Rigid bodies are prohibited by relativity

Luboš Motl, Thursday, August 20, 2015

A vast majority of the answers I have posted on Stack Exchange did fine but I have experienced a highly unexpected opposition today – one about a basic problem in special relativity.

The question by seeking_infinity was:

Refer, «The classical theory of Fields» by Landau lifshitz (Chap 3). Consider a disk of radius \(R\), then circumference is \(2\pi R\). Now, make this disk rotate at velocity of the order of \(c\) (speed of light). Since velocity is perpendicular to radius vector, the radius does not change according to the observer at rest. But the length vector at boundary of disk, parallel to velocity vector will experience length contraction. Thus, the circumference-to-radius difference is smaller than \(2\pi\) when the disk is rotating. But this violates rules of Euclidean geometry. What is wrong here?

It is clearly a totally rudimentary problem in special relativity. It has its own name and if you search for Ehrenfest paradox, you quickly find out that there’s been a lot of debates in the history of physics – relatively to what one would expect for such a basic high school problem in classical physics. Born, Ehrenfest, Kaluza, von Laue, Langevin, Rosen, Eddington, and Einstein have participated, among many others.

My obviously correct answer immediately got at least two downvotes.

What is wrong is the idea that one can actually make the disk rotate; and it will remain perfectly rigid.

In reality, what this correct argument shows is that relativity doesn’t admit the existence of any perfectly rigid bodies. Be sure that despite all the confused users’ negative votes, this is a perfectly basic, settled, and indisputable textbook material that every mature physicist knows. The first sentence of this paragraph contains a link to the Gravity Probe B website.

When one takes a solid disk and makes it rotate, it will do all kinds of things resulting from the «imperfection of the material». It will tear apart by the centrifugal force, and if it won’t, it will either tear basically along radial lines, or it will bend (the disk won’t be planar anymore) because the circumference really shrinks by the Lorentz factor. If there existed a material that is perfectly rigid and cannot stretch or bend or tear, then it would be impossible to make it spin.

However, the non-existence of such a material may be shown even microscopically. It is not possible to «order» any solid object to keep the proper distances at every moment because the distance between two atoms (or points on the solid object) may only be measured with a delay \(\Delta t = \Delta x / c\) simply because no information may move faster than light. That’s why it’s always possible to squeeze any rod on one end and the opposite end of the rod won’t move at least for this \(\Delta t = \Delta x / c\).

In fact, the delay will be much larger than that, dictated basically by the speed of sound, not by the speed of light. Whatever material you have, relativity guarantees that it can be squeezed as well as stretched as well as bent.

Many people don’t like for some reason!

There actually exists a more popular answer that claims that rigid bodies are perfectly OK and possible in relativity – and the paradox is cured by the «usual» reason, namely by the relativity of simultaneity.

But it cannot be cured in this way. Why? Because you may have a point \(A\) on a rotating disk at the distance \(R\) from the axis of rotation, and a nearby point \(B\) on the non-rotating table beneath the disk. The point is that the points \(A,B\) repeatedly touch each other as the disk rotates. It’s because the angular coordinate along the circumference is periodic.

So the observers sitting at \(A\) and \(B\) may talk to each other about their measurements of the circumference. The guy on the table will see the circumference shrink but it can’t be shrunk because the laws of the Euclidean geometry are still valid in this inertial system (gravity and GR curvature is negligible) and they imply that the circumference is \(2\pi R\). On the other hand, the guy on the rotating disk may distribute his numerous equally rotating friends along the circle of radius \(R\). They may measure their proper distances in the local frames and the sum of these distances up to the guy \(A\) will clearly be \(2\pi R \sqrt{1-v^2/c^2}\).

This would be a real paradox. There is no paradox because some of the assumptions is wrong. And the wrong assumption is that a perfectly rigid object may exist and may be bring to rotation. It just can’t. This thought experiment is a macroscopic proof of the non-existence of rigid bodies. However, as I mentioned, one may also easily present the microscopic proofs.

A rod can’t be «unbendable» or «unsqueezable» or «unstretchable» because it would mean that there is something in the rod that guarantees its prescribed proper length at all times. If you bend this rod or push it or pull it, it would immediately have to change the shape across the length. But that can’t happen because in this way, you could use the rod to send signals faster than light. With some extra thinking, you can convince yourself that the actual speed by which the squeezing or stretching or bending is spread through the rod is the speed of sound – something that is guaranteed to be lower than the speed of light (and is lower roughly by a factor of 1 million in the real world).

This non-existence of perfectly rigid rods in relativity should be totally obvious for rods. But it holds for disks, too. If you push the disk (a vinyl record) at a particular point and want to make it spin, the material gets squeezed «in front» of your finger and stretched «behind» your finger. Sound waves will be moving along the vinyl record. Sound waves mean that pieces of the disk may stretch or squeeze and that’s what will happen. The disk may also bend and become non-planar to adapt to the Lorentz-contracted circumference. At some high enough speed, it may crack along radial cracks. Or it may tear apart by the centrifigal force along concentric cracks. Or something in between.

At any rate, the non-existence of perfectly rigid bodies is undoubtedly a characteristic, almost defining, implication of relativity.

I am pretty amazed that even in 2015, 110 years after Einstein presented his relativity, this very simple point remains controversial. Well, I am convinced that at least since 1911, almost all good physicists have agreed what the correct answer basically is.

Well, in 1909, Max Born introduced the «rigid motion». Yes, decades before quantum mechanics! 😉 In the same year, Ehrenfest presented the «paradox» – which is often named after him – and gave the right basic basic solution. Motions of extended objects can simply almost never be Born rigid! In 1910, Gustav Herglotz and Fritz Noether correctly argued that only 3 degrees of motion may be picked for rigid objects. But the disk has many more (infinitely many, as Max von Laue argued in 1911) so it’s impossible to make the disk spin, as Ehrenfest had previously correctly said. (Von Laue was the main guy who offered the proof of Ehrenfest’s conclusion using the impossibility to send faster-than-light signals.)

In 1910 Max Planck pointed out that one has to distinguish a non-rotating disk described in various coordinate systems (which is OK, of course), a permanently rotating disk constructed to rotate and observed in many frames (which is also OK, as long as you don’t want it stop), and the actual change of the angular velocity of the rigid disk (which is not allowed). He correctly said that the last problem among these three does require one to study elasticity etc. in the real world.

In the same year, Theodor Kaluza just made a comment without any argument that the disk itself has the geometry of the hyperbolic plane. Well, it depends from which side you look at it. The 2+1D disk embedded in 3+1D is actually spherical (positive curvature) if you allow it to bend. If you don’t allow it to bend, it’s obviously flat and the flatness of a 2+1D manifold embedded in the 3+1D space does not depend on whether or not you choose a rotating coordinate system or skeleton inside this 2+1D space! So Kaluza was pretty much wrong, as far as I can say.

In 1916, Einstein began to combine the insights with the new general relativity. He realized that GR allows the space to be curved or Riemannian – and it is basically useful to use it for the rotating frame, too. Well, a problem with that is that a Minkowski space is flat – curvature-free – in all coordinate systems. Being flat is a coordinate-independent property. But if you eliminate the Coriolis force and the «mixed components» of the metric tensor and only acknowledge the centrifugal force, the rotating disk is a great model for the gravitational field, one from which Einstein had derived the correct (in the leading approximation) gravitational red shift, too.

In the 1920s, 1930s etc. people like Eddington and Langevin began to add complications, allowed both the radius and the circumference to adapt, and introduced new coordinate systems etc. etc. Various people introduced some errors, fixed other people’s real errors, and claimed that other people had errors that were not actually errors. It became a messy history. But the basic Ehrenfest problem is still very simple and the answer is totally indisputable.

A simple Internet search finds lots of pages, books, and papers that say that perfectly rigid bodies aren’t allowed according to relativity. But for certain reasons, this obviously true, fundamental, and catchy slogan isn’t generally known and appreciated and if you say it and watch the reactions, you might even think that it’s controversial! It’s crazy.

I think that one reason why many laymen – including third-class physicists – don’t like the correct answer is that the answer says that «one can’t do something». They prefer the moronic «yes we can» answers to every question, even if these answers are incorrect. They confuse science – which impartially looks for the truth (and the true answer to every Yes/No question can a priori be both Yes or No) – with some kind of never disappearing faith in one’s omnipotence, neverending self-confidence, or with some sort of predetermined wishful thinking. It’s the same misunderstanding of the «truth in science» that turns people into fanatic fans of the cold fusion, warp drives, and tons of similar garbage. Sorry, such attitudes are not scientific and many «conclusions» that this attitude produces are demonstrably incorrect according to the scientific method.

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