On the Relativistic Twisting of a rotating cylinder (Max von Laue)

[AVentura]

I'll start with link describing the phenomenon:

https://books.google.com/books?id=W...&q=relativity twist rotating cylinder&f=false

Quote:

"A cylinder rotating uniformly about the x' axis in the frame S' will seem twisted when observed in the usual second frame S, in which it not only rotates but also travels forward."​

Now picture that instead of a cylinder we have a helix in S'. The helix in S' has an integer number of curls in S', so its center of gravity is on the x' axis. It rotates independently in S'. If its pitch in S' is such that the twisting seen in S perfectly straightens it out then an observer in S sees a rod that is not on the x-axis rotating around the x-axis all by itself.

How can that be? Or, where have I gone wrong?

 

[PeterDonis]

If its pitch in S' is such that the twisting seen in S perfectly straightens it out

That can't happen; the twisting in question does not change the extent of the cylinder (or helix) perpendicular to the direction of motion.

 

[AVentura]

That can't happen; the twisting in question does not change the extent of the cylinder (or helix) perpendicular to the direction of motion.

I'm sorry, what do you mean by extent?

 

[PeterDonis]

will seem twisted

Note the key word "seem". What is actually going on here is relativity of simultaneity: in the frame in which the cylinder is moving forward as well as rotating, surfaces of constant time no longer cut the cylinder exactly perpendicular to its axis; instead they cut it at an angle. That's what creates the appearance of a twist. There is no actual stress applied to the cylinder (since all we're doing is changing coordinates, and changing coordinates can't change any physical observables).

 

[PeterDonis]

what do you mean by extent?

The size of the cylinder or helix in the ##y## and ##z## directions (assuming the relative motion is in the ##x## direction).

 

[AVentura]

The size of the cylinder or helix in the ##y## and ##z## directions (assuming the relative motion is in the ##x## direction).

I don't think it needs to. The apparent twisting changes the apparent pitch by itself.

Note the key word "seem". What is actually going on here is relativity of simultaneity: in the frame in which the cylinder is moving forward as well as rotating, surfaces of constant time no longer cut the cylinder exactly perpendicular to its axis; instead they cut it at an angle. That's what creates the appearance of a twist. There is no actual stress applied to the cylinder (since all we're doing is changing coordinates, and changing coordinates can't change any physical observables).

Agreed. But if we stick to a cylinder, and just paint a helix on it in S', that painted line should have a different apparent pitch in S, correct? If not, then what exactly is meant by "seems twisted"?

 

[PeterDonis]

The apparent twisting changes the apparent pitch by itself.

It can't change a helix into a straight line, which is what you were claiming, without changing its extent in the ##y## and ##z## dimensions.

if we stick to a cylinder, and just paint a helix on it in S', that painted line should have a different apparent pitch in S, correct?

Yes. But that doesn't imply that the painted helix will become a straight line if we pick S with a large enough relative velocity.

 

[Dale]

That can't happen

I think that it could happen. I would have to work it out, but I am ~80% confident. Is it the integer number of turns in the center frame?

 

[AVentura]

It can't change a helix into a straight line, which is what you were claiming, without changing its extent in the ##y## and ##z## dimensions.
Yes. But that doesn't imply that the painted helix will become a straight line if we pick S with a large enough relative velocity.

The pitch of a painted line is arbitrary. The required pitch in S' for a straight line in S is given in the OP link.

 

[PeterDonis]

I am ~80% confident

I am at least that confident that it can't happen. The reason is simple: a Lorentz transformation in the ##x## direction does not affect distances in the ##y## and ##z## direction. Making a helix into a straight line would require changing distances in the ##y## and ##z## directions--points which have nonzero separation in those directions in the original frame would have to have zero separation in the transformed frame.

 

[AVentura]

I am at least that confident that it can't happen. The reason is simple: a Lorentz transformation in the ##x## direction does not affect distances in the ##y## and ##z## direction. Making a helix into a straight line would require changing distances in the ##y## and ##z## directions--points which have nonzero separation in those directions in the original frame would have to have zero separation in the transformed frame.

I can paint two different helices on the same cylinder

 

[PeterDonis]

The required pitch in S' for a straight line in S is given in the OP link.

Where?

 

[PeterDonis]

I can paint two different helices on the same cylinder

So what? We are talking about one helix, not two.

 

[PeterDonis]

The pitch of a painted line is arbitrary.

Yes, but it has to be finite if the painted line is a helix. And as far as I can tell, a Lorentz transformation can only change one finite pitch to another finite pitch; it can't change a finite pitch into an infinite pitch, which is what "straight line" means.

 

[PeterDonis]

If its pitch in S' is such that the twisting seen in S perfectly straightens it out

Another item to think about: in a frame in which the helix is moving, it length contracts; so it seems like the pitch of the helix should increase (turns getting squeezed together), not decrease (turns getting stretched out) in a frame in which it is moving.

 

[Dale]

a Lorentz transformation in the ##x## direction does not affect distances in the ##y## and ##z## direction. Making a helix into a straight line would require changing distances in the ##y## and ##z## directions

Any two points on the helix have different x coordinates, so distances between them can change under the Lorentz transform.

 

[AVentura]

Where?

gamma*omega*v/c^2

Yes, but it has to be finite if the painted line is a helix. And as far as I can tell, a Lorentz transformation can only change one finite pitch to another finite pitch; it can't change a finite pitch into an infinite pitch, which is what "straight line" means.

This phenomenon is in addition to lorentz contraction

 

[PeterDonis]

Any two points on the helix have different x coordinates, so distances between them can change under the Lorentz transform.

I didn't say the total distances between them couldn't change; I said the distances in the ##y## and ##z## directions couldn't change. Or, to put it more simply, the ##y## and ##z## coordinates of a given point on the helix can't change. But turning a helix into a straight line requires changing the ##y## and ##z## coordinates of at least some of its points.

 

[PeterDonis]

gamma*omega*v/c^2

That is the twist per unit length in S of something that is a stack of circles in S'. It is not the pitch in S' of something that is a straight line in S.

 

[AVentura]

I didn't say the total distances between them couldn't change; I said the distances in the ##y## and ##z## directions couldn't change. Or, to put it more simply, the ##y## and ##z## coordinates of a given point on the helix can't change. But turning a helix into a straight line requires changing the ##y## and ##z## coordinates of at least some of its points.

I think "twisting" implies that they do, but stay on the surface of the cylinder. If not, then what is twisting? Also remember there is more going on than Lorentz transformation. The surface of the cylinder is not a rest frame.

 

[AVentura]

That is the twist per unit length in S of something that is a stack of circles (a "helix" with zero pitch) in S'. It is not the pitch in S' of something that is a straight line (infinite pitch) in S.

if you twist a helix the exact magnitude of its pitch, but in the opposite direction, you get a line.

BTW, you don;t need to make it a straight line for the question to remain. If you change it enough that its center of mass is not on the x-axis the helix should wobble.

 

[PeterDonis]

it seems like the pitch of the helix should increase (turns getting squeezed together), not decrease (turns getting stretched out)

I got this backwards: squeezing the turns together means decreasing pitch, and stretching them out means increasing pitch.

 

[PeterDonis]

if you twist a helix the exact magnitude of its pitch, but in the opposite direction, you get a line

Um, what? I have no idea what you're talking about here.

Part of the problem might be the term "pitch". What are you using it to mean? The standard definition is the height of one complete turn, measured parallel to the axis. So twisting the helix "the exact magnitude of its pitch, but in the opposite direction", if anything, means squeezing all the turns into the same plane. That doesn't give you a line; it gives you a circle.

 

[AVentura]

Um, what? I have no idea what you're talking about here.

Part of the problem might be the term "pitch". What are you using it to mean? The standard definition is the height of one complete turn, measured parallel to the axis. So twisting the helix "the exact magnitude of its pitch, but in the opposite direction", if anything, means squeezing all the turns into the same plane. That doesn't give you a line; it gives you a circle.

if the cylinder has a helix on it with one complete clockwise turn, and you grab the ends and twist it counterclockwise one turn, you get a straight line.

more importantly, if you turn it any, it will no longer have the same integer number of turns on the surface

 

[PeterDonis]

if the cylinder has a helix on it with one complete clockwise turn, and you grab the ends and twist it counterclockwise one turn, you get a straight line.

I see what you are describing now, but it is not what a Lorentz transformation does. I think we are at the point where we really need to look at the math. I don't have time to try it now, but I strongly advise you to do it before posting further.

 

[AVentura]

From the original link: The apparent twist per unit length is γ*ω*v/c²
ω is the angular speed in S'

the helix drawn on the cylinder in S' happens to have a twist per unit length of -γ*ω*v/c²

An observer in S sees a straight line on the cylinder

Note: the surface of the cylinder is under acceleration, Lorentz transformation should not describe it.

 

[PeterDonis]

From the original link: The apparent twist per unit length is γ*ω*v/c2

the helix drawn on the cylinder in S' happens to have a twist per unit length of -γ*ω*v/c2

An observer in S sees a straight line on the cylinder

I understand this is what you think will happen. But you have not done the Lorentz transformation to prove it happens. You need to explicitly do the math. A good start would be to do a more explicit version of what is done only sketchily in the book you linked to. I don't think the fully explicit version will look the way you are implicitly assuming it will.

 

[PeterDonis]

a Lorentz transformation in the ##x## direction does not affect distances in the ##y## and ##z## direction

I see something I forgot here: since we are dealing with rotation, i.e., points are moving in the ##y-z## plane, the statement quoted above is not necessarily true, because of relativity of simultaneity.

 

[PeterDonis]

If its pitch in S' is such that the twisting seen in S perfectly straightens it out then an observer in S sees a rod that is not on the x-axis rotating around the x-axis all by itself.

I still don't have time to do the math, but in view of what I just posted in #28, I do see a possible resolution. Consider: for the case described in the book linked in the OP, viewed in frame S' (i.e., the frame in which all the "clock hands" are synchronized and there is no twist), what does the motion of all the tips of the "clock hands", taken together, look like?

 

[AVentura]

They make a straight line in S'. And for the observer in S they make a helix.

You reversed the situation.

 

[PeterDonis]

They make a straight line in S'.

Yes. And what does the motion, in S', of that straight line look like? What happens to that straight line, as a function of time, in S'?

 

[AVentura]

It goes around and around the x' axis. If they were pointing to a rod with mass the rod's motion would be impossible independently (according to an observer in S').

 

[PeterDonis]

It goes around and around the x' axis.

Yes.

If they were pointing to a rod with mass the rod's motion would be impossible independently (according to an observer in S')

Why?

 

[AVentura]

Without external forces things can only rotate around their center of mass?

But its a different story for an observer in S, who sees the rod as a helix (in our current discussion which is reversed from the OP). The length of the rod in S' is arbitrary. What if it were just the right length so in S the helix is an integer number of turns. The center of mass is on the x-axis and the motion is possible.

 

[PeterDonis]

Without external forces things can only rotate around their center of mass?

"Center of mass" is not a relativistic concept; more precisely, it's not relativistically invariant by itself. (There is an invariant that includes it, which comes into play in the relativistic treatment of angular momentum.)

In the case of the helix described in your OP, you stipulated that it had an integral number of turns in frame S'. But if we consider a succession of frames between S' and S, with continuously varying boosts relative to S', the number of turns of the helix in those frames will vary continuously from one to zero (which is how many it has in frame S). This means the center of mass of the helix, which is on the x' axis in frame S', will move off the "x" axis under these boosts--and in frame S, it will be at the center of the rod which is a straight line in that frame.

At least, that description is the straightforward implication of what we've said so far. However, there is an additional implication: that if the center of mass moves off the "x" axis in these different frames, the axis of rotation of the helix can no longer be the "x" axis in these different frames; it must also move. Heuristically, it would have to move such that, in frame S, it becomes the rod itself. But that doesn't seem consistent with what appears to be happening to the tips of the clock hands in the cylinder case.

So now I'm even more insistent that we need to do the actual math, since we have arrived by what seems to be obvious reasoning at different mutually inconsistent descriptions of what is going on.