikipedia.org/wiki/Wikipedia:Requests_for_comment/Rod_Ball to see why I would think that fact is significant.
It's a pity to see things like this happening.

I do see now how much time and effort Chris has devoted to write all these
articles. Don't let this get you down Chris!


Regards, Hans
 
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I would naturally expect anyone who wishes to contradict my assertions to be familiar with the citations in that review, and to be familiar with the mathematical techniques I used in the analysis I presented. (See the book by Eric Poisson for a recent introduction to the kinematical decomposition wrt a congruence. See the old book by Harley Flanders, Differential Forms and their Application to Physics for a readable introduction to frame fields.) I know that puts a heavy burden on you and MeJennifer, but the techniques at least or worth learning for many other reasons.
For the good order Chris, my questions were simply made from a perspective of clarification and improving my understanding not from a position of trying to contradict you.
I am here to learn and in the best of my ability, which is not that great, to help others. :smile:
 
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pervect

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Hi, pervect,



I agree that you wrote down a pressure-free stress tensor, and I agree that this is often appropriate in a weak-field situation (and in this problem we are of course neglecting gravitation entirely). On the other hand, those authors who wish to carefully consider whether or not internal stresses in the disk play a decisive role cannot assume these stresses are neglible (see some of the papers cited by Gron), so depending upon context, it may or may not be appropriate to try to evade (or to confront!) the issue of stresses.
I think one of the more modest points I want to make is that if we consider rotating disks that we can actually make out of materials like steel, or perhaps carbon nanotubes and glue, that we can say that the stresses can't contribute very much to the mass or energy of the disk because the materials simply won't support levels of stress which could make a difference.

Unfortunately, if we restrict ourselves to normal materials like this, we'll probably find we can't actually build relativistic disks, that they'll fail before we get to relativistic velocities.

We might try to imagine spinning up an atomic nucleus, or perhaps some other form of matter that's held together by something much stronger than chemical bonds - but I'm not sure how to write down the stress-energy tensor of such a material.

pervect said:
Once we have this notion, this means that we can talk about "spinning up" a disk in an unambiguous fashion.
Not sure I entirely agree, but never mind, since:
Shucks - here I thought we could have an argument and maybe I could learn something.

I was re-reading the Dixon article, and I'll have to admit that there is some "fine print" about bodies "not being too big" that I don't really follow. While I could be fooling myself, I think the point of this fine print is to avoid some problems with extreme conditions like Tippler cylinders, which don't quite fit into the neat model I've outlined.


For this reason, and because of the difficulty I seem to be encountering in conveying the importance that everyone be studying the same version of the same WP article, and because I have no wish to repeat here my bad experience in Wikipedia, I'll happily hand this thread over to you now.
Gee, thanks :-)
 

Demystifier

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pervect

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For what it's worth, I don't believe I've contributed to the Ehrenfest paradox article, just the Bell Spaceship paradox article (which is where I became acquainted with Rod Ball).

As far as the Ehrenfest paradox goes, my personal favorite explanation is along the lines of http://arxiv.org/abs/gr-qc/9805089

The point, put simply, is that it's not possible to globally splice space-time so that space is always orthogonal to time on a rotating disk. Unfortunately, I get a lot of "blank looks" (or their electronic equivalent) when I offer this explanation. But I really don't know how to say it in a manner that's any more simple and also close to being correct. The above explanation may even already be slightly over-simplifed.

This is close to the Wikipedia article's resolution, which makes much the same point graphically in http://en.wikipedia.org/wiki/Image:Langevin_Frame_Cyl_Desynchronization.png

Note that talking about the "circumference" of the helix in the above article isn't really well defined, because it just isn't a geometrically closed object.

I haven't previously encountered the Nikolic articles above, (I have encountered articles by the same author with respect to the Bell spaceship paradox). However, some remarks in the abstract tend to make me believe that there isn't any serious disagreement between the Tartaglia view and these articles, specifically

The generally accepted formula for the space line element in a non-time-orthogonal frame is found inappropriate in some cases.
Thus Nikolic recognizes the non-orthogonality problem, which is where I think the key difficulty lies.

I have yet to give the above two URL's a really close read (they look interesting).

I will also agree with Chris's remarks that there are unfortunately a lot of confused papers out there on the topic, so that the student who picks a paper at random on the rotating disk, seeking enlightenment, risks being further confused. Tartaglia makes much the same point that Chris and I do about the existence of many confused papers on the topic in the literature.
 

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