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

[maline]

Well, sure enough! Here is an example of a helix that can rotate freely around its axis:
The helix has $n$ complete turns, parametrized as above by $\vec r (\theta) = ( \cos \theta , \sin \theta, p \theta)$ where $-n\pi \leq \theta \leq n\pi$. Start with a uniform mass distribution of $\lambda$ units per radian. As calculated there, if this rotates with $\vec \omega = \omega \hat z$, the angular momentum has a non-parallel component $J_y =\lambda \int_{-n\pi}^{n\pi} \, d\theta\, y(\theta) \omega z(\theta) =(-1)^n 2\pi n \omega p \lambda$. To fix this, we can simply add two point masses to the helix, at points $\theta = \pm \left( n- \frac 3 2 \right) \pi,$ with masses of $\frac n {n-\frac 3 2} \lambda.$ Their contribution to the integral is equal and opposite to the above value, so $J_y =0.$ They lie in the $yz$ plane, so they have no effect on $J_x=0$, nor on the $x$ component of the center of mass. Their $y$ coordinates are equal and opposite $(\pm 1)$, so the center of mass remains on the $z$ axis, and free rotation is possible.

So the paradox remains! In the frame boosted by $\vec v = - \frac 1 {\omega p} \hat z$, (which exists provided $\omega > \frac 1 p$, in turn requiring $p>1$), we have a straight rod, including the point masses, revolving around an axis parallel to itself, seemingly a violation of linear momentum conservation!

 

[Dale]

Very interesting work @maline!

 

[jartsa]

It is true that a helix with a whole number of turns and a uniform mass distribution cannot rotate freely around its axis. Its principal direction (of minimal moment of inertia) is slanted toward the direction a quarter-turn away from each endpoint of the helix, so if it is rotated around the axis, the angular momentum vector will be slanted away from that principal direction, and the rotation will precess around the AM vector.
One way to avoid this is if the helix is infinitely long. In that (obviously very hypothetical) case, the rotation is possible, and there will indeed be an inertial frame in which we have a straight rod revolving around an axis parallel to itself. Momentum will not be conserved, but that is normal for mass distributions that extend to infinity. In particular, the shear tension in the rod is a component of the stress tensor that remains nonzero at infinity, so we can describe a sourceless "flow of force" from infinity that accelerates the rod toward the axis.
However, I'm not sure the OP's paradox has been adequately resolved. What about situations with nonuniform mass distributions along a finite helix? Is it true that there is no possible mass distribution that will allow free rotation around the axis? That seems quite surprising, but it seems like AVentura has proved it using relativity! Does anyone have a direct proof, a counterexample, or another resolution to the paradox?

Let's consider what happens when we make finite helix rotate non-freely using rocket motors. At the ends of the helix we attach two precession preventing rockets which apply the same forces that an infinite helix would apply at the end of the short helix. And along the helix we attach million rotation initiating rockets (the helix is very long).

Now we observe the rotation initiation process. In the helix frame the rockets apply a torque and the helix gains angular momentum. In another frame rockets start non-simultaneously, which straightens the helix. Somehow the straight rod is able to gain angular momentum, and possesses angular momentum.

What happens if the rod/helix breaks to small segments? In the helix frame the parts fly apart. In the rod frame different parts of the rod start the linear motion at different times, and because of that the parts fly to different directions.

So I think the important problem here is: How does the rod have all the different momentums?

 

[AVentura]

I'm trying to follow you jartsa but I'm having trouble. Is the question how could such a helix come to have such a rotation? I'm not sure we should bother thinking about the rod. We know it cannot do what it appears to here.

If maline's analysis is correct there must be something else preventing the rotation. I don't really have the math skills to apply the Herglotz-Noether theorem, but I'd like to have a layman's understanding of it. I only understand born rigidity in the sense of linear acceleration (and nothing can naturally achieve it, correct?)

It seems there should be a general theorem concerning how a relativistic center of mass (?) transforms between frames. Similar to how angular momentum in the direction of motion is unchanged in boosts, but perpendicular AM is.

 

[maline]

We know it cannot do what it appears to here.

Is that completely clear? I'm a bit confused about this. The off-diagonal spatial components of the stress tensor are nonzero, so doesn't that turn into a "hidden momentum" in the boosted frame? Perhaps our rod always has zero momentum in the $x,y$ directions, despite its revolution? I'm probably talking nonsense, so somebody please help clear this up.

I tried a bit to work out what the stress tensor looks like, but I got stuck because in a one dimensional body, there are (IIUC) only three degrees of freedom for the spatial components of the tensor, so we get a system of three linear ODE's in three variables (the force density at a point is the derivative of the stress along the length of the body), but then we can only satisfy three boundary conditions, and we have six because all components of the stress must vanish at both ends of the body. It seems our helix needs to be three dimensional, and that makes it pretty scary to work with.

(and nothing can naturally achieve it, correct?)

I think you are referring to the fact that any angular acceleration must distort a body & violate Born rigidity, because different parts of the body length-contract differently.

It seems there should be a general theorem concerning how a relativistic center of mass (?) transforms between frames.

Yes, I would also love a clear reference on this topic!

 

[PeterDonis]

It seems there should be a general theorem concerning how a relativistic center of mass (?) transforms between frames. Similar to how angular momentum in the direction of motion is unchanged in boosts, but perpendicular AM is.

The general theorem, to the extent that there is one, says that you have to use the relativistic angular momentum tensor. I posted some computations of that for the helix earlier in this thread.

 

[AVentura]

I think you are referring to the fact that any angular acceleration must distort a body & violate Born rigidity, because different parts of the body length-contract differently.

Yes. But we don't have angular acceleration in this case. I can see how that could affect how the helix came to be in this state though.

The general theorem, to the extent that there is one, says that you have to use the relativistic angular momentum tensor. I posted some computations of that for the helix earlier in this thread.

I see, thanks.

 

[jartsa]

I'm trying to follow you jartsa but I'm having trouble. Is the question how could such a helix come to have such a rotation? I'm not sure we should bother thinking about the rod. We know it cannot do what it appears to here.

If maline's analysis is correct there must be something else preventing the rotation. I don't really have the math skills to apply the Herglotz-Noether theorem, but I'd like to have a layman's understanding of it. I only understand born rigidity in the sense of linear acceleration (and nothing can naturally achieve it, correct?)

It seems there should be a general theorem concerning how a relativistic center of mass (?) transforms between frames. Similar to how angular momentum in the direction of motion is unchanged in boosts, but perpendicular AM is.

Well, I think a helix can have such a rotation - with just a little help from two rockets at the ends of the helix. (such rotation is possible for an infinitely long helix, so we take a finite clip of that helix and emulate the two removed parts by two rockets)

But, a real rod can not have such sideways motion (acceleration) as the 'rod', because we know that a rod inside a helix can not get out of the helix by moving sideways.

I guess the above is really hard to follow. I was saying that if we accelerate an axially moving real rod sideways, the rod will either experience axial stress or the rod will turn. But that does not apply to a 'rod' - an object that is a helix in its rest frame. (it does not apply to that odd motion that the 'rod' does according to me, and according to you can not do)

So (according to me) a 'rod' and a real rod don't follow the same laws, so maybe it's not so surprising if a 'rod' moves around a little bit oddly?

 

[maline]

The post by maline in #61 seems to increase the feeling of confusion in this thread.

Well yes! I showed in that thread that the suggested resolution of the paradox fails. I am hoping someone will come up with something, otherwise we have a serious contradiction on our hands!

 

[Dale]

I think that @PeterDonis proposed method may still work. Since it is based on the kinematics rather than the dynamics it would hold for any mass distribution. So I wouldn't go to "serious contradiction" yet. I would go to "requires a fully relativistic treatment".

 

[maline]

I think that @PeterDonis proposed method may still work.

I must admit that if Peter suggested a solution, it went right over my head! What was the direction? Is the rotating helix impossible, or is the transformation incorrect, or is the revolving rod somehow legal?

 

[Dale]

I must admit that if Peter suggested a solution, it went right over my head! What was the direction? Is the rotating helix impossible, or is the transformation incorrect, or is the revolving rod somehow legal?

He suggested writing down the congruence and then calculating the shear and expansion tensor. If those are not 0 then the motion cannot be performed by an object in a free motion.

 

[PeterDonis]

He suggested writing down the congruence and then calculating the shear and expansion tensor.

More precisely, the expansion tensor, which includes the expansion scalar (the trace of the tensor) and the shear tensor (the symmetric traceless part of the tensor).

I have done some calculations along these lines, but the problem I am having is how to distinguish the helix congruence from the "cylinder" congruence (the one describing a cylinder rotating about its axis). So far every way I have found of writing down the helix congruence gives me the same (zero) expansion tensor as the one for the cylinder (since the worldlines in the helix congruence are a subset of those in the cylinder congruence). I'm not sure how to capture in the math the fact that the helix congruence "twists" around the cylinder.

 

[maline]

So far every way I have found of writing down the helix congruence gives me the same (zero) expansion tensor as the one for the cylinder (since the worldlines in the helix congruence are a subset of those in the cylinder congruence).

Right, that's why I didn't even get that you thought there might be a solution in this direction. How could the rotating helix be kinematically impossible, when a helix can be considered as just a cylinder with a particular (singular) mass distribution?

 

[PeterDonis]

How could the rotating helix be kinematically impossible, when a helix can be considered as just a cylinder with a particular (singular) mass distribution?

No, a helix is not a cylinder with a particular mass distribution. A cylinder is symmetrical about its axis; that's what makes its free rotation kinematically possible. A helix is not symmetrical about its axis; that should make a difference somewhere in the physical model we use to describe it. I just have not figured out where.

 

[maline]

No, a helix is not a cylinder with a particular mass distribution. A cylinder is symmetrical about its axis; that's what makes its free rotation kinematically possible. A helix is not symmetrical about its axis;

From a purely kinematic perspective, can't you just imagine that we have a full cylinder, but that its mass density is zero except at the points making up the helix?

 

[PeterDonis]

From a purely kinematic perspective, can't you just imagine that we have a full cylinder, but that its mass density is zero except at the points making up the helix?

That might help for something like the angular momentum tensor, but it wouldn't help for the kinematic decomposition (expansion, shear, and vorticity), because that uses the 4-velocity, not the 4-momentum.

 

[maline]

That might help for something like the angular momentum tensor, but it wouldn't help for the kinematic decomposition (expansion, shear, and vorticity), because that uses the 4-velocity, not the 4-momentum.

That's exactly what I'm trying to say. The 4-velocity of the helix, at every point, is identical to that of the corresponding point in a rotating cylinder. The differences are only a question of mass distribution. So if a rotating cylinder is possible, that already includes a rotating helix, along with any other subset of the cylinder's worldlines.

 

[PeterDonis]

if a rotating cylinder is possible, that already includes a rotating helix, along with any other subset of the cylinder's worldlines

No, it doesn't. The general case you are thinking of is a given 4-velocity field with no restrictions on the mass distribution. A rotating cylinder--the case for which we know free motion is possible--is a particular instance of this general case in which the mass distribution is symmetrical about the axis (the usual assumption is that it is constant). Any mass distribution that is not symmetrical about the axis (which a helix is not) is not a rotating cylinder; it is a different particular instance of the general case, and we cannot conclude that it is possible as a free motion just because a rotating cylinder is.

 

[maline]

The general case you are thinking of is a given 4-velocity field with no restrictions on the mass distribution. A rotating cylinder is a particular instance of this general case in which the mass distribution is symmetrical about the axis (the usual assumption is that it is constant). Any mass distribution that is not symmetrical about the axis (which a helix is not) is not a rotating cylinder; it is a different particular instance of the general case.

But doesn't the possibility of a rotating cylinder show that the general case- simple rotation around an axis- is always kinematically possible?

 

[PeterDonis]

doesn't the possibility of a rotating cylinder show that the general case- simple rotation around an axis- is always kinematically possible?

Kinematically possible? Sure, with appropriate external forces applied. But the question I'm trying to answer is what is possible as a free motion, with no external forces applied. The rotating cylinder--mass distribution symmetrical about the axis--is possible as a free motion. I don't think the helix is, because of the unsymmetrical mass distribution. But I haven't been able to figure out how to model that asymmetry mathematically in order to test my conjecture.

 

[Dale]

From a purely kinematic perspective, can't you just imagine that we have a full cylinder, but that its mass density is zero except at the points making up the helix?

You can but then it is no longer axisymmetric

 

[maline]

The rotating cylinder--mass distribution symmetrical about the axis--is possible as a free motion. I don't think the helix is, because of the unsymmetrical mass distribution.

Oh, ok. So you are working with mass distributions, i.e. dynamics and not just kinematics. I think Dale misunderstood this:

Since it is based on the kinematics rather than the dynamics it would hold for any mass distribution.

That's what threw me off.

But isn't it the case that for a rigid cylindrical shell with arbitrary mass distribution, rotating in its rest frame around the axis, the conditions for relativistic free rotation reduce to those of the nonrelativistic case, because all points on the shell have the same "gamma factor"? Is that's correct, then the distribution I mentioned in post #61, which can rotate freely in Newtonian physics, should be able to do so in SR as well.

 

[Dale]

@maline I don't know of any argument against your reasoning

 

[PeterDonis]

the distribution I mentioned in post #61, which can rotate freely in Newtonian physics

This claim would seem to be a "paradox" in Newtonian physics, since as you said in post #61, it appears to violate linear momentum conservation, which is a valid conservation law in Newtonian physics. So maybe we first need to figure out whether this claim is actually true in Newtonian physics, or whether there is in fact some flaw in the reasoning in post #61 (or some other factor involved that that post does not address).

 

[maline]

This claim would seem to be a "paradox" in Newtonian physics, since as you said in post #61, it appears to violate linear momentum conservation, which is a valid conservation law in Newtonian physics.

The apparent violation of momentum conservation is in the relativistic boosted case, because the helix becomes a rod. In Newtonian physics, and seemingly in the SR rest frame as well, the motion is unproblematic.

In Newtonian physics this is not usually considered a problem or an "unsurmountable paradox" because the "infinitely long" mathematical idealization is an accepted ordinary procedure

This is not relevant at all; the rigid body we are discussing is finite (although its mass distribution happens to be singular, which may or may not be important).

we can see how a general rigid body like a helix(without the particular symmetries of a disk or a cylinder that allow the rotation to remain a Killing motion) is not allowed to freely rotate with translation in SR

We only need to show that free rotation without translation, i.e. in the rest frame, is possible. The Lorentz transformation will tell us what happens in other frames. Only in our scenario, it seems to be giving an absurd answer.

 

[maline]

That's what we're trying to figure out.
The theorem says that any Born rigid motion with nonzero vorticity must be a Killing motion. But that doesn't help unless we can verify that the helix motion in question is or is not a Killing motion. That's what I asked if you have a mathematical proof of; you said of the helix that "its points don't describe a Killing motion when rotating", i.e., that the helix motion is not a Killing motion. Do you have a mathematical proof of that? The Herglotz-Noether theorem is not such a proof because it doesn't tell you whether or not a particular motion with nonzero vorticity is or is not a Killing motion; it just says that if it isn't a Killing motion, it can't be Born rigid.

Hold it. Why is this whole question about Born rigidity, Killing fields, and Herglotz-Noether still an issue? I thought we were in agreement, in Peter's post #85, that our rotating helix definitely is both Born rigid and Killing, being that its velocity field is simply a subset of the standard Killing field for rigid rotation, namely, in cylindrical coordinates $(t,r,\theta,z)$, the field $u^\alpha =(\gamma (r),0,\gamma(r) r \omega,0)$, where $\omega$ is a constant, $\gamma(r)=(1-r^2\omega^2)^{-1/2}$, and $0\leq r<1/\omega$.

Peter agreed there that the only doubt was about the dynamical issue of whether such a motion can be free, for any particular mass distribution. Herglotz-Noether has nothing to say about that question- it is a purely kinematic theorem giving conditions for rigidity, i.e whether the motion distorts the body. Issues of mass, momentum and force do not enter the theorem at all.

So can we please lay Born rigidity to rest for the remainder of this thread, and focus on the aspects of angular momentum and center of energy?

 

[pervect]

Hold it. Why is this whole question about Born rigidity, Killing fields, and Herglotz-Noether still an issue? I thought we were in agreement, in Peter's post #85, that our rotating helix definitely is both Born rigid and Killing, being that its velocity field is simply a subset of the standard Killing field for rigid rotation, namely, in cylindrical coordinates $(t,r,\theta,z)$, the field $u^\alpha =(\gamma (r),0,\gamma(r) r \omega,0)$, where $\omega$ is a constant, $\gamma(r)=(1-r^2\omega^2)^{-1/2}$, and $0\leq r<1/\omega$.

I'd agree with this. This also implies that expansion and shear for the congruence should be zero, because it's rigid. Which matches the calculations, I gather.

So can we please lay Born rigidity to rest for the remainder of this thread, and focus on the aspects of angular momentum and center of energy?

As I recall, center of energy is just frame dependent, and this is an example of said frame dependence. Angular momentum is still conserved, of course.

 

[PeterDonis]

I thought we were in agreement, in Peter's post #85, that our rotating helix definitely is both Born rigid and Killing

I originally thought it was, but in post #85 I only said I thought it was "kinematically possible...with appropriate external forces applied". Not all motions that meet that description are Born rigid Killing motions.

being that its velocity field is simply a subset of the standard Killing field for rigid rotation

Yes, but one of the potential issues with this is how to define derivatives of the velocity field if the subset we select is not continuous. And you have to have well-defined derivatives of the velocity field in order to evaluate Killing's equation.

 

[AVentura]

If the helix had a minimum width tangentially this width would look wider in the frame where it is translating (slower angular velocity but same radius). This would move the center of energy back towards the original axis. Just brainstorming.