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

[PeterDonis]

Their contribution to the integral is equal and opposite to the above value

Looking back at this, I see how it is true for $n = 1$, but that's because the masses turn out to be negative. I don't see how it is true for $n > 1$, where the masses are positive; it seems to me that that makes their contributions to $J_y$ of the same sign as those of the uniform mass distribution in the original helix.

 

[maline]

Looking back at this, I see how it is true for $n = 1$, but that's because the masses turn out to be negative. I don't see how it is true for $n > 1$, where the masses are positive; it seems to me that that makes their contributions to $J_y$ of the same sign as those of the uniform mass distribution in the original helix.

AVentura got it right:

Is the problem with n=1 just that there is nowhere on the helix to place the needed counterweights (using positive mass)?

Visual, for z<0 you can only put the weight on one side (say top). For z>0 you can only put on the bottom. With more curls you can put one on any side you want, then find the needed weight.

In other words, you need to put positive masses at points where the sine function makes the sign of the contribution as desired. It doesn't work for $n=1$ because $x\sin x$ is positive for $-\pi\leq x\leq \pi$.

 

[maline]

I also want to point out that up to this point, the classical and relativistic treatments are essentially identical. We relabeled the "center of mass at time 0" and the angular momentum 3-vector as components of a rank-2 4-tensor, but that makes no difference until we actually carry out a Lorentz boost. The integral expressions for the components are the same as in the Newtonian case except that the mass density, in each expression, is multiplied by $\gamma(r)=(1-r^2\omega^2)^{-1/2}$. But since we are dealing with a cylindrical shell, $r=R$ is just a constant and so is $\gamma$, so it just gets factored out of the integrals.

 

[Dale]

classical and relativistic treatments are essentially identical

I wouldn't say that. I think that we haven't done a fully relativistic treatment which I think would involve the stress energy tensor on the helix. We have instead done a classical treatment and some mixed treatments. This problem seems to be finding a flaw in the mixed analysis.

 

[PeterDonis]

you need to put positive masses at points where the sine function makes the sign of the contribution as desired. It doesn't work for $n=1$ because $x\sin x$ is positive for $-\pi\leq x\leq \pi$.

Yes, I see that. I'm working now on how to capture that, or something equivalent, in the relativistic formulation.

 

[PeterDonis]

I think that we haven't done a fully relativistic treatment which I think would involve the stress energy tensor on the helix

I'm not sure that's necessary; I think we can get away with just a variable mass distribution $\mu(z)$ along the helix, without having to do the full stress-energy tensor treatment. But I won't know for sure until I've tried it.

 

[AVentura]

Is the stress in the parts of the helix that are connected to any point masses infinite?

 

[maline]

Is the stress in the parts of the helix that are connected to any point masses infinite?

Good question. I think the answer is no: the 4-force density is the derivative (4-divergence) of the stress-energy tensor, $f^\alpha={T^{\alpha \beta}}_{,\beta}$. So the point force, being a "delta function" in the force density, can be caused by a jump discontinuity in the stress, whose derivative is a delta of the appropriate magnitude.

 

[pervect]

I'm afraid I haven't had time to keep up with this thread, but a couple of comments. First on the issue of rigidity.

The expansion scalar is a rank-0 tensor. The shear tensor is a rank 2 tensor. These tensors are both zero in some coordinate system, the Born coordinate system of a rigidly rotating disk or cylinder. <<wiki link for Born coordinates>>

A tensor that is zero in one coordinate system will be zero in all coordinate systems.

Therefore an object that is Born rigid in one coordinate system will be Born rigid in all coordinate systems, because one of several equivalent definitions of rigidity is that an object have a vanishing expansion and shear. And we know that a tensor that is zero in one coordinate system is zero in all.

One application of this is noting that changing the state of motion of the rotating cylinder by boost its motion along it's axis of symmetry can be regarded just as a coordinate transformation - the same cylinder viewed in two different coordinate systems moving relative to one another. Then we conclude that this boosting operation doesn't and can't change the rigidity of system.

I don't see what there is to argue about. Hopefully the argument is really over at this point, except possibly for a few holdouts who don't seem to be actually carrying out the math. Hopefully the argument will be sufficient to convince the moderators and science advisors - and anyone actually doing the math, which is my primary concern at this point.

The second point is that it might be useful to construct a fully relativistic model of the situation out of point masses, and massless strings, which exert the forces that hold the masses in place, in some convenient set of coordinates.

The point to realize is that the assumption that the strings are massless IS coordinate dependent. The tools we need to demonstrate and understand this are just an understanding of the stress energy tensor, and it's transformation properties. For a textbook discussion of some of the issues, see a discussion of the weak energy condition in your favorite textbook, specifically why a massless string under tension is considered to be "exotic matter" which has a negative energy density in some coordinates.

So, one convenient approach is to first create a system consisting of point masses and massless strings in a convenient coordinate system.

Evaluating the mass of the string after one changes coordinates will give one an idea of whether the contribution of the stresses in the string to the angular momentum of the system can be ignored in the new coordinates. In the coordinates in which the mass of the string is zero, the string will not contribute to the linear or angular momentum, but the same is not true (hopefully, this is obviously not true) in coordinates in which the mass of the string is not zero.

 

[PeterDonis]

I don't see what there is to argue about.

I don't think there's any dispute about the rotating cylinder--there appears to be general agreement that this is a Born rigid motion and that it is a possible "free" motion, i.e., its angular momentum and angular velocity vectors are parallel, so that no external forces are required to maintain the motion.

The open question is about the case of the rotating helix, considered as a subset of the cylinder. There appears to be agreement that, if the mass distribution is uniform along the helix, this motion, while it is Born rigid, is not a possible "free" motion, because the angular momentum and angular velocity are not parallel; external forces would be required to maintain this motion. However, we have not resolved the question of whether there is a non-uniform mass distribution along the helix that would make the motion "free", by removing the non-parallel component of the angular momentum. @maline gave a Newtonian scenario that appears to have this property; I have been working on a relativistic model but have not yet come up with one that has the property.

If there is a relativistic model that has the above property--a helix rotation about the axis of the helix that is a "free" motion--then we have a possible "paradox" in the relativistic case: there will be some frame in which the helix transforms into a straight rod that is off of the axis of rotation (basically, imagine painting the helix on the rotating cylinder and then transforming into a frame in which the painted-on helix is "unwound" so that it is just a straight line down the cylinder that rotates with the cylinder). The latter motion does not seem like it could possibly be a "free" motion, but the property of being a "free" motion, requiring no external forces to maintain, should be invariant under Lorentz transformations.

it might be useful to construct a fully relativistic model of the situation out of point masses, and massless strings

That's not the approach I am taking. The approach I am taking is to model the helix as a congruence of worldlines (a subset of the cylinder congruence), paramterized by $z$, the axial coordinate along the cylinder in the center of mass frame (the frame in which the cylinder/helix only rotates, with no translation). The 4-position and 4-velocity vectors of this congruence as a function of $z$ are known and have been posted earlier in this thread. I am then modeling the mass distribution as a function $\mu(z)$, whose units are mass per unit length, along the helix. If I am able to find such a function that makes the relativistic angular momentum in this frame have only one nonzero component, $M^{23}$ (in the x-y plane, perpendicular to the axis of rotation), I am then going to boost the corresponding solution into the moving frame that, kinematically, "unwinds" the helix, to see what things look like in that frame.

 

[PeterDonis]

Post #106 has the relativistic analysis I have done so far, for the case of a constant mass distribution:

https://www.physicsforums.com/threa...inder-max-von-laue.913600/page-6#post-5762960

 

[Dale]

The connection with the constraints is that the time reversibility and foliability of the geometry that they enforce guarantee the vanishing of the momentum transfer(killing motion:the rotating cylinder is an example of this).

Got it. Those conditions guarantee that there exists some coordinate system such that the stress energy tensor is diagonal. Of course, you can choose poorly adapted coordinates but the nice coordinates exist.

 

[PeterDonis]

As a first step at analyzing the case with a non-constant mass distribution, consider the following modifications to the model I gave in post #106.

First, we extend the model to cover more than one turn of the helix. A simple way to do that is to extend the limits of integration to $\pm NZ / 2$, where $N$ is the number of turns.

Second, we observe that the nonzero integrals are even functions over an even domain, which means we can rewrite them as integrals from $0$ to $NZ / 2$ and multiply by $2$. This leaves the final answers the same but helps with the next step.

Next, we allow for a non-constant mass distribution. Instead of trying to work with a generic function $\mu(z)$, I'm first going to try a simpler model. View the helix as composed of "segments", which appear in the integrals (rewritten as in the second step above) as pieces each occupying an interval of $Z / 2$, so that piece $k$ goes from $(k - 1) Z / 2$ to $k Z / 2$, and $k$ runs from $1$ to $N$. (Note that, except for $k = 1$, each "piece" actually represents two disconnected segments of the helix, one in the negative $z$ range and one in the positive $z$ range, which happen to make identical contributions to the integrals as written according to the first step above, i.e., before changing the limits as in the second step above. "Piece" $k = 1$ represents the one-turn helix that was modeled in post #106.) We then give each piece $k$ a "weight" $W_k > 0$ that represents its contribution to the mass of the helix. We won't try at this point to normalize the weights or worry about the exact units; their relative magnitudes are all that will matter in what follows. This is equivalent to assuming that, within each piece, the mass distribution is uniform, i.e., the value of $\mu$ is constant within each piece, but its value can vary from one piece to another.

(Note: This is not the same as the point mass method that @maline used in an earlier post, placing additional mass in points lying on the $x$ axis to avoid making $J_x$ nonzero. I think it still works because the mass distribution of each piece is symmetrical about the $x$ axis, so it should not affect the $M^{13}$ total angular momentum component--note that I am using the $t, z, x, y$ coordinate ordering, so $x$ is coordinate 2, perpendicular to the 13 plane.)

With these adjustments, the two integrals under consideration are modified as follows:

$$
M^{12} = \gamma \omega R \frac{Z^2}{\pi} \sum_{k=1}^N \left( -1 \right)^k \left( 2 k - 1 \right) W_k
$$

$$
M^{23} = \gamma \omega R^2 Z \sum_{k=1}^N W_k
$$

Notice that the only changes from before are the sums over the weights.

We can see, first, that $M^{23}$ remains positive and is basically unaffected by all this, since the sum of the weights still has to give the overall mass of the helix (again, without worrying about normalization or exact units). We can also see that, for $N = 1$, $M^{12}$ is unchanged from before (as expected, since our assumption about the mass distribution leaves the case of one turn the same--but even allowing the mass distribution to vary within a piece would not allow any cancellation in $M^{12}$ for the case $N = 1$, since the sign of all contributions within a given piece is the same). But for the case $N > 1$, we can now see the possibility of arranging the weights so as to make $M^{12}$ vanish. For example, for the case $N = 2$, if we have $- W_1 + 3 W_2 = 0$, or $W_1 = 3 W_2$, then $M^{12}$ vanishes. This matches up with what @maline posted earlier in the thread in the Newtonian case.

What I think I was missing before is that, even though the mass distribution $\mu(z)$ is even, which means it can't make any of the integrals that were zero before nonzero, it can still, for the case $N > 1$, make a nonzero integral zero, if there is the possibility of different segments of the helix making contributions of different signs. The above analysis shows that that is indeed possible.

The next step will be to transform the $X^a$ and $P^a$ vectors into a frame in which the helix is "unwound" (kinematically), and see what the angular momentum tensor looks like in that frame. I'll address that in a follow-up post.

 

[maline]

Excellent! Now we know that the paradox is not due to using point masses. Not only that, but we can easily extend this example to a fully three-dimensional one: just multiply this distribution by an arbitrary function of $r$, defined on an interval $R_1\leq r\leq R_2$, and by another arbitrary function of $\theta$ defined on $\theta_1\leq\theta \leq\theta_2$. We may as well choose the angular function to be constant, but choosing different radial functions may be useful.

I think that we haven't done a fully relativistic treatment which I think would involve the stress energy tensor on the helix.

I think we can get away with just a variable mass distribution mu(z) along the helix, without having to do the full stress-energy tensor treatment.

If it's the case that all we need to do is check that the center of energy and the angular momentum are both on the z-axis and therefore are maintained by the rotation, in other words that $M^{tx}=M^{ty}=M^{xz}=M^{yz}=0$, then I agree with Peter. The full expression for $M$ is: $$M^{\alpha \beta}=\int_V\, (X^\alpha T^{\beta \gamma}-X^\beta T^{\alpha \gamma})\,d\Sigma_\gamma$$ where $V$ is the spacelike hypersurface representing the "present" and $d\Sigma_\gamma$ is the volume element (one-form) normal to it. Since we are considering surfaces of constant $t$, $d\Sigma_\gamma$ is purely in the $t$ direction, so we can set the index $\gamma$ as $t$. Then only the $T^{\alpha t}$ components of the tensor contribute, and those are the mass and momentum densities we have been considering.

So we now have clear examples where the rotation does conserve $M^{\alpha \beta}$, regardless of the stress. Is that sufficient for free motion to be possible, or are there other conditions that may involve the stress?

BTW, I need to correct what I wrote before to @AVentura:

the 4-force density is the derivative (4-divergence) of the stress-energy tensor

That is clearly wrong, ${T^{\alpha \beta}}_{,\beta}=0$ always; that is conservation of energy & momentum. What I meant to say is $\frac \partial {\partial t} T^{i t}=-{T^{i j}}_{,j}$ (the momentum part of the conservation equation). In words: the rate of change of the momentum density is minus the 3-divergence of the stress. Note that this time derivative is by coordinate time, unlike 4-force which is a derivative by proper time, so its value in our case is $-\omega^2 R \hat r$ times the mass density, without the $\gamma$ factor.

 

[PeterDonis]

I'm not seeing how $M^{12}$ can be made to vanish

It can't be made to vanish unless the helix has more than one full turn. What having multiple turns does is add segments of the helix that make contributions to $M^{12}$ that are of opposite signs. With a uniform mass distribution, this can't make $M^{12}$ vanish because the contributions from the segments are of different magnitudes; but allowing a different mass for each segment let's you adjust the masses so the contributions of opposite sign cancel. I gave a specific illustration for the case $N = 2$, a helix of two full turns.

 

[maline]

Note that this time derivative is by coordinate time, unlike 4-force which is a derivative by proper time, so its value in our case is −ω2R^r−ω2Rr^-\omega^2 R \hat r times the mass density, without the γγ\gamma factor.

No, this doesn't work either. The $t$ derivative of momentum density at a constant spatial position $x^i$ does not include the acceleration of the mass elements as they move, so in particular the centripetal acceleration will not show up there at all. The force required for that must be expressed as a proper time derivative $\frac {dT^{\alpha t}} {d\tau}={T^{\alpha t}}_{,\beta}u^\beta$, also known as the Langrangian derivative of the 4-momentum density.
But I'm afraid I now am confused about how to work with this "force". What are its transformation properties? What is the correct way to write it as a derivative of stress? Can someone help me out here?

 

[PeterDonis]

the centripetal acceleration will not show up there at all

The centripetal acceleration can be derived from the kinematics alone--just compute the proper acceleration from the 4-velocity field. That gives the same answer for the helix as it does for the cylinder, because the 4-velocity fields are the same (unless there is something to my previous thought that there might be some issue with derivatives in the helix case because it is a discontinuous subset of the cylinder--but as I said before, I have no idea how to deal with that if it's an issue).

I now am confused about how to work with this "force". What are its transformation properties? What is the correct way to write it as a derivative of stress?

I'm not sure about how to work with this either; in fact I'm not sure how trying to work with the full stress-energy tensor in general will tell us anything new. That's why I've been trying to stick to the simplified relativistic model I've been using. I'm hoping that boosting that model into a frame that kinematically unwinds the helix will shed some light.

 

[pervect]

I don't think a rotating helix is a force-free motion, considering it as a purely classical non-relativistic problem. But I could be making a mistake. Hopefully it's obvious that if the motion isn't force free, then a good analysis of angular (or linear) momentum must consider the external torques and forces on the system.

To set up the problem as a purely classical problem, I consider a rigid central shaft, with tension bearing strings keeping the rotating masses rotating in the helix. This satisfies the force balance equation of the masses, we now need to consider the force balance equation on the central shaft. To have the motion of the rigid central shaft be force-free, we need two conditions. The total force on the central shaft must be zero, and the total torque on the central shaft must be zero.

Either condition alone appears to be possible to satisfy, but not both simultaneously.

To set up the problem, If we let the axis of symmetry of the cylinder be the z-axis, we can write the forces on the central shaft due to the strings as

$$f_x = -\cos kz \quad f_y = -\sin kz \quad f_z = 0$$

For convenience, we can set the pitch factor k equal to 1.

To maintain symmetry, we'll let z vary from -a to a. Then the force free condition (with the condition that the pitch factor k=1, which we will assume from this point onwards) turns out to be 2*sin(a) = 0, which has the obvious solutions a=$\pi, 3\pi$ etc, as expected.

The torques should be given by $\vec{f} \times \vec{z}$, the components of which should be:

$$ \begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \\ -\cos z & -\sin z & 0 \\ 0 & 0 & z\end{vmatrix}$$

Carrying out a similar integral for the torques as we did for the forces, we get the torque free condition as:

$$2 \sin a - 2\,a\,\cos a = 0$$

The solutions of the total force being zero don't satisfy the conditions that the torque be zero. The conditions that the torque be zero does has solutions (interestingly enough), but these solutions don't make the total force on the central shaft zero.

 

[PeterDonis]

To set up the problem as a purely classical problem, I consider a rigid central shaft, with tension bearing strings keeping the rotating masses rotating in the helix.

This is different to how I was doing it, but I think I can do a similar analysis in the model I've been using. I'll see.

 

[PeterDonis]

I think I can do a similar analysis in the model I've been using. I'll see.

This is interesting enough to sidetrack me from boosting my relativistic model, so I'll tackle it first.

The 4-position of the helix is given by

$$
X^a = \left( t, z, R \cos ( \omega t + k z ), R \sin (\omega t + k z ) \right)
$$

The 4-velocity field of the helix is given by

$$
U^a = \left( \gamma, 0, - \gamma \omega R \sin ( \omega t + k z ), \gamma \omega R \cos (\omega t + k z ) \right)
$$

where $k = 2 \pi / Z$ and $Z$ is the axial length (along $z$) of a single turn of the helix. The proper acceleration for this 4-velocity field is

$$
A^a = \left( 0, 0, - \gamma^2 \omega^2 R \cos ( \omega t + k z ), - \gamma^2 \omega^2 R \sin ( \omega t + k z ) \right)
$$

What we need to do now is to sum the force and torque over the entire helix and see whether they both vanish, which must be the case if the motion is a valid free motion. The two integrals in question are

$$
F^a_{\text{total}} = \int_{- NZ / 2}^{NZ / 2} \mu(z) A^a(z) dz
$$
$$
T^{ab}_{\text{total}} = \int_{- NZ / 2}^{NZ / 2} \mu(z) \left( X^a (z) A^b (z) - X^b (z) A^a (z) \right) dz
$$

Note that I have modeled the torque as an antisymmetric tensor rather than as a vector, for the same reason I did that for the angular momentum. Also, we evaluate the integrals at $t = 0$.

The force integral only has two nonzero components, $x$ and $y$; and the $y$ integral is an odd function over an even domain, which is zero. So we have, using the same simplifications as I used for the angular momentum in my earlier post:

$$
F^x_{\text{total}} = - 2 \gamma^2 \omega^2 R \int_{0}^{NZ / 2} \mu(z) \cos \left( \frac{2 \pi z}{Z} \right) = - 2 \gamma^2 \omega^2 R \sum_{k = 1}^N \sin \left( \pi k \right) W_k
$$

Since the $\sin$ will be zero for all terms, the weights don't matter and we have that the total force on the helix does vanish.

The torque integral turns out to also have only one component that is not an odd function over an even domain (note that $T^{23}$, the torque in the x-y plane, is an integral of the product $\cos k z \sin k z$, which is odd and therefore vanishes):

$$
T^{13} = - 2 \gamma^2 \omega^2 R \int_0^{NZ/2} \mu(z) z \sin \left( \frac{2 \pi z}{Z} \right)
$$

This turns out to be the same integral (except for some slightly different constant factors) as $M^{12}$, and therefore can be made to vanish by the same method as I showed in my previous post, by choosing appropriate weights for each piece of the helix.

So it looks to me like the total force and total torque on the helix can be made to vanish, so we can't rule out that it is a possible free motion on those grounds.

 

[PeterDonis]

I don't think a rotating helix is a force-free motion, considering it as a purely classical non-relativistic problem.

Looking at the relativistic analysis I just posted, the only force on an individual element of the helix (in my model, where there are no massless strings or massless central shaft) is the centripetal force that keeps the piece rotating about the $z$ axis with angular velocity $\omega$. This force is exerted by neighboring pieces of the helix: it is true that those pieces also exert sideways forces, but those should cancel leaving only the centripetal net force. That, at any rate, is what the kinematics, the proper acceleration vector, says. In this respect the helix appears to work exactly like a rotating ring, for which the centripetal force is also exerted by neighboring pieces of the ring.

 

[pervect]

Looking at the relativistic analysis I just posted, the only force on an individual element of the helix (in my model, where there are no massless strings or massless central shaft) is the centripetal force that keeps the piece rotating about the $z$ axis with angular velocity $\omega$. This force is exerted by neighboring pieces of the helix: it is true that those pieces also exert sideways forces, but those should cancel leaving only the centripetal net force. That, at any rate, is what the kinematics, the proper acceleration vector, says. In this respect the helix appears to work exactly like a rotating ring, for which the centripetal force is also exerted by neighboring pieces of the ring.

I'd agree that the force on an element is the centripetal force that keeps the piece rotating about the z axis.

But I don't think this matters to the analysis. The situation is one of dynamic balance, as seen for instance in tires. A tire can be perfectly balanced statically, but it must also be dynamically balanced to rotate without wobbling. So the question is - would a helix shaped "tire" be dynamically balanced, or not? The static balance of the helix is insufficient to answer this question.

According to wiki, the physical condition for torque free motion is that the axis of rotation be a principal axis of the moment of inertia tensor. This sounds right to me, I haven't tried to look it up in Goldstein (or another text) to confirm.

Whenever a rotor is forced to rotate about an axis that is not a principal axis, an external torque is needed.

https://en.wikipedia.org/w/index.php?title=Tire_balance&oldid=779604153

So, in this approach, we need to ask what the moment of inertia tensor of a helix is - specifically, we want to know if the z-axis of the cylinder is one of the principle axes of the moment of inertia tensor. If it is one of the principle axes, then the moment of inertia tensor should be diagonal.

We can write the moment of inertia tensor as

$$I^{ij} = \sum x^i x^j dm $$

where dm is a mass element.

If the off-diagonal elements of this tensor are zero, then the principle axes are aligned with the coordinate axes. Parameterizing the helix as we did before we have:

$$x = \cos s \quad y = \sin s \quad z = s$$

To maintain static balance, we'll let s range from $-\pi$ to $\pi$.

We can write $I^{yz}$ as being proportional to

$$\int_{-\pi}^\pi s \, \sin s \, ds$$

which is nonzero, demonstrating that the z-axis is not a principle axis of the moment of inertia tensor, and thus demonstrating that we need to apply an external torque to make a helix rotate around the z-axis.

 

[PeterDonis]

To maintain static balance, we'll let s range from $-\pi$ to $\pi$.

This is the case of one turn, which we already know is not a possible free motion. Try it for $s$ ranging from $- 2 \pi$ to $2 \pi$, which describes a helix of 2 turns; this gives a zero integral with appropriate weights for the segments (it's the same integral that comes up in the $M^{12}$ tensor component for the angular momentum).

We can write $I^{yz}$ as being proportional to
$$
\int_{-\pi}^\pi s \, \sin s \, ds
$$

You left out the mass distribution, which means you're assuming a uniform mass distribution. We already know this case is not a possible free motion. If you put in a non-uniform mass distribution with appropriate weights, as I did in my last few posts, you can get the components that are not in the x-y plane to vanish, since, as above, the integral that arises is the same one that has come up in my previous posts.

 

[PeterDonis]

the physical condition for torque free motion is that the axis of rotation be a principal axis of the moment of inertia tensor

I think what we have been showing in our last few exchanges is that there are at least three equivalent ways of stating this condition:

(1) The axis of rotation must be a principal axis of the moment of inertia tensor;

(2) The axis of rotation must be perpendicular to the plane of the angular momentum tensor;

(3) The total torque on the body, summed over all elements of the body, must be zero.

It looks, from my calculations, as if all three of these conditions give rise to the same integral (in this case the integral of $s \sin s$), so they will all three be satisfied, if they can be satisfied, by the same mass distribution and kinematics (where the latter means the same 4-position, 4-velocity, and 4-acceleration vectors).

 

[PeterDonis]

Here is a first stab at boosting my relativistic solution into a frame that unwinds the helix. We boost by a velocity $- v$, so that the helix in the new frame is moving at $v$ in the positive $z$ direction (note that this basically means we are treating the original frame, in which the helix center of mass is at rest, as the "primed" frame, even though the notation I am using labels it as unprimed and the new, boosted frame in this post as primed). Thus we have $t' = g(t + vz_0)$, $z' = g(z_0 + vt)$, where $g = 1 / \sqrt{1 - v^2}$, and I have used the notation $z_0$ to emphasize the fact that, in the original frame, each worldline in the helix congruence has a constant $z$ coordinate. This will help when we start looking at integrals, since we can still integrate over $z_0$ as the worldline label.

The inverse transform is then $t = g(t' - v z')$, $z_0 = g(z' - vt')$. We use the second equation to find $z'$ as a function of $t'$, and the first to express the arguments of the trig functions as functions of $t'$. This gives us a 4-position vector in the new frame:

$$
X^a = \left( t', v t' + \frac{z_0}{g}, R \cos ( \frac{\omega}{g} t' - \frac{v}{g} z_0 + k z_0 ), R \sin ( \frac{\omega}{g} t' - \frac{v}{g} z_0 + k z_0 ) \right)
$$

If we set $k = v / g = v \sqrt{1 - v^2}$, then the trig functions no longer depend on $z_0$ and we have unwound the helix. The only constraint is that, for this to be possible, we must have $k \le 1/2$, since we have $0 < v < 1$ and the function $v \sqrt{1 - v^2}$ has a maximum value of $1/2$ in that interval, at $v = 1 / \sqrt{2}$.

With the helix unwound, the 4-position, 4-velocity, and 4-acceleration vectors are

$$
X^a = \left( t', v t' + \frac{z_0}{g}, R \cos ( \frac{\omega}{g} t' ), R \sin ( \frac{\omega}{g} t' ) \right)
$$
$$
U^a = \left( g \gamma, g \gamma v, - \gamma \omega R \sin ( \frac{\omega}{g} t' ), \gamma \omega R \cos ( \frac{\omega}{g} t' ) \right)
$$
$$
A^a = \left( 0, 0, - \gamma^2 \omega^2 R \cos ( \frac{\omega}{g} t' ), - \gamma^2 \omega^2 R \sin ( \frac{\omega}{g} t' ) \right)
$$

This does indeed look like a straight rod that is off axis and is rotating as it translates, which intuitively does not seem like a possible free motion. However, I'll save further comment until I've done the computations of the tensor components.

 

[PeterDonis]

Following on from my last post, here are the nonzero tensor components for an individual element of the helix at $t' = 0$. First, the angular momentum tensor $M^{ab} = \mu (z_0) \left( X^a U^b - X^b U^a \right)$:

$$
M^{01} = - \mu (z_0) \gamma z_0
$$
$$
M^{02} = - \mu (z_0) g \gamma R
$$
$$
M^{12} = - \mu (z_0) g \gamma v R
$$
$$
M^{13} = \mu (z_0) \frac{\gamma \omega R}{g} z_0
$$
$$
M^{23} = \mu (z_0) \gamma \omega R^2
$$

Already we can see that something is going on, since $M^{02}$ and $M^{12}$ are just constants times the even mass distribution function, with no $z_0$ in them at all, so when we integrate over the whole helix (or rod in this frame), they won't vanish, no matter what mass distribution $\mu$ we try. $M^{02}$ and $M^{13}$ will vanish because they are odd functions of $z_0$, but that's no help.

Next, the force vector $F^a = \mu(z_0) A^a$:

$$
F^1 = - \mu (z_0) \gamma^2 \omega^2 R
$$

This also won't vanish when integrated over the whole helix/rod, regardless of the mass distribution.

Finally, the torque tensor $T^{ab} = \mu (z_0) \left( X^a A^b - X^b A^a \right)$:

$$
T^{12} = - \mu (z_0) \frac{\gamma^2 \omega^2 R}{g}
$$

This also won't vanish.

So now we have a puzzle: in the original frame, we were able to choose a mass distribution that made all the necessary things vanish; but in this new boosted frame, we apparently aren't. So something is wrong somewhere. The question is: where?

 

[AVentura]

Is this possibly a manifestation of the Trouton–Noble paradox?

https://arxiv.org/abs/1206.4487
https://arxiv.org/abs/physics/0603110

 

[PeterDonis]

If we set $k = v / g = v \sqrt{1 - v^2}$, then the trig functions no longer depend on $z_0$ and we have unwound the helix

On going back over this, I realized that this part is wrong. It should be $k = \omega v$ to unwind the helix. This means we have to boost by $v = k / \omega = 2 \pi / \omega Z$. Everything else in my post should still be ok.

 

[PeterDonis]

Is this possibly a manifestation of the Trouton–Noble paradox?

That might be a piece of the puzzle, yes. At the very least you've reminded me that my equations for the force are not correct. I need to rework those with the correct relativistic relationships.

 

[PeterDonis]

The weights $W_k$ are all positive. The contributions to the integrals for different pieces of the helix have different signs because the trig functions involved change sign from one piece to another. But that only happens for $N > 1$.

It might be helpful for me to briefly expand on how this works. Here are the key steps. We start with the following integral (where I have removed irrelevant constant factors to simplify things for this post):

$$
I = \int_{- N \pi}^{N \pi} \mu(z) z \sin z dz
$$

We first rewrite the integral by observing that the integrand is even and the integral is over an even domain; that let's us do this:

$$
I = 2 \int_{0}^{N \pi} \mu(z) z \sin z dz
$$

Next we separate the integral into $k$ pieces, as follows:

$$
I = 2 \sum_{k=1}^{N} \int_{\left( k - 1 \right) \pi}^{k \pi} \mu(z) z \sin z dz
$$

Now we adopt a particular form for $\mu(z)$; we assume that it is a constant $W_k > 0$ for each piece, but can vary from piece to piece. That gives us:

$$
I = 2 \sum_{k=1}^{N} W_k \int_{\left( k - 1\right) \pi}^{k \pi} z \sin z dz
$$

Finally, we evaluate the integral in each piece, which gives

$$
I = 2 \sum_{k=1}^{N} W_k \left[ z \cos z + \sin z \right]_{\left( k-1\right) \pi}^{k \pi}
$$

The $\sin z$ term vanishes, and the $z \cos z$ term gives a factor $\left( 2 k - 1 \right) \left( -1 \right)^k$, for a final result:

$$
I = 2 \sum_{k=1}^{N} \left( -1 \right)^k \left( 2 k - 1 \right) W_k
$$

Obviously the condition $I = 0$ has no solution for $N = 1$, but it can be solved for any $N > 1$.

 

[PeterDonis]

At the very least you've reminded me that my equations for the force are not correct. I need to rework those with the correct relativistic relationships

Actually, on looking back over these, I think they are correct. There is no force along the direction of the boost (the $z$ direction), so issues with how forces transform don't come into play.

 

[AVentura]

Is there a possible false assumption that force and acceleration are the same direction? I know nothing about this, I ask because this has been presented as a resolution to Trouton–Noble.

Also, in the first paper I linked the abstract claims a "suitable treatment of angular momentum and simultaneity" as a resolution. Sounds awfully related.

 

[PeterDonis]

Is there a possible false assumption that force and acceleration are the same direction?

If a component of the force is parallel to the boost direction (which is the $z$ direction in this case), then yes, the force and acceleration will not be parallel except in one particular frame. In the Trouton-Noble case (also called the right angle lever paradox), one of the forces is parallel to the boost direction, so the resolution of the paradox does require properly taking into account that the force and the acceleration are not parallel in all frames.

But that isn't the case in this problem--the force is entirely in the $x$ and $y$ directions, so it remains parallel to the proper acceleration in any frame which is boosted only in the $z$ direction. So unfortunately I don't think the resolution of Trouton-Noble will help here. It was a good thought though.

 

[PeterDonis]

I realized on reading back over the thread that I never actually computed the kinematic decomposition for either the cylinder or the helix. So I'm going to do that now. As we'll see, this only adds to the confusion.

The key thing we want is the tensor $\theta_{ab} + \omega_{ab} = A_a U_b + U_{a, b}$. (Normally this tensor is split up into its trace--the expansion scalar, its symmetric traceless part--the shear tensor, and its antisymmetric part--the vorticity tensor. We'll look at that very briefly once we have computed it.) Note the lower indexes; lowering the index is easy in Cartesian coordinates, since it just flips the sign of the $0$ component (I'm using the $-+++$ signature convention, which is easier to work with).

For the cylinder, we have

$$
U_a = \left( - \gamma, 0, - \gamma \omega R \sin ( \omega t + \phi ), \gamma \omega R \cos ( \omega t + \phi ) \right)
$$
$$
A_a = \left( 0, 0, - \gamma^2 \omega^2 R \cos ( \omega t + \phi ), - \gamma^2 \omega^2 R \sin ( \omega t + \phi ) \right)
$$

The nonzero components of the tensor are

$$
\theta_{20} + \omega_{20} = A_2 U_0 + U_{2, 0} = \left( \gamma^3 - \gamma \right) \omega^2 R \cos ( \omega t + \phi )
$$
$$
\theta_{30} + \omega_{30} = A_3 U_0 + U_{3, 0} = \left( \gamma^3 - \gamma \right) \omega^2 R \sin ( \omega t + \phi )
$$
$$
\theta_{22} + \omega_{22} = A_2 U_2 = \gamma^3 \omega^3 R^2 \cos ( \omega t + \phi ) \sin ( \omega t + \phi )
$$
$$
\theta_{33} + \omega_{33} = A_3 U_3 = - \gamma^3 \omega^3 R^2 \cos ( \omega t + \phi ) \sin ( \omega t + \phi )
$$
$$
\theta_{23} + \omega_{23} = A_2 U_3 = - \gamma^3 \omega^3 R^2 \cos^2 ( \omega t + \phi )
$$
$$
\theta_{32} + \omega_{32} = A_3 U_2 = \gamma^3 \omega^3 R^2 \sin^2 ( \omega t + \phi )
$$

I'm not going to try to interpret this in detail; I think it ends up showing zero expansion and shear (the zero expansion is evident from the fact that the $22$ and $33$ components cancel each other) and nonzero vorticity, as expected, but I'm not familiar enough with how to interpret this tensor to be able to read that off from the above. The only comment I have is that the idea is that this tensor at a given event should be orthogonal to the 4-velocity at that event, and the 4-velocity in this case has a $\theta$ component as well as a $t$ component, so the spacelike surface orthogonal to it is not just a surface of constant $t$.

The key thing is to compare the above with the tensor we get from setting $\phi = k z$ in the above formulas for $U$ and $A$. This means there are now some more nonzero components that weren't there before, while all of the components above are still there and still look the same:

$$
\theta_{21} + \omega_{21} = U_{2, 1} = - \gamma \omega k R \cos ( \omega t + k z )
$$
$$
\theta_{31} + \omega_{31} = U_{3, 1} = - \gamma \omega k R \sin ( \omega t + k z )
$$

What I'm wondering is if those extra components mean that the congruence describing the helix, unlike the one describing the cylinder, has nonzero shear, because of the "twist" of the helix (basically, that a given point on the helix does not have neighboring points in all directions, but only in two directions, and those directions are at an angle to the direction of motion due to rotation). I don't have an answer to this question, but I wanted to post the computations above (which also could stand some checking) to at least make it clear what questions I think we're dealing with.

 

[maline]

I'd like to elaborate on a point that I made a few days ago, without sufficient explanation:

we can easily extend this example to a fully three-dimensional one: just multiply this distribution by an arbitrary function of $r$, defined on an interval $R_1≤r≤R_2$, and by another arbitrary function of $\theta$ defined on $\theta_1\leq\theta \leq\theta_2$. We may as well choose the angular function to be constant, but choosing different radial functions may be useful.

The reason I am interested in making the body three-dimensional is that I still expect that working out the stress-energy tensor will turn out to be necessary for a resolution of our paradox. If so, our one-dimensional helix model will not suffice, because our helix must support shear stress (i.e. resist bending), and this is impossible for a truly one-dimensional body.

The reason shear stress is required (unlike, for instance, the case of a rotating ring) is that our body has free endpoints. The net force on the endpoints, like that on the rest of the body, must be centripetal, in the $-r$ direction, which is perpendicular to the helix curve. Along the interior of the helix this can be provided by tensile stress, because the change in the tangent direction along the curve is indeed in the $-r$ direction. Therefore, as with a rotating ring, taking the derivative of a tensile stress of constant magnitude gives a net centripetal force. At the endpoints, however, the tensile stress must go to zero- otherwise we have a jump discontinuity in the stress which implies a delta-function of force, while the mass distribution remains finite. Even if we added point masses to absorb that force, it would cause them to accelerate in the direction tangent to the helix, ruining our rigidity. It seems clear that we should indeed let the tension go to zero at the ends, and that the centripetal force will be provided, at least near the ends, by a shear stress. In other words, a rotating ring can be replaced by a flexible wire and still retain its shape, but a rotating helix cannot- it will "unwind" under the centrifugal force.

The reason you can't have shear stress in a 1D body is that the stress tensor is symmetrical. If, for instance, the $T^{yx}$ term is nonzero then so is the $T^{xy}$ term. But in a one-dimensional body, lying along the $x$ axis, we can't have $x$-momentum flowing in the $y$ direction- there's nowhere for it to go! Trying to calculate the total $x$-force on any point, we would have to take a derivative in the $y$ direction. We would be stuck with the unphysical "derivative of the delta function".
The following is a bit of an aside, but here is an example of how shear stress works. Let the body lie along the $x$- axis at $-1\leq x\leq1$, and suppose two forces, each $+1\hat y$, are applied at $x=\pm 1$ while a force $-2\hat y$ is applied at the origin. It is clear that the total force and total torque are both zero, so assuming the body remains rigid, it will not move.

But momentum conservation is primarily a local law (in fact in GR, according to most formulations it's only a local law), and we can think of each force as a constant local "input" of momentum. The rod is "absorbing" $y$-momentum at the ends and must transport this momentum to the origin, where it is canceled by the negative force. This "flow of momentum" is (one way of looking at) stress, and the material must have strength to support it. Since the momentum being carried is not in the direction of the flow, it is shear stress rather than tensile stress. It shows up in the stress-energy tensor as a nonzero $T^{yx}$ term- a flow of $y$-momentum in the $x$ direction. In this example it will be $+1$ for negative $x$ and $-1$ for positive $x$. But it can't work in one dimension!

To make the shear stress possible, the body must have thickness in the $y$ direction, say $0\leq y\leq 0.1$. At the top and bottom surfaces, we will indeed have $T^{yx}=T^{xy}=0$ to avoid jump discontinuities in the $y$ direction causing infinite $x$-force. Within the thickness, T^{yx}=T^{xy} will be positive for negative $x$ and negative for positive $x$.

Now let's consider the local force balance. At every point except where the forces are applied, we must have zero net force, meaning $\frac \partial {\partial x} T^{xx} +\frac \partial {\partial y} T^{xy} =0$ and $\frac \partial {\partial x} T^{yx} +\frac \partial {\partial y} T^{yy} =0$. We do not need $T^{yx}$ to change in the $x$ direction except at $x=0$, so we can set $\frac \partial {\partial x} T^{yx} =\frac \partial {\partial y} T^{yy} =0$. But $T^{yx}$, and therefore also $T^{xy}$, does change in the $y$ direction. For negative $x$, $\frac \partial {\partial y} T^{xy}$ is positive in the lower part of the body and negative in the upper part, and the opposite for positive $x$. Therefore $\frac \partial {\partial x} T^{xx}$ must have the opposite sign at each point. $T^{xx}$ must vanish at the ends of the body unless outside forces are applied. So along the bottom edge, $T^{xx}$- the tensile stress in the $x$ direction, will become more and more negative going from the ends toward the center. This stress can reach values much higher than the $T^{yx}$ term that is actually handling the force!

Sure enough, our body- let's say a wooden shelf supporting a weight that is concentrated in the center- is most likely do deform by stretching in the horizontal direction, at the point on the bottom surface directly beneath the weight. There is also an equally large positive term along the top surface, but wood is more likely to fail under tension than under compression.
Note that the maximum tension is greater the longer the shelf is, and less the thicker the shelf is, because $x$, $\frac \partial {\partial y} T^{xy}$ is smaller. This corresponds to experience.

I also should point out that stress is usually expressed as a 3-tensor $\sigma^{ij}$, which is basically the same as the stress-related part of the spatial terms in $T^{\alpha \beta}$, but with opposite sign: positive for tension and negative for compression.