Spinning disk and the speed of light

[billiards]

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?

 

[ranger]

Hi billiards,

There was a similar discussion sometime ago. Mayb reading it may give you an idea:
https://www.physicsforums.com/showthread.php?t=75563

 

[disregardthat]

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]

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".

 

[Demystifier]

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

 

[Chris Hillman]

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.)

 

[billiards]

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.

 

[disregardthat]

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]

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 \hbar/(mc)
has the following angular momenta for various gamma:

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

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" Regards, Hans

 

[MeJennifer]

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 written in the referenced article.

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

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?

 

[Chris Hillman]

Which version of which article?

Hi, MeJennifer,

I have some questions with regards to the Ehrenfest paradox as written 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 2 \pi r /\sqrt{1 - v^2}.

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.

 

[MeJennifer]

Sorry Chris, my mistake, I was reading http://en.wikipedia.org/w/index.php?title=Ehrenfest_paradox&oldid=58681705"

 

[Hans de Vries]

Sorry Chris, my mistake, I was reading http://en.wikipedia.org/w/index.php?title=Ehrenfest_paradox&oldid=58681705"

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 (2\pi r/\sqrt{1-v^2/c^2} ) 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 2\pi /\sqrt{1-v^2/c^2} instead of 2\pi.
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% in his
reference frame. Regards, Hans

 

[Chris Hillman]

Yes, but what version?

Hi, MeJennifer,

Sorry Chris, my mistake, I was reading http://en.wikipedia.org/w/index.php?title=Ehrenfest_paradox&oldid=58681705"

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.

 

[Chris Hillman]

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 (2\pi r/\sqrt{1-v^2/c^2} ) 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 2\pi /\sqrt{1-v^2/c^2} instead of 2\pi.
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.

 

[Chris Hillman]

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]

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]

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 ( \hat{t},\hat{r},\hat{\theta},\hat{\phi} )

<br /> \begin{pmatrix}<br /> E(r) &amp; 0 &amp; P(r) &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 0\\<br /> P(r) &amp; 0 &amp; 0 &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 0<br /> \end{pmatrix}<br />

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.

 

[Hans de Vries]

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.

 

[Chris Hillman]

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]

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

 

[MeJennifer]

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.

 

[pervect]

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.

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]

Hope you like the articles! (I believe that pervect has worked on at least one of these more recently, by the way.)

I like them only partially. Since I was not completely satisfied with the existing resolutions of the Ehrenfest paradox, I made my own resolution:
http://arxiv.org/abs/gr-qc/9904078
See also
http://arxiv.org/abs/gr-qc/0307011

 

[pervect]

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.

 

[MeJennifer]

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.

While I do not disagree that a 360 degree rotation in the accelerating frame is not identical to a complete circumnavigation of the circle of the inertial frame I do not quite understand what you are saying here with regards to a geometrically closed object.
Could you please explain what exactly is not closed here?

Note that the accelerating observer is technically circumnavigating a non inertial circle surrounding the rotating disk.

 

[pervect]

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.

Let me expand a bit on these remarks and say that one might well be interested in inquiring what sort of effects could be predicted from STR about rotating disks we can actually build out of real materials. (I don't necessarily expect that such effects will be experimentally observable with current technology due to their very small magnitude).

In such an effort, I feel that Hooke's law could be used and would be useful to calculate observables such as total energy of the disk as a function of its angular velocity omega (possibly including its thermodynamic temperature T if needed), and the total angular momentum of the disk as a function of omega (and possibly T). I am also sticking my neck out a tiny bit by saying that these functions exist, as I remarked earlier. I think this is where the interesting question lies, but I'm not sure there is much more to be said that hasn't been said. Chris Hillman has expressed some doubts about the self-consistency of this interpretation, my arguments for its existence are primarly physical and intuitive rather than mathematical and rigorous. More rigorous treatments do exist in the literature (i.e. Dixon), but there is still the question of the "fine print" in these more rigorous argments.

Note that in the above context the plastic flow of a material comprising a physical disk that we could actually build would not be taken as a disproof of relativity, but as a non-ideal effect that would have to be accounted for in the "error budget" of the experiment, placing some further constraints on how accurate of an experiment we could actually build.

 

[pervect]

While I do not disagree that a 360 degree rotation in the accelerating frame is not identical to a complete circumnavigation of the circle of the inertial frame I do not quite understand what you are saying here with regards to a geometrically closed object.
Could you please explain what exactly is not closed here?

Look at http://en.wikipedia.org/wiki/Image:Langevin_Frame_Cyl_Desynchronization.png

This is a 3-d space-time diagram with time running "up" the center axis. Thus we have two space dimensions, one time dimension, in this 3-d space-time diagram.

Note that the set of points regarded as "simultaneous" as defined by Einstein clock synchronizaation do not form a closed set. They form a non-closed helix.

Look at http://en.wikipedia.org/wiki/Image:Relativity_of_simultaneity.png for a 2-d version of the "line of simultaneity" of a moving observer. The "line of simultaneity" is just a plot of the set of points regarded as "simultaneous" by some specific observer as defined by Einstein clock synchronization.

 

[MeJennifer]

Are you referring to http://en.wikipedia.org/w/index.php?title=Born_coordinates&oldid=53957524 ?

I read it.

It is fascinating to see how an apparent and simple movement such as a rotating disk creates such complexities! Science is most certainly not boring!

One question though, at one point you write:

If they measure larger distances, regardless of which method of measurement they use, they will most likely obtain results inconsistent with any Riemannian metric. In particular, in the simple case of radar distance, owing to various effects such as the asymmetry already noted, they will conclude that the "geometry" of the disk is not only non-euclidean, it is non-Riemannian.

That seems to be true to me only if those observers are unwilling to let go of the idea of isotropic speed of light for accelerating observers. Or am I mistaken in this?

 

[MeJennifer]

This is a 3-d space-time diagram with time running "up" the center axis. Thus we have two space dimensions, one time dimension, in this 3-d space-time diagram.

Correct, I realize that.

Note that the set of points regarded as "simultaneous" as defined by Einstein clock synchronizaation do not form a closed set. They form a non-closed helix.

Yes the set of points of the rotating ring (the edge of the disk), from the perspective of an observer on one point of this ring, cannot possibly be clock synchronized using the Einstein synchronization method. I agree with that. But what I do not understand is how you connect your observation that this set forms a non-closed helix with your observation that its circumference is not well defined.

Clearly the circumference is very well defined. It really is 2 \pi radians of a circle in flat Minkowski space surrounding the rotating disk.
Needless to say that for the accelerating observer it seems much more than 2 \pi radians because of the simultaneity situation you just highlighted. But that is not a surprise, Einstein was clear in his postulates that the laws of physics are not (necessarily) identical in an accelerated and inertial frame.

At least that is how I see the issue, am I wrong?

By the way, can you give me an example of a set that forms a closed helix?

 

[pervect]

If we assume we have a long cable, with tensile strength T, and density rho, a Newtonian analysis gives me the following, for an estimate of "how fast can we actually spin a disk made out of the strongest known materials".

I'm assuming that the disk in this case is a single cable joined end-end. (Perhaps this isn't the optimum configuration, however) and that the cable fails when its tensile strength is exceeded.

For a small segment of cable dtheta

total radial force = T A dtheta

where A is the cross-sectional area of the cable, and T is the tensile strength (in pascals, i.e Newton/m^2)

mass of cable segment = rho A R dtheta

where rho is the density (kg/m^3), A is the cross sectional area, and R is the radius of the circle formed by the cable.

This means that the radial acceeration, R omega^2 = T / rho R

where omega is the angular frequency of rotation (radians/sec)

Thus omega = sqrt(T/rho)/r, or r omega = sqrt(T/rho)

So the r*omega at failure is (interestingly enough) independent of the radius for this configuration. r*omega is, of course, the linear velocity.

This means if we take a bucky cable, with an estimated theoretical yield strength of 150 gigapascals, and a density of 1300 kg/m^3 (figures from a random website, http://www.islandone.org/LEOBiblio/SPBI1MA.HTM) that we get a velocity of about 10 km/sec before the cable fails, i.e around 3*10^-5 of the speed of light.

I think I've seen estimates as high as 300 gigapascals, but that would only improve the velocity by a factor of sqrt(2).

 

[pervect]

Let's continue on with the spinning hoop analysis a bit.

Using http://en.wikipedia.org/wiki/Young's_modulus and a value of about 600 giga-pascal for the Young's modulus for a buckytube, we can say that the strain at the tensile limit is rouglyh delta-L/L = .25

So the elastic energy per unit volume for the stressed buckytube at its yield point is just
.5 * 600 Gpa * (.25^2) = 20e9 joules/m^3

This uses Hooke's law, but I think this is a perfectly fair application of Hooke's law.

This compares to about 10^29 joules/m^3 for the rest energy based on a density of 1300 kg/m^3 So the ratio of elastic energy to rest energy is about 2e-19.

The ratio of rotational kinetic energy to rest mass is just

(.5 m v^2) / (m c^2) = .5 (v/c)^2. With v=10,000 m/s and c=3e8 m/s, this gives a ratio of 5e-10

Therfore the elastic energy is negligible compared to the rotational energy in this example.

I'd also like to point out that for a thin cable, the "tension" in the cable is only present by the engineering defintion. In the GR sense of the stress energy tensor, we can say that for a thin cable there are no tension terms in the stress-energy tensor.

[add]
I should be more specific. What I mean is that in the non-rotating frame, i.e. the inertial frame, there are no tension terms in the stress-energy tensor. It is only in the co-rotating frame that there are tension terms in the stress-energy tensor.

So the stretching of the cable (force*distance) contributes to its energy, this is the elastic energy we computed above, but there is no contribution in the sense of (rho+3P), i.e. in the sense of the tensions contributing to the Komar mass [add] when we carry out our analysis in the inertial frame, the obvious choice because of the metric coefficients are Minkowskian.

Because the total momentum of the hoop is zero, it's mass (which will be its invariant mass, it's relativistic mass, AND it's inertial mass), will be E/c^2, where E can be thought of as the sum of the rest energy, rotational energy, and elastic energy as calculated above (and the elastic energy turns out to be by far the smallest contribution).

 

[pervect]

Correct, I realize that.


Yes the set of points of the rotating ring (the edge of the disk), from the perspective of an observer on one point of this ring, cannot possibly be clock synchronized using the Einstein synchronization method. I agree with that. But what I do not understand is how you connect your observation that this set forms a non-closed helix with your observation that its circumference is not well defined.

Clearly the circumference is very well defined. It really is 2 \pi radians of a circle in flat Minkowski space surrounding the rotating disk.
Needless to say that for the accelerating observer it seems much more than 2 \pi radians because of the simultaneity situation you just highlighted. But that is not a surprise, Einstein was clear in his postulates that the laws of physics are not (necessarily) identical in an accelerated and inertial frame.

At least that is how I see the issue, am I wrong?

By the way, can you give me an example of a set that forms a closed helix?

There is certainly a circumference for the disk in a non-rotating Minkowskian frame. However, there isn't really any such thing as the circumference in a "rotating frame". And the helix example shows why. Or rather it should show why. You seem not to get the point I'm trying to make, and I don't understand the difficulty. And I don't seem to be able to think of a way to explain it better than I already have.

And no, there is no such thing as a closed helix, unless one draws the helix on a torroid (i.e. a donught).

 

[pervect]

I read it.

It is fascinating to see how an apparent and simple movement such as a rotating disk creates such complexities! Science is most certainly not boring!

One question though, at one point you write:

That seems to be true to me only if those observers are unwilling to let go of the idea of isotropic speed of light for accelerating observers. Or am I mistaken in this?

I think you may be starting to get the idea. The only thing that may be missing is some background on why people don't want to get rid of the isotropic speed of light assumption. Getting rid of this idea has consequences.

The clock synchronization that makes the speed of light isotropic is the one and only "fair" clock synchronization. The fundamental issue here is isotropy. Getting rid of isotropy plays some merry havoc with the idea of momentum. One basically has to give both the velocity of an object AND the direction that it is moving to compute its momentum when one has anisotropically synchronized clocks. This is because if the speed of light going east-west is different from the speed of light going west-east, the momentum of a physical object with a measured velocity of v going west-east is different from the momentum of a physical object with a measured velocity of v going east-west.

Extremely anisotropic clock synchronizations can have even worse consequences, such as ariving before one arives. For a slightly less extreme example, imagine, for instance taking "daylight savings time" clock synchronizations as a method for measuring the velocity of jet aircraft. One would have to explain why jets were so very very fast flying east-west, and so much slower flying west-east.

Imagine taking this idea seriously, and explaining why the "physics" of east-west travel was different in principle from the "physics" of west-east travel for some idea of the unpleasantness of this idea.

It's generally a lot simpler to use a "fair" clock synchronization scheme. Unfortunately fair clock synchronization schemes aren't really compatible with rotating frames.

 

[pervect]

I like them only partially. Since I was not completely satisfied with the existing resolutions of the Ehrenfest paradox, I made my own resolution:
http://arxiv.org/abs/gr-qc/9904078
See also
http://arxiv.org/abs/gr-qc/0307011

Sometimes I'm a little slow. Are you actually claiming to *be* H Nikolic? If so, a hearty welcome to PF, and many cyber kow-tows.

https://www.physicsforums.com/showpost.php?p=1196920&postcount=11

makes me think this is in fact the case. Of course, if you're just, to use the idiomatic expression, yanking our chains, I take it all back...

 

[MeJennifer]

There is certainly a circumference for the disk in a non-rotating Minkowskian frame. However, there isn't really any such thing as the circumference in a "rotating frame".

But there is!
If you stand on the edge of a disk and I stand right next to you and you start to rotate on it and I stay inertial we will meet again.
Needless to say that we would not agree at all on both the duration and distance of the trip but nevertheless we would both agree our wordlines formed a closed loop!

And the helix example shows why. Or rather it should show why. You seem not to get the point I'm trying to make, and I don't understand the difficulty. And I don't seem to be able to think of a way to explain it better than I already have.

The helix example shows something else, it shows that if we insist on splitting space-time in 3+1 for an observer on the disk we will have many troubles ahead to put it mildly.

You seem not to get the point I'm trying to make, and I don't understand the difficulty.

Actually I think do get your point but I am lost as to why your point has anything to do with: "...the "circumference" of the helix in the above article isn't really well defined, because it just isn't a geometrically closed object."

It seems that your point is that there is no proper plane of simultaneity describing the complete ring (let alone the disk) for an accelerating observer on the ring, and thus one cannot calculate the circumference in terms of space only. And I do not disagree with that! I am fully aware of that.
But that does not mean that there is no circumference!

 

[Demystifier]

Sometimes I'm a little slow. Are you actually claiming to *be* H Nikolic? If so, a hearty welcome to PF, and many cyber kow-tows.

https://www.physicsforums.com/showpost.php?p=1196920&postcount=11

makes me think this is in fact the case. Of course, if you're just, to use the idiomatic expression, yanking our chains, I take it all back...

Well, I cannot prove this, but yes, I am H. Nikolic.
By the way, I am not a native english speaker, so plese explain me the expressions "kow-tows" and "yanking our chains, I take it all back"! Thanks!

 

[pervect]

Hi, and welcome to Physics forums!

A kow-tow is a gesture of respect, literally a deep bow - a detailed description is in the wikipedia article http://en.wikipedia.org/wiki/Kowtow

To yank someone's chain is to decieve or manipulate somoene. A google search turns up http://www.thefreedictionary.com/yank+chain

 

[Chris Hillman]

Clarification needed?

Oh dear! In trying to walk the fine line between avoiding the kind of "discussion" which drove me out of sci.physics.* and then out of WP, while not entirely ducking my responsibility to try to correct serious errors should I spot them, I may have once again stepped on several toes.

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.

Jennifer, don't worry, I have enjoyed your questions and I do think you are trying to learn, which I much appreciate!

Be aware that some of the same individuals who played a role in driving me out of sci.physics.* and later out of WP have PF accounts, and naturally I wish to avoid encouraging them to try to drive me out of PF too! (To avoid confusion: at the moment of my post, I don't think any of them have shown up in this thread, but I wish to duck out before that happens. And I certainly with to avoid naming names or to encourage guessing, but the only way to do that appears to be for me to quit this thread, or even PF.)

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!

Thanks! I really appreciate that! My hope is that those who see them and are not yet familiar with frame fields, the kinematical decomposition of a timelike congruence (acceleration vector, expansion tensor, vorticity tensor) will be motivated to learn these concepts/techniques so that they can read the articles with full appreciation.

I like them only partially. Since I was not completely satisfied with the existing resolutions of the Ehrenfest paradox, I made my own resolution:
http://arxiv.org/abs/gr-qc/9904078
See also
http://arxiv.org/abs/gr-qc/0307011

Which I studied, along with many dozens of other papers or arxiv eprints, when I was writing "my" versions of the WP articles I cited. (Again, these were at least somewhat collaborative, with User:Pjacobi making important contribs to at least one article.) I made a conscious decision not to cite all the research papers I studied (dating back to the original papers) because I believe that general encyclopedia articles should simply cite a review paper in preference to repeating the citations in such a review. (It is safe to assume I read all the arXiv eprints and all the papers cited by Gron, for example.)

Well, I cannot prove this, but yes, I am H. Nikolic.

Well, you guess correctly that I consider some of your papers correct and others somewhat misleading, but I plead exhaustion. That whole experience was so horrific that I doubt I'll ever to seriously discuss this topic again--- in my initial comments I was simply trying to point some members toward my own previous attempt to summarize some of the best previous contributions. While in happier times I would have been eager to learn from knowledgeable and insightful comments on my own writings, I think I'll have to let the cited versions of these articles speak for themselves, and to avoid any further discussions, especially in fora filled with pseudonymous "personas".

Like pervect, I was not aware until you spoke up that you are Nikolic, but thanks for clarifying this. For what it is worth, I was not thinking of you when I mentioned the large number of awful papers/eprints in this subject: I was thinking of authors whose papers I consider entirely without any redeeming features whatsoever, the kind of author who ignores the vast pre-existing literature, either out of laziness or out of the assumption that their own "brilliant insights" render all prior efforts moot. Paraphrasing the late Lloyd Bentsen (an American politician), all I have to say to those people is "You, sir, are no Einstein!". I don't think they are here yet but (having said what I just said) I should leave before they turn up!

Shucks - here I thought we could have an argument and maybe I could learn something.

As could I! And in happier times there would have been nothing I would have enjoyed more. Since you cited the RfC in which my comment rather obviously reflects my distress over my bitter disillusionment with WP (a volunteer project I joined with perhaps overly idealistic enthusiasm, because its stated goal, of bringing good information, understanding and appreciation to the masses, is so dear to my heart) and my anger over the way in which leadership failures (in regard to policy and procedure) fostered an environment so hostile to scholars that one stubborn ignoramus succeeded in destroying, apparently permanently, my former enjoyment of writing about math/physics, I hope and expect that you realize that my inability to continue to participate in this thread has nothing to do with you. Utopian social experiments have a long history of destruction, and apparently WP is no exception to this rule.

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.

Thanks for that!

I think our current disagreement is a judgement of degree regarding how much effort one should expect to expend in learning this subject well enough to make well-informed and thoughtful comments.

My position is that this area is extraordinarly subtle, full of conceptual pitfalls. In fact, one could say (paraphrasing Misner) that this topic may be a universal mindtrap offering ample scope for every conceptual error it is possible to make in relativistic physics.

(A predatory philosopher could go on a most satisfying rampage here. I find it frustrating to observe that young philosophers of physics seem to invariably be guided into revisiting the same tired ground endlessly contested by their elders, when there is so much fertile territory for ripping careless argumention to shreds in the very next feedlot, as it were.)

I have two suggestions for a compromise:

1. A related and very interesting topic (analyzing "fishing" in Rindler or Schwarzschild models) has recently been discussed at length by Greg Egan http://gregegan.customer.netspace.net.au/SCIENCE/Rindler/RindlerHorizon.html where one can begin to see some of the subtlety of trying to treat deformed materials relativistically, even if one tries to avoid developing a fully fledged general theory of elasticity. In my experience, those who haven't attempted a detailed analysis of an idealized model (and who haven't critically examined about their results) tend to vastly underappreciate this subtlety.

2. Give me some time (months? years?) to get a bit less depressed over this topic and PM me to arrange a one-one discussion--- maybe bye and bye I will find this topic somewhat less distressing. My only request would be that you get some experience using frame fields and kinematic decomposion first, since my analysis was based on these techniques and I worked hard to make that analysis clear to anyone with this background knowledge, and in invading unknown territory it is best to avoid splitting up the expeditionary force, as it were. (C.f. Stanley's notorious mission to "rescue" Emin Pasha--- which I see is quite inadequately explained in his WP wikibio.)

 

[pervect]

2. Give me some time (months? years?) to get a bit less depressed over this topic and PM me to arrange a one-one discussion--- maybe bye and bye I will find this topic somewhat less distressing. My only request would be that you get some experience using frame fields and kinematic decomposion first, since my analysis was based on these techniques and I worked hard to make that analysis clear to anyone with this background knowledge, and in invading unknown territory it is best to avoid splitting up the expeditionary force, as it were. (C.f. Stanley's notorious mission to "rescue" Emin Pasha--- which I see is quite inadequately explained in his WP wikibio.)

I'm also sorry if I have stepped on any toes - I think PF badly needs more knowledgeable people, especially in the relativity forums, and I'd like to recruit as many as possible - and definitely I don't want to drive any away, or contribute to driving any away.

I'm reasonably familiar with frame fields. However, I'm not as familiar with the vorticity tensor, the expansion tensor, or the Raychahurdi equation as I would really like, though I've at least heard of them. (I gather that the Raychahurdi equation is what you are referring to as a "kinematic decomposition"?). Also, I have not yet had a chance to read Gron - it's on my list to grab the next time I make it to a library with access. But if it will take you years to feel like talking about it again, I guess I have plenty of time to fetch it :-). (I should also add that I'm not particularly fond of E-mail, but of course that's not your problem, it's a personal quirk of mine, one of many, and I'm not opposed to it enough to miss a good physics discussion).

I'd really like your opinion on the spinning buckytube wire analysis I posted. (I have the depressing thought that your reply will show that it's not as clearly written as I could hope for, though I've tried to clarify a few points and I've made minor edits to it for hopefully improved clarity).

I'll expand on the analysis with a little conceptual overview. First off, one can view the wire as a swarm of atoms. And one can regard the atoms as connected to each other by springs, which are an abstract representation of the electromagnetic fields.

My first point is that in physically known materials like the buckycable, the amount of energy in the "springs", i.e. the fields holding the atoms together, is basically negligible. I go through some calculations to hopefully demonstrate this, using Hooke's law (which I feel is applied well within its bounds of applicability) - and to come up with a number for the magnitude of the energy stored in this form. I point out that this has an elementary interpretation in terms of non-relativistic mechanics, that the energy stored in these interatomic fields is just the work done in stretching the wire.

My next point is that because the energy in the fields is neglgible, we can basically justify the approximation of the wire as n atoms, each of which has a velocity v. Because the velocity is so low, we can use the Newtonian formula for kinetic energy. Because the energy in the fields is so small we are justified in ignoring the amount of energy it takes to set these fields in motion (it will be present, but small). We then have a complete reasonably accurate accounting for the total energy of the spinning wire. This is just the number of atoms in the wire (which is a relativistic invariant), multipled by the energy (rest mass*c^2) of each atom, multiplied by the gamma factor. In this case the velocity is so low that one can equally well use the Newtonian formula mc^2 + .5 m v^2 instead of multiplying by gamma. The energy in the mechanical stretching of the wire is also present, but it's basically so small in magnitude as to be irrelevant.

I could try to make the presentation a bit more formal, by actually writing down the stress-energy tensor that I expect in the non-rotating frame explicitly. But to do this I would need Poisson's ratio for the bucky material or equivalent information - i.e. information on how much the width of the wire changes when we stretch it under tension. I think it is simpler to talk about the total energy of the wire, instead and avoid the issue. It's simplest to focus on the number of atoms (which does not change) - which we can do, since the contribution to the stress-energy tensor of the fields holding the atoms together is so small.

 

[pervect]

I've been reading "Relataivisitc contraction and related effects" in non-inertial frames, http://arxiv.org/abs/gr-qc/9904078, and I have a few comments

The way that the "generally accepted line element" is determined is to integrate the Lorentz interval along a joining line of constant time. "Constant time" is defined by the coordinate system used.

This seems to be pretty much universal - for instance, this is how the cosmologists arrive at what they call, variously, proper distance or sometimes comoving distance - by integratng the Lorentz interval along a line joining the two points of constant cosmological time.

This then gives the formula quoted in the paper as the "conventional formula" - because t is constant, all the terms with dt in them are zero in the line element for distance.

The proposal to replace the line element seems to me to be equivalent to integrating the Lorentz interval over some other curve to define a distance measure, i.e. some curve other than a curve of constant coordinate time. I haven't quite worked out the details of exactly what curve this corresponds to, though I have some suspicions which I won't write down at this point.

 

[Chris Hillman]

We're cool, man, we're cool!

Hi, pervect,

I'm also sorry if I have stepped on any toes - I think PF badly needs more knowledgeable people, especially in the relativity forums, and I'd like to recruit as many as possible - and definitely I don't want to drive any away, or contribute to driving any away.

I wasn't worrying about Jennifer driving me away, or H. Nikolic driving me away, or in fact about anyone who has yet participated in this thread. And
I certainly am not worrying about you driving me away! You're one of the main reasons why I bother to turn up here at all! (That's a convoluted compliment.) I'll take the unusual step of explaining a bit of background by PM.

I'm reasonably familiar with frame fields. However, I'm not as familiar with the vorticity tensor, the expansion tensor, or the Raychahurdi equation as I would really like, though I've at least heard of them. (I gather that the Raychahurdi equation is what you are referring to as a "kinematic decomposition"?).

Actually, it is what you start with to derive the Raychaudhuri equation. Namely, the invariant decomposition of any timelike vector field into acceleration vector, expansion tensor, and vorticity tensor. The expansion tensor is designed exactly to answer questions like "is this thing rigid", so recoiling in fear of the unfamiliar is like me telling someone who wants to compute volume "futzing with complicated and confusing sequence of simplicial approximations is all fine and well, but there's this thing called integral calculus which you really outta learn forthwith". Once you become familiar with this stuff I am confident you will agree with my little analogy! This is an absolutely fundamental concept, useful for a gazillion things, many having nothing to do with gtr or even physics.

(For example, the Hopf fibration is a lovely thing: as an exercise, you should compute the expansion and vorticity tensor of the Clifford congruence on S^3, which consists of a family of great circles which fills up the sphere; each pair is linked but they maintain constant distance while twisting around one another in the sense of vorticity; this is possible because the geodesic convergence due to positive curvature exactly compensates for the increase in distance from the "twisting"; compare a pair of Clifford circles with a pair of skew lines in euclidean three-space.)

Incidently, gr-qc/??12140 (oh no, a bit error, I've been zapped by a cosmic ray or something) by Costa and Herdeiro, has a nice remark about the so-called Raychaudhuri scalar, which in terms of the electrogravitic or tidal tensor is analogous to Gauss's law with a pressure term added to the mass density; see Baez and Bunn's well known expository eprint, "The Meaning of the EFE".