Unveiling the Observations of a Spinning Disc Approaching Light Speed

[Denton]

Suppose you have a very strong, spinning disc with a diameter of say 10 km in length. At the centre the centripetal velocity is approaching the speed of light, how would we observe the outer edge of the disc to be.

 

[DaveC426913]

You will never get the disc to spin up that fast. As the disc's edge speed approaches c, it will require more and more energy to further accelerate the disc, ultimately requiring infinite energy.

 

[Dale]

I think this question would require a relativistic version of Hooke's law in order to properly analyze, but I agree with Dave.

 

[Chris Hillman]

Do We Really Need to Reopen That Can of Ex-Mouse-Eating Maggots?

Suppose you have a very strong, spinning disc with a diameter of say 10 km in length. At the centre the centripetal velocity is approaching the speed of light, how would we observe the outer edge of the disc to be.

This is a FAQ; see for example this Physics FAQ article and these previous PF threads:

• the locked thread rotational kinematics from April 2004,
• the thread [thread=149176]Spinning disk and the speed of light[/thread] from Dec 2006,
• the locked thread (about a closely related and also "faux-contentious" and long-ago resolved "paradox") [thread=153770] Why is the Wikipedia article about Bell's spaceship "paradox" disputed at all?[/thread] from January 2007,
• the very long and contentious thread [thread=168121] Stress-energy tensor of a wire under stress[/thread] (about stresses in a relativistically spinning hoop) from April 2007.

What dave said is correct and I hope that will satisfy all readers.

If not, to reformulate the question:

"What is the geometry of a relativistic spinning disk? Especially, a rigidly rotating disk? What is the physical experience of observers riding on the disk?"

Please understand that this is exactly the topic of a huge and mostly ill-informed/incorrect discussion* in the research literature since Ehrenfest first raised the subject in 1907. The discussion was resolved c. 1927 but bad physicists have kept the "debate" alive out of failure to read and understand prior work and failure to master now standard mathematical tools. As usual, Einstein's intuition (expressed in private correspondence) was basically correct, but he was smart enough to recognize that he lacked the relevant mathematical tools (in particular, the kinematic decomposition of a certain non-geodesic timelike congruence, the world lines of particles in spinning disk, or equivalently spinning cylindrical slug) required to most conveniently analyze the situation, and consequently shared his thoughts only with some close friends.

[*I exempt good review papers and good expositions from my criticism of the recent arXiv literature on this topic, since the issues were all resolved by about 1930, but bad/cranky eprints continue to appear, incorrectly claiming that "it's all perfectly simple" or that the mainstream explanations are incorrect.]

I stress that mathematically speaking, there is no doubt about what str predicts, but to understand the mainstream (mathematically correct!) analysis of what str predicts, one needs to be familiar with not only the kinematic decomposition but also quotient manifolds versus submanifolds, the existence of multiple operationally significant notions of "distance in the large" even in flat spacetime, careful discussion of what we mean by "observe", and dozens of other topics which are probably too advanced for this forum.

As typically happens when someone reopens a can of worms, Denton, you have already committed several all-too familiar "errors of discourse" which would have to be patiently corrected in order to discuss this topic. To mention just one, you failed to specify that you are asking for the answer given by standard relativistic physics in flat spacetime (str), assuming that is what you had in mind. (Gtr is not required unless you believe--- largely incorrectly--- that gravitational phenomena are relevant here, but techniques often first encountered in gtr courses, such as the kinematic decomposition, are essential to avoid endless and pointless confusion over mathematical trivialities.)

See [thread=168121]this long PF thread[/thread] (which also discussed the closely related so-called "Ehrenfest's paradox" and "Bell's paradox", which are of course not paradoxical at all)--- in particular, please see my Post #27 in that thread, which warns

it's not nearly that simple!

Please see also the excellent book by Poisson, A Relativist's Toolkit, Cambridge University Press, the quartet of Wikipedia articles in the versions I cited, and the invaluable review paper by Oyvind Gron in the book edited by Rizzi and Ruggiero, Relativity in Rotating Frames, Kluwer, 1994.

I think this question would require a relativistic version of Hooke's law in order to properly analyze

If you want to carefully compare rigidly spinning disks with different constant angular velocities (say $\omega = 0$ vice $\omega=0.01$), you will probably need to model a "spinup phase". This was extensively discussed in the thread just cited, but that discussion will go over the heads of anyone who has not mastered a good deal of relativistic physics at the research level. At the very least, I believe it is reasonable to demand that no-one discuss "relativistic spinup scenarios" who has not previously very carefully analyzed Newtonian spinup scenarios using the theory of elastic materials, and perhaps also "microscopic models" using Hooke's law (as I suggested in the cited thread).

Everyone: please don't raise this subject again until you have at least carefully studied the sources cited above :grumpy: Plus kinematic decomposition (see the book by Poisson just cited) and the Langevin congruence and other topics discussed in the cited thread. Pervect, myself, and greg egan all worked hard to clarify numerous conceptual subtleties which invariably cause confusion if they are not recognized and overcome, so naturally I at least have no wish to reslay the slain!

And everyone: please don't resurrect long-dormant threads, and please don't illustrate the phenomenon discussed in [thread=200063]this thread[/thread]! :yuck: Thanks to all in advance for their cooperation!

 

[pervect]

Suppose you have a very strong, spinning disc with a diameter of say 10 km in length. At the centre the centripetal velocity is approaching the speed of light, how would we observe the outer edge of the disc to be.

I'm not quite sure what aspect of the relativistic rotating disk you are interested in. Probably a reasonable place to start would be with this FAQ entry, which might inspire some more specific questions.

The FAQ above does assume a certain amount of pre-existing knowledge of relativity. I would advise someone just learning about relativity to study other topics than the rotating disk first, because it can be very confusing to the newcommer :-(.

 

[Denton]

I do have a pre existing knowledge of relativity, however I am reaching my limit of understanding for now - I've not yet delt with the complex math of general relativity what with these 'tensors' and whatnot which is vaguely confusing, I understand in principle but not in math yet. I however hope to 'catch' up to all of this once I start University.

 

[jimgraber]

Hi everyone, but particularly Chris:
I'd like a simple (yes-no)answer to a simple question:
Consider the circular train in the current Wikipedia article "Ehrenfest paradox".
Replace the bungee cords with rigid threads, as suggested.
Gently accelerate to 0.6 c.
Do the threads break or not?
TIA.
Jim Graber

 

[Fredrik]

Hi everyone, but particularly Chris:
I'd like a simple (yes-no)answer to a simple question:
Consider the circular train in the current Wikipedia article "Ehrenfest paradox".
Replace the bungee cords with rigid threads, as suggested.
Gently accelerate to 0.6 c.
Do the threads break or not?
TIA.
Jim Graber

That isn't a simple question at all. We have to make some assumptions to make sense of it. In particular, we have to say how the train cars are to be accelerated. These are the assumptions I've made:

* When you say "rigid threads" you mean "threads that will break rather than stretch when a force is applied".

* Each train car has its own engine, controlled by a computer in that train car, and all the computers are running the same program, at the same time, according to clocks in the the train cars that were synchronized in the stationary frame before the acceleration started.

* The acceleration is such that the train cars are "Born rigid" (in the direction of motion) to a good approximation.

With these assumptions, the answer is "yes". From a stationary observer's point of view, the train cars will get smaller (because of Lorentz contraction). This means that distance between the points where the ends of a string are attached will increase, so the strings will break immediately.

The reason I object to the word "rigid" is that there are no rigid bodies in SR. Any two particles in a rigid body is supposed to stay the same distance from each other at all times. SR makes that impossible. If two particles in an accelerating object stay the same distance from each other in one frame, they don't in other frames.

An accelerating object is "Born rigid" if any two parts of it stay the same distance from each other in comoving inertial frames.

 

[pervect]

I do have a pre existing knowledge of relativity, however I am reaching my limit of understanding for now - I've not yet delt with the complex math of general relativity what with these 'tensors' and whatnot which is vaguely confusing, I understand in principle but not in math yet. I however hope to 'catch' up to all of this once I start University.

My favorite paper, which may be a difficult read, is Tartaglia's

http://arxiv.org/abs/gr-qc/9805089

It does mention tensors at various points (right near the beginning for instance). but understanding tensors is not absolutely necessary to understanding the point that the paper, though the use of the T word may scare some readers away :-(.

The basic issue is how one splits space-time into space+time on a rotating disk, i.e. from the abstract of the paper

It is often taken for granted that on board a rotating disk it is possible to operate a global 3+1 splitting of space-time, such that both lengths and time intervals are uniquely defined in terms of measurements performed by real rods and real clocks at rest on the platform.

This is a rather complex statement. A simpler and closely related statement is that it is not possible to synchronize all the clocks on a rotating platform via Einstein's clock synchronization method. You can start at one point on the rim, and synchronize points to this going around clockwise. When you get back to the starting point, however, your last two clocks will not be synchronized.

This synchronization error is what prevents the "global 3+1 split".

The point is that while there is a local split of space-time into space+time that respects Einstein's clock synchronization principle, there isn't any such global split. I.e. there is a commonly held notion that any observer has a coordinate system. Unfortunately, the fine print says "this coordinate system may not cover all of space-time, it may only be good sufficiently close to the observer".

While this is my personal favorite paper, there is at least one entire book on the topic. The particular book that comes to mind, which is a collection of papers, has a website with much of the material online. The website states that the version of the papers online are the "early" versions, so there may be editing which is done in the published work.

The website for this is at:

http://digilander.libero.it/solciclos/

I haven't read every paper in this collection, which is quite large. This is one reason why people are warning you that this can be a very big topic. There is a lot of material written about it, even this *large* book isn't complete.

Note: the website also has some commentary. While I haven't read all of the commentary, either, it looks like it could be quite useful. For instance, Gron talks about Klauber's paper:

Comment on Klauber’s article: “Toward a Consistent Theory of Relativistic Rotation
Klauber [1] analyses rotation with relativistic velocities from several points of view. The analysis is unconventional and controversial, but not uninteresting. Klauber argues against Einstein’s conclusion that the spatial geometry in a rotating reference is not Euclidean, and claims that the conventional relativistic analysis or rotating reference frames is not consistent.
One can think of Klauber’s article as representing the point of view of “The Devil’s Advocate”, and it gives an opportunity to defend the point of view that is about to be “canonized”. In the present note I shall try to show that the relativistic analysis is indeed consistent.

So this may give you a better idea of what's going on in this book. The book's authors have tried to collect a snapsshot of the literature about rotating disks in relativity in one collection. Some of the viewpoints are mainstream, others are not, so you have to have enough knowledge to tell which (the commentary may aid you in this task).

It's been noted that even the professionally written papers on this particular topic are of varying quality, one can't necessarily assume that any given paper is free from subtle conceptual errors or represents a "mainstream" view.

This makes the topic quite challenging. But it's not hopeless, though it is rather involved.

 

[pervect]

That isn't a simple question at all. We have to make some assumptions to make sense of it.

A very good post, and I agree with the answer and the cautions. One thing that's interesting is that assuming

Each train car has its own engine, controlled by a computer in that train car, and all the computers are running the same program, at the same time, according to clocks in the the train cars that were synchronized in the stationary frame before the acceleration started.

and that the magnitude of the accelerations of all the trains set by the program are the same, that the string connecting the cars also breaks even on a straight track. This is called, however, "Bell's spaceship paradox", and is also unfortunately another common source of confusion and false controversy.

 

[jimgraber]

I would like to thank Fredrik very much for a straight forward answer, which even the original poster, who is "not yet at University" can probably understand.
I agree with all three of Fredrik's starred assumptions, and now I would like to ask a more complicated and possibly more difficult question, which fits nicely with the second assumption about how the trains are programmed. "Is there any way the trains can be programmed to accelerate to 0.6 c without breaking at least one thread?
I assume the answer is no, but I want to be sure.
Thanks again.
Jim Graber

 

[pervect]

I've had some spare time, and have been reading more of the online reference I mentioned earlier.

I believe that Cantoni's arguments, as discussed by Gron

http://www2.polito.it/ricerca/relgrav/solciclos/gron_d.pdf

demonstrate that there is no way to program the trains to not break at least one thread. While I can't duplicate the figures in the above article, I'll quote the relevant section to enable the interested reader to find it:

“One can give a consistent definition of the length of the whole “circumference” relative to the rotating reference frame K as the length of the curve PP’ everywhere orthogonal to the world-lines of the reference points of K, starting from an event P and ending on an event P' on the world-line of the same particle of the edge of the disk, and winding once around the axis of rotation. Notice that such a curve is not closed in space-time, and distant events on it, such as P and P’, can in no sense be regarded as simultaneous.

While other authors might use a different definition of circumstance, I think that Cantoni's results demonstrate that with this definition, the circumference increases.

Note that I am assuming that Cantoni's is the proper definition of circumference to use when considering whether or not the threads "break", which I believe is a correct assumption.

[add]I should probably explain more

What one does is take the limiting case where the trains are all points, all of which move at the specified velocity.

Because the trains are now points, there isn't any concern about rigidity of the trains. The trains have essentially been abstracted out of the problem. I think this is OK, because the focus of interest is really on the string, not on the trains.

The length of the string connecting all these points together is a defined quantity, though note that the resulting geometrical figure is not closed. This is an important point which is frequently overlooked. See the diagram in the original link for more information on why the geoemtrical figure is not closed.

So, by considering the case where the trains are replaced with points, one can come up with a figure for the total length of string required to "wrap around" the length of the train so that all the points are connected together. And this length can be shown to increase when the collection of points move, and the acceleration history is not needed to perform this calculation, one needs know only the final velocity.

 

[yoyoq]

if this is also "not that simple" feel free to ignore, but
why wouldn't the "threads" also be lorentz contracted?

in other words C < 2 pi r for rotating objects

 

[yoyoq]

reading up on the related 'Bell spaceship paradox'
from wikipedia:
"...Bell pointed out that length contraction of objects as well as the lack of length contraction between objects in frame S can be explained physically, using Maxwell's laws..."

but that is counter to the relativistic explanation of magnetic field from a current,
where the space between the charges contract and increase the charge density
compared the the 'currents' rest frame

 

[Alfi]

And everyone: please don't resurrect long-dormant threads, and please don't illustrate the phenomenon discussed in [thread=200063]this thread[/thread]! :yuck: Thanks to all in advance for their cooperation!

You seemed to not want this topic rehashed without your mentioned conditions.
In your post you pointed out why, by directing the reader to information sources.
With all the hyperlinks and links to previous threads, I find it difficult to keep track of the posting time stamps. What is to some, a dormant thread is to some others a first time read and may invoke a question or comment.
Some times the choice is to reopen a previous thread or to start a new thread on an old topic. If the later, then the posts contain hyperlinks and links to previous threads.
I find it difficult to keep track of the posting time sta...

sorry, bit of a loop there.

What would you suggest as guide lines to revisiting topics?
If we didn't revisit some ideas, the world would still be flat.

 

[yoyoq]

talking mainly to myself, but i think i accept the standard resolution now.

this site really helped

grenouille-bouillie.blogspot.com/2007/10/how-to-teach-special-relativity.html