Question relating to the Ehrenfest Paradox

[Hansiman]

My question relates to the “Ehrenfest paradox”. I try to grasp it.

“In its original formulation as presented by Paul Ehrenfest 1909 in the Physikalische Zeitschrift, it discusses an ideally rigid cylinder that is made to rotate about its axis of symmetry. The radius R as seen in the laboratory frame is always perpendicular to its motion and should therefore be equal to its value R0 when stationary. However, the circumference (2πR) should appear Lorentz-contracted to a smaller value than at rest, by the usual factor γ. This leads to the contradiction that R=R0 and R<R0.”
(Source italic text = wikipedia)

Why can one not just simply accept that the contracted circumference of the rotating cylinder has a smaller and different radius than the static cylinder at rest?
In other words that a radius is always abstract and inherent to the circumference.
In other words that a radius cannot be thought of without the concept of a circumference. Why should the radius of the circumference of a rotating cylinder then ever be equal to the circumference of the cylinder at rest?
Because of this I fail to understand the paradox and see it more as logical trick played upon me.

Could anyone explain in a relatively simple way where I’m wrong? This would help me to grasp the complexity of the problem of the Ehrenfest Paradox which I’m really keen to fully understand.

 

[PAllen]

Imagine the radius as a piece of string on the across the cylinder. You rotate it to relativistic speed. The string is everywhere orthogonal to the direction of motion, so it doesn't change in length. The outermost circumference (for example) shrinks. Yet there is still perfect axial symmetry, so the outermost circumference must still be a circle, equal 2 pi R (length of string). Contradiction.

Note that there is no problem if you allow the cylinder to become un-rigid during speedup, then rigid again when at constant angular velocity. You simply posit the the circumferential atoms move farther apart (in their own ultra-local comoving frame) during speed up, so contracted length remains the same in the lab frame.

 

[Hansiman]

Imagine the radius as a piece of string on the across the cylinder. You rotate it to relativistic speed. The string is everywhere orthogonal to the direction of motion, so it doesn't change in length. The outermost circumference (for example) shrinks. Yet there is still perfect axial symmetry, so the outermost circumference must still be a circle, equal 2 pi R (length of string). Contradiction.

Note that there is no problem if you allow the cylinder to become un-rigid during speedup, then rigid again when at constant angular velocity. You simply posit the the circumferential atoms move farther apart (in their own ultra-local comoving frame) during speed up, so contracted length remains the same in the lab frame.

Thank you for your helpful and quick response. I (think I) understand the line of reasoning of Ehrenfest better now. But my problem in understanding the paradox remains exactly the same as before. Exactly because I’m not able to imagine any ‘radius’ R as a tangible real life (rigid or unrigid) object (as is a piece of string across the cylinder). Once the radius is presented as a tangible piece of string, independent from the circle it originally derives from, I cannot think of it anymore as a mathematical ‘radius’ of a circle. It feels as if there is somewhere a strange switch between a real physical entity and and an abstract idea as a radius. Therefore I feel/felt that a logical feat is played upon me. I'm still troubled. But I will give it some more thought. I’m a bit of slow type in understanding. Thank you for having replied once again!

 

[PAllen]

Thank you for your helpful and quick response. I (think I) understand the line of reasoning of Ehrenfest better now. But my problem in understanding the paradox remains exactly the same as before. Exactly because I’m not able to imagine any ‘radius’ R as a tangible real life (rigid or unrigid) object (as is a piece of string across the cylinder). Once the radius is presented as a tangible piece of string, independent from the circle it originally derives from, I cannot think of it anymore as a mathematical ‘radius’ of a circle. It feels as if there is somewhere a strange switch between a real physical entity and and an abstract idea as a radius. Therefore I feel/felt that a logical feat is played upon me. I'm still troubled. But I will give it some more thought. I’m a bit of slow type in understanding. Thank you for replying once again!

Remove all abstraction. Imagine attached to the cylinder's circumference is rigid ring that can neither be compressed or stretched. Imagine the radius is a band of the same rigid material. Spin it. The rigid ring must get shorter and yet remain a circle of the same radius (because the radius band has not shrunk).

To be precise, since there are no rigid object in relativity, replace rigid with Born rigid, and you have rigorous statement of the problem. Born rigidity gets around fininte speed sound by positing that individual 'atoms' can accelerate independently so as to remain stationary relative to their ultra-local neighbors. It is perfectly feasible to retain Born rigidity during linear acceleration. It is impossible to do while 'spinning up' a cylinder.

 

[Fredrik]

A spacetime diagram and some thought about what the term "rotating coordinate system" could mean, make it pretty obvious that both the radius and the circumference are the same as in the inertial coordinate system where the disc was previously at rest.

However, if we imagine that the radius and circumference are being measured by really tiny radar devices and mirrors attached at many different points on the disc, then the result for the circumference will add up to 2πRγ. The reason is that each radar device is measuring the proper length of a short geodesic segment whose endpoints aren't simultaneous in the inertial coordinate system.

The key to understanding the rotating disc is the following: Imagine that the disc is initially at rest, and that you give it a spin by turning a rod through its center. I'm specifying this detail to make it obvious that every point on the edge of the disc will have the same speed in the inertial coordinate system, at all times. What the radius of the disc will be when it has reached relativistic speeds is hard to say, since it depends on its elasticity. So just suppose that we have somehow constrained it to have have the same radius at all times. Now consider two adjacent points on the edge of the disc. (Close enough to have a velocity difference that's small enough to be neglected). They have changed their velocity in the inertial coordinate system. This means that in their new co-moving inertial coordinate system, they are a different distance from each other than they were in their old co-moving inertial coordinate system before we gave the disc a spin. This means that the disc has been forcefully stretched along the circumference!

 

[Quinzio]

Once the radius is presented as a tangible piece of string, independent from the circle it originally derives from, I cannot think of it anymore as a mathematical ‘radius’ of a circle. It feels as if there is somewhere a strange switch between a real physical entity and and an abstract idea as a radius. Therefore I feel/felt that a logical feat is played upon me. I'm still troubled.

Anything related to SR leaves the reader with a certain sense of confusion..., but anyway... you may think of this modified version of the experiment, and it somehow will be clearer.

You have a transmission belt, which is shaped as a square. At the corner of the square the belt passes over 4 pulleys/cogs which are nailed to a rigid frame. The belt is rapidly accelerated up to relativistic speed.
We are in the same situation of the Ehrenfest paradox.

The 4 pulleys are at rest, according to the original inertial frame, so their distances cannot be changed. The belt length is shorter, according to SR, so a paradox arises.
It is likely that the belt will experience a higher tension, and it may finally break if the tension becomes too high.

 

[Bill_K]

It is likely that the belt will experience a higher tension, and it may finally break if the tension becomes too high.

No, you've got it backwards. Instead of a rubber belt, imagine N fireflies going around in a circle a distance 2∏R/N apart. The proper distance between consecutive fireflies will be less, not more. If there were springs between the flies the springs would be compressed, not stretched.

 

[Quinzio]

It seems like you don't get what "length contraction" is.

Ehrenfest considered an ideally Born rigid cylinder that is made to rotate. Assuming that the cylinder does not expand or contract, its radius stays the same. But measuring rods laid out along the circumference 2πR should be Lorentz-contracted to a smaller value than at rest, by the usual factor γ. This leads to the paradox that the rigid measuring rods would have to separate from one another due to Lorentz contraction; the discrepancy noted by Ehrenfest seems to suggest that a rotated Born rigid disk should shatter.
http://en.wikipedia.org/wiki/Ehrenfest_paradox

 

[harrylin]

My question relates to the “Ehrenfest paradox”. I try to grasp it.

“In its original formulation as presented by Paul Ehrenfest 1909 in the Physikalische Zeitschrift, it discusses an ideally rigid cylinder that is made to rotate about its axis of symmetry. The radius R as seen in the laboratory frame is always perpendicular to its motion and should therefore be equal to its value R0 when stationary. However, the circumference (2πR) should appear Lorentz-contracted to a smaller value than at rest, by the usual factor γ. This leads to the contradiction that R=R0 and R<R0.”
(Source italic text = wikipedia)

Why can one not just simply accept that the contracted circumference of the rotating cylinder has a smaller and different radius than the static cylinder at rest?
In other words that a radius is always abstract and inherent to the circumference.
In other words that a radius cannot be thought of without the concept of a circumference. Why should the radius of the circumference of a rotating cylinder then ever be equal to the circumference of the cylinder at rest?
Because of this I fail to understand the paradox and see it more as logical trick played upon me.

Could anyone explain in a relatively simple way where I’m wrong? This would help me to grasp the complexity of the problem of the Ehrenfest Paradox which I’m really keen to fully understand.

Hi Hansiman, If I correctly understand you, if the circumference stresses to contract then logically, this necessarily results in a reduction of the diameter - right? That's correct of course. And there is nothing "wrong" with not seeing a problem that doesn't really exist; many paradoxes are logical tricks.

However, this paradox is not a logical trick but a demonstration: Ehrenfest's showed with his illustration that "Born rigidity" is not applicable to rotation, because that leads to a contradiction. This is made clear in the original paper (which is shorter than the discussion here!):
- http://en.wikisource.org/wiki/Uniform_Rotation_of_Rigid_Bodies_and_the_Theory_of_Relativity

Cheers,
Harald

 

[Hansiman]

@ PAllen, Fredrik, Quinzio, BillK
Thank you all very much for your helpful replies. It appears I understood the paradox completely wrong. I did not understand that it was mainly aiming at the fact that ‘Born Rigidity’ is not applicable to rotation.

@ Harald/Harrylin
You understood in detail my difficulties with the paradox. I indeed failed to see a problem that according to me did not really exist. I always wondered why this Ehrenfest-paradox was so fundamental.
Erroneously I always thought the paradox was mainly about the fact that R=R° and (2π)R < (2π)R° for a (rigid) rotating cylinder (and to a lesser extent about Born Rigidity). I erroneously thought Ehrenfest wanted to make a statement about the tension that exists between the physics of special relativity and pure Euclidian math.

So you (and all the others who replied on my question) are very right in pointing out that I failed to see that the paradox does illustrate something specific, it demonstrates that Born rigidity is not applicable to rotation. I understand that the word ‘rigid’ is key in understanding the whole paradox. I was thinking it too abstract.

@All
So thank you all, all the replies clarified some essential things to me about understanding the problem that the paradox wants to demonstrate.

 

[Hansiman]

Hi Hansiman, If I correctly understand you, if the circumference stresses to contract then logically, this necessarily results in a reduction of the diameter - right? That's correct of course. And there is nothing "wrong" with not seeing a problem that doesn't really exist; many paradoxes are logical tricks.

It was precisely that what confused me. I initially thougt Ehrenfest meant to say with his paradox that a rigid circle that rotates from a mathematical point of view does not comply anymore to the classical geometry of the circle (2RPi). I thought he meant that because the radius of the circle is perpendicular to the direction of the rotation of the circle and therefore would not shrink as the circumference does when rotating. That position was impossible to understand for me as I thought that the intangible diameter (or intangible radius) shrinks together with the circumference as you point out above, so it would be a non problem.
Now understand that Ehrenfest meant to demonstrate with his paradox the problem regarding Born Rigidity.

 

[PAllen]

It was precisely that what confused me. I initially thougt Ehrenfest meant to say with his paradox that a rigid circle that rotates from a mathematical point of view does not comply anymore to the classical geometry of the circle (2RPi). I thought he meant that because the radius of the circle is perpendicular to the direction of the rotation of the circle and therefore would not shrink as the circumference does when rotating. That position was impossible to understand for me as I thought that the intangible diameter (or intangible radius) shrinks together with the circumference as you point out above, so it would be a non problem.
Now understand that Ehrenfest meant to demonstrate with his paradox the problem regarding Born Rigidity.

That's an interesting observation. If instead of a disk or a cylinter, one spins up an infinitely thin circular wire, it *can* maintain (Born) rigidity, thus shrinking in both circumerence and radius in the lab frame.

 

[harrylin]

That's an interesting observation. If instead of a disk or a cylinter, one spins up an infinitely thin circular wire, it *can* maintain (Born) rigidity, thus shrinking in both circumerence and radius in the lab frame.

Thanks for the correction - I should have written "rotation of solid objects". Indeed a very thin rotating rim can (almost) freely contract.

Harald