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

[PeterDonis]

We need to take partial derivatives by all the spatial coordinates, so $U$ needs to be defined in an open set and not just on a cylindrical shell.

This would resolve my concern about derivatives, yes. I had thought of this but have not tried to work it out in detail yet. After confirming the kinematics, we would also need to redo the dynamics, including the mass distribution, with everything being functions of the coordinates; that gets messy because there are delta functions involved. I might try it in parallel with the other subthread that is going on.

 

[TSny]

I have rewritten my notes for the analysis of the linear momentum of the three particle system. It only involves the shear stress in the rod and is not too hard to work out. Here is a link to the notes
https://www.dropbox.com/s/2lhrbipzvs2exyv/Analysis of Linear Momentum.docx?dl=0
There are likely to be "typo" errors, but hopefully there are no severe conceptual errors. Let me know of any.

If there is a better way to share notes on the forum let me know. I'm not tech savvy.

The angular momentum is messier and involves normal stress as well as shear stress. I can work on cleaning up my notes on this if anyone would like to see them.

 

[maline]

Thank you @TSny !
So the idea seems to be that as the shear stress in the crossbar transfers momentum along the direction of motion in the primed (translating) frame, it contains a density of "hidden momentum". Since in this frame the crossbar is twisted, the integral of the hidden momentum turns out to be perpendicular to the plane containing the masses, the strings, and the axis of rotation. It exactly cancels the momentum of the masses, which are moving perpendicularly to that plane in their revolution around the axis. Relativity is weird!

This basic idea probably works for the helix case as well, except that it will be harder to work out.

We are still left with a revolution around an axis not containing the center of mass. Even if there is no violation of momentum conservation, is it really legal for the CoM to just wander around? Intuitively, where is the lateral motion of the mass "coming from"? I would appreciate seeing your work on the angular momentum tensor. Perhaps that will shed some more light.

One technical point is that you modeled the shear stress as being uniform throughout the cross-section of the bar, which is not really possible- it must vanish at the edges in the $y$ direction. But apparently this detail turns out to be irrelevant, so more power to you for not getting tangled with it!.

 

[TSny]

So the idea seems to be that as the shear stress in the crossbar transfers momentum along the direction of motion in the primed (translating) frame, it contains a density of "hidden momentum". Since in this frame the crossbar is twisted, the integral of the hidden momentum turns out to be perpendicular to the plane containing the masses, the strings, and the axis of rotation. It exactly cancels the momentum of the masses, which are moving perpendicularly to that plane in their revolution around the axis. Relativity is weird!

Yes, if $\vec{g} \,’$ is the density of momentum due to the shear stress in the primed frame, then $\vec{g} \,’$ points perpendicular to the axis of the rod and twists in direction as you move along the rod with a sudden change in direction at the center. The red arrows in the picture below show $\vec{g} \,’$ for different cross sections of the rod at time $t’ = 0$. (I hope I got these directions right.) You can see that when you add all the red vectors you get a net momentum in the negative $z'$ direction, as you said. The twisting of directions is of course due to relativity of simultaneity and the sudden change in direction in the middle of the rod is due to the discontinuous change in sign of the shear stress there.

[Picture edited June 5, 2017, to make a correction in the direction of the arrows]

We are still left with a revolution around an axis not containing the center of mass. Even if there is no violation of momentum conservation, is it really legal for the CoM to just wander around? Intuitively, where is the lateral motion of the mass "coming from"? I would appreciate seeing your work on the angular momentum tensor. Perhaps that will shed some more light.

It appears that you can actually show with a calculation that the centroid of the entire system in the primed frame is located on the x-axis! So, it does not revolve around the axis. Somewhat amazingly, there is a contribution to the location of the centroid of the system from the normal stress component $t_{xx}$ in the rod that “balances out” the location of the centroid of the three particles.

One technical point is that you modeled the shear stress as being uniform throughout the cross-section of the bar, which is not really possible- it must vanish at the edges in the $y$ direction. But apparently this detail turns out to be irrelevant, so more power to you for not getting tangled with it!.

My knowledge of elastic stress is shallow and sketchy. So, I could very well be overlooking something here.

 

[maline]

It appears that you can actually show with a calculation that the centroid of the entire system in the primed frame is located on the x-axis! So, it does not revolve around the axis. Somewhat amazingly, there is a contribution to the location of the centroid of the system from the normal stress component txxtxxt_{xx} in the rod that “balances out” the location of the centroid of the three particles.

Indeed, it definitely does not make sense for the center of energy to "wander". Just as in the classical case, the motion of the center of energy in anyone Lorentz frame must indeed equal the total momentum in that frame, divided by the total energy. Here is a proof: the $i$ component of the center of mass at time $t_0$is given by $$\int_{t=t_0}\,dxdydz\, T^{tt} X^i$$
Take the partial derivative by $t$, and use local conservation of energy: $$ \frac \partial {\partial t} \left \{ \int_{t=t_0}\,dxdydz\, T^{tt} X^i \right \}=\int_{t=t_0}\,dxdydz\, {T^{tt}}_{,t} X^i=-\int_{t=t_0}\,dxdydz\, {T^{tj}}_{,j} X^i$$.

Now note that for any field $A^j$, we have $(A^jX^i)_{,j}={A^j}_{,j}X^i+A^j\delta^i_j$, so $$-\int_{t=t_0}\,dxdydz\, {T^{tj}}_{,j} X^i=-\int_{t=t_0}\,dxdydz\, {(T^{tj}X^i)}_{,j} +\int_{t=t_0}\,dxdydz\, T^{ti}$$
The first term is a 3-divergence, so using Gauss's theorem we can convert it to a boundary integral over a 2-surface surrounding our 3-volume at $t=t_0$. At long as our distribution in bounded in space, the boundary integral can be made to give zero. By symmetry of $T$, the second term is$$\int_{t=t_0}\,dxdydz\, T^{ti}=\int_{t=t_0}\,dxdydz\, T^{it}=P^i$$ which is what we wanted to show.

So if the only energy density in the system is at the point masses, they cannot move while the total momentum is zero. You are saying that there is an additional energy density somewhere in our massless, perfectly rigid crossbar? How so?

 

[TSny]

You are saying that there is an additional energy density somewhere in our massless, perfectly rigid crossbar? How so?

Yes. Consider the Lorentz transformation from the unprimed to primed frame: $T^{t't'} = \Lambda^{t'}_\mu \Lambda^{t'}_\nu T^{\mu \nu}$.

Using $T^{tx} = 0$ in the rest frame, you find $T^{t't'} = \gamma^2 (T^{tt} + T^{xx}v^2/c^4)$. For a "massless" rod, I think we can take $T^{tt} = 0$ in the rest frame of the rod. So, $T^{t't'} = \gamma^2 v^2/c^4 \, T^{xx}$.

So, the i ^th coordinate of the centroid in the primed frame will contain a contribution from the normal stress component $T^{x'x'}$ in the rod given by $\gamma^2 v^2/c^4 \int dx'dy'dz' X^{i'} T^{x'x'}$ divided by the total energy.

At time $t' = 0$, the masses are vertically above the rod. So, the integral of interest is $\gamma^2 v^2/c^4 \int dx'dy'dz' y' T^{x'x'}$. I find this to evaluate to a negative value that cancels the contribution of the three particles.

 

[TSny]

From post #61:

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

Shouldn't the $(-1)^n$ be $(-1)^{(n+1)}$? If so, then unfortunately the required point masses will not lie on the helix, but rather on the opposite side of the axis.

 

[maline]

I see. The part of the bar with maximal compression stress is directly opposite the middle mass, so in the primed frame, it is on the side opposite all the masses. The side facing the masses will be under tension, so that will give a negative energy density with a positive $y$ value that also helps our balance.

Of course, negative energy is unphysical, but that comes from the idealizations of masslessness and perfect rigidity, as Pervect mentioned back in post #114. In real materials, the various elastic moduli must be bounded above by the mass density (up to constants) so that the speed of sound should be less than 1. Then any stress produces deformation inversely proportional to the relevant modulus, and the deformation stores (positive) energy quadratically, so the lower the mass density, the higher the minimal deformation energy. The total of (mass + deformation energy + energy from boosting the stress) will be positive in all frames and at all points.

But now, just as I feared, we have a resolution that cannot work for the helix case! There, it seems quite unavoidable that the center of mass will be within the body, away from the axis of rotation. We do not need to assume infinite rigidity, so the energy distribution should be purely positive.The revolution should be impossible!

From post #61:
Shouldn't the $(-1)^n$ be $(-1)^{(n+1)}$? If so, then unfortunately the required point masses will not lie on the helix, but rather on the opposite side of the axis.

No, the sign is correct. I assume you are asking because in post #60, the final result was $-\cos n\pi$ which equals $(-1)^{(n+1)}$. However, that calculation was for the integral of the second term in $d \vec J = [\vec \omega r^2 - \vec r (\vec \omega \cdot \vec r)]\rho (\vec r) dV$, which contributes negatively to the angular momentum. In that post I only needed to check whether the result was zero, so I didn't clarify the sign of the actual contribution. Good eye though!
Anyway, if you want to actually try working with a helix, Peter's model in post #118 is better than mine from post #61. It doesn't require point masses, so you can give the helix a finite thickness and have all the stresses at least be continuous.

 

[PeterDonis]

I had thought of this but have not tried to work it out in detail yet

Here is the first installment of my working it out in detail. I'm not going to give all of the calculations behind all this; I'm mainly just going to state the results. (I might write up an Insights post on the general technique, since the kinematic decomposition is a valuable tool and I can't be the only one that could use more practice at it.)

We have three vector fields on Minkowski spacetime:

$$
X^a = \left( t, z, x, y \right)
$$
$$
U^a = \left( \gamma, 0, - \gamma \omega y, \gamma \omega x \right)
$$
$$
A^a = \left( 0, 0, - \gamma^2 \omega^2 x, - \gamma^2 \omega^2 y \right)
$$

where $\gamma = 1 / \sqrt{1 - \omega^2 x^2 - \omega^2 y^2}$. For help in computations to follow, we note that $\gamma_{, x} = \gamma^3 \omega^2 x$ and $\gamma_{, y} = \gamma^3 \omega^2 y$.

We now compute the tensor $D_{ab} = \theta_{ab} + \omega_{ab} = A_a U_b + U_{a, b}$. (Note, as before, the lower indexes; since we are working in Minkowski coordinates, that just means the signs of the $0$ components of all vectors are reversed.) This gives:

$$
D_{ab} = \begin{bmatrix}
0 & 0 & - \gamma^3 \omega^2 x & - \gamma^3 \omega^2 y \\
0 & 0 & 0 & 0 \\
\gamma^3 \omega^2 x & 0 & 0 & - \gamma^3 \omega \\
\gamma^3 \omega^2 y & 0 & \gamma^3 \omega & 0
\end{bmatrix}
$$

This is obviously antisymmetric, so we have zero expansion and shear and nonzero vorticity. So far so good.

Next we add a mass distribution, which we will first write simply as a function $\rho (t, z, x, y)$ which gives the mass density as a function of the coordinates. (We will leave out the functional dependence for simplicity.) We then have the 4-momentum density $P^a = \rho U^a$ and the 4-force $F^a = \rho A^a$. Note that these two expressions assume that $d\rho / d\tau = 0$, i.e., that along each worldline in the congruence, the mass density is constant; it only varies from one worldline to another. This assumption is valid for the scenario we are considering, but it's important to recognize that it is there.

We next compute the angular momentum density tensor $M^{ab} = \rho \left( X^a P^b - P^a X^b \right)$. This gives:

$$
M^{ab} = \rho \begin{bmatrix}
0 & - \gamma z & - \gamma \left( x + \omega t y \right) & - \gamma \left( y - \omega t x \right) \\
\gamma z & 0 & - \gamma \omega z y & \gamma \omega z x \\
\gamma \left( x + \omega t y \right) & \gamma \omega z y & 0 & \gamma \omega \left( x^2 + y^2 \right) \\
\gamma \left( y - \omega t x \right) & - \gamma \omega z x & - \gamma \omega \left( x^2 + y^2 \right) & 0
\end{bmatrix}
$$

Finally, we compute the torque density tensor $T^{ab} = \rho \left( X^a A^b - X^b A^a \right)$. This gives:

$$
T^{ab} = \rho \begin{bmatrix}
0 & 0 & \gamma^2 \omega^2 t x & \gamma^2 \omega^2 t x \\
0 & 0 & - \gamma^2 \omega^2 z x & - \gamma^2 \omega^2 z y \\
- \gamma^2 \omega^2 t x & - \gamma^2 \omega^2 z x & 0 & 0 \\
- \gamma^2 \omega^2 t y & - \gamma^2 \omega^2 z y & 0 & 0
\end{bmatrix}
$$

More to come in follow up posts.

 

[PeterDonis]

For this second installment, I'm going to do the same computations as before, but in a frame in which the center of mass is moving in the positive $z$ direction with velocity $v$. We then have the vector fields as follows:

$$
X'^a = \left( t', z', x', y' \right)
$$
$$
U'^a = \left( g \gamma, g \gamma v, - \gamma \omega y', \gamma \omega x' \right)
$$
$$
A'^a = \left( 0, 0, - \gamma^2 \omega^2 x', - \gamma^2 \omega^2 y' \right)
$$

where $g = 1 / \sqrt{1 - v^2}$. Note that $g$ is constant everywhere.

The tensors are as follows:

$$
D'_{ab} = \begin{bmatrix} \\
0 & 0 & - g \gamma^3 \omega^2 x' & - g \gamma^3 \omega^2 y' \\
0 & 0 & g \gamma^3 \omega^2 v x' & g \gamma^3 \omega^2 v y' \\
g \gamma^3 \omega^2 x' & - g \gamma^3 \omega^2 v x' & 0 & - \gamma^3 \omega \\
g \gamma^3 \omega^2 y' & - g \gamma^3 \omega^2 v y' & \gamma^3 \omega & 0
\end{bmatrix}
$$

This is still obviously antisymmetric, so we have zero expansion and shear and nonzero vorticity, as expected since a Lorentz transformation should not affect the symmetry properties of a tensor.

For the other two tensors, we write the mass distribution as $\rho' (t', z', x', y')$, indicating that we have to transform it into the new coordinates. We'll save details of that for future posts.

$$
M'^{ab} = \rho' \begin{bmatrix}
0 & - g \gamma \left( z' - v t' \right) & - g \gamma \left( x' + \omega t' y' \right) & - \gamma \left( y' - \omega t' x' \right) \\
g \gamma \left( z' - v t' \right) & 0 & - g \gamma \left( v x' + \omega z' y' \right) & g \gamma \left( - v y' + \omega z' x' \right) \\
g \gamma \left( x' + \omega t' y' \right) & g \gamma \left( v x' + \omega z' y' \right) & 0 & \gamma \omega \left( x'^2 + y'^2 \right) \\
g \gamma \left( y' - \omega t' x' \right) & - g \gamma \left( - v y' + \omega z' x' \right) & - \gamma \omega \left( x'^2 + y'^2 \right) & 0
\end{bmatrix}
$$

$$
T'^{ab} = \rho' \begin{bmatrix}
0 & 0 & g \gamma^2 \omega^2 t' x' & g \gamma^2 \omega^2 t' x' \\
0 & 0 & - g \gamma^2 \omega^2 z' x' & - g \gamma^2 \omega^2 z' y' \\
- g \gamma^2 \omega^2 t' x' & - g \gamma^2 \omega^2 z' x' & 0 & 0 \\
- g \gamma^2 \omega^2 t' y' & - g \gamma^2 \omega^2 z' y' & 0 & 0
\end{bmatrix}
$$

More to come.

 

[TSny]

The helix has $n$ complete turns, parametrized as above by $\vec r (\theta) = ( \cos \theta , \sin \theta, p \theta)$

$J_y =\lambda \int_{-n\pi}^{n\pi} \, d\theta\, y(\theta) \omega z(\theta) =(-1)^n 2\pi n \omega p \lambda$.

Please bear with me. If $y(\theta) = \sin \theta$ and $z(\theta) = p \theta$, then for the case $n = 1$ we have the integral $\int_{-\pi}^{\pi} \, \theta \sin \theta \, d\theta$ which is positive. But $(-1)^n = -1$, which is negative.

 

[PeterDonis]

Following on from my previous posts, I now want to try out some specific assumptions for the mass distribution $\rho$. I'll start with the cylinder. We assume that the cylinder is of radius $R$ and axial length $Z$, and has total mass $m$. The mass distribution will be:

$$
\rho = \frac{m}{2 \pi R Z} \delta \left( x^2 + y^2 - R^2 \right) \theta \left( \frac{Z}{2} + z \right) \theta \left( \frac{Z}{2} - z \right)
$$

where $\delta$ is the Dirac delta function and $\theta$ is the Heaviside step function. (We could also model the $z$ dependence by just limiting the integration over $z$ to the interval $- Z / 2 \le z \le Z / 2$ by hand.)

Let's look at how this behaves when we integrate the angular momentum and torque over the entire cylinder for the instant $t = 0$. We first observe that we can integrate separately over $z$ and over $x, y$, and that the $z$ dependence of $\rho$ is even; so any component that is odd in $z$ will integrate to zero. Next, we observe that when integrating over $x$ and $y$, the distribution given by $\rho$ is even in both variables, so any term that is odd in both variables (for example, $x$ alone with no $y$ factor, or vice versa) will integrate to zero. That gets rid of every term except the 2-3 term in the angular momentum, which integrates to

$$
M^{23}_{\text{total}} = \frac{m}{2 \pi R Z} \int dz \, \theta \left( \frac{Z}{2} + z \right) \theta \left( \frac{Z}{2} - z \right) \int dx dy \, \delta \left( x^2 + y^2 - R^2 \right) \gamma \omega \left( x^2 + y^2 \right) = m \gamma \omega R^2
$$

Now we look at how this goes in the primed frame. Here we have to transform the density $\rho$, which becomes

$$
\rho' = \frac{m g^2}{2 \pi R Z} \delta \left( x'^2 + y'^2 - R^2 \right) \theta \left( \frac{Z}{2g} + z' - v t' \right) \theta \left( \frac{Z}{2g} - z' + v t' \right)
$$

where $g = 1 / \sqrt{1 - v^2}$, as before. At the instant $t' = 0$, the only nonzero term is still the 2-3 term in the angular momentum, and it integrates, if I've gotten the factors of $g$ right, to $m g \gamma \omega R^2$.

The next installment will look at how this goes for the helix.

 

[PeterDonis]

For this installment, we'll look at a mass distribution for the helix and see how it goes. First we'll look at uniform mass distributions.

We will assume that a single turn of the helix has axial length $Z$ and that there are $N$ turns, and that the radius of the helix is $R$. The mass distribution I propose, in the center of mass frame, is:

$$
\rho = \frac{m}{N \sqrt{4 \pi^2 R^2 + Z^2}} \delta \left( x^2 + y^2 - R^2 \right) \delta \left( \tan^{-1} \frac{y}{x} - \omega t - \frac{2 \pi z}{Z} \right)
$$

where for simplicity I have left out the step functions in $z$ and just assumed that we limit the integration over $z$ appropriately. The easiest way to integrate this is to change variables from $x, y$ to $r, \theta$, which gives

$$
\rho = \frac{m}{N \sqrt{4 \pi^2 R^2 + Z^2}} \delta \left( r - R \right) \delta \left( \theta - \omega t - \frac{2 \pi z}{Z} \right)
$$

Now we can separate the integrals as before, and in fact we will end up with the same integrals that I did earlier (modulo some constant factors), with the same results. But let's try it now in the primed frame, picking just the right velocity $v$ to unwind the helix:

$$
\rho' = \frac{m g^2}{N \sqrt{4 \pi^2 g^2 R^2 + Z^2}} \delta \left( x'^2 + y'^2 - R^2 \right) \delta \left( \tan^{-1} \frac{y'}{x'} - \frac{\omega}{g} t' \right)
$$

Once again, it looks like we can change variables as above and separate the integrals so that things work out as before.

In the next installment I'll look at non-uniform mass distributions.

 

[maline]

Please bear with me. If $y(\theta) = \sin \theta$ and $z(\theta) = p \theta$, then for the case $n = 1$ we have the integral $\int_{-\pi}^{\pi} \, \theta \sin \theta \, d\theta$ which is positive. But $(−1)n=−1$, which is negative.

Again, $\lambda \omega p \theta \sin \theta$ is the $y$ component of the term $\rho (\vec \omega \cdot \vec r) \vec r$. But the angular momentum density is given by $d \vec J = \rho [r^2 \vec \omega - (\vec \omega \cdot \vec r)\vec r]dV$, so you get the opposite sign.

 

[maline]

Let's look at how this behaves when we integrate the angular momentum and torque over the entire cylinder for the instant t=0.

I don't think the local angular momentum tensor in the form you gave, $M^{\alpha \beta}=\rho (X^\alpha U^\beta-X^\beta U^\alpha)$, can be integrated over a hypersurface to give a tensor. To integrate over constant $t$, what you need is the $t$ components of the rank-3 angular momentum density tensor, $M^{\alpha \beta t}=X^\alpha T^{\beta t}-X^\beta T^{\alpha t}$. But the "kinetic" part of $T^{\alpha \beta}$ is given by $\rho U^\alpha U^\beta$, so the $t$ components $T^{\alpha t}=\rho U^\alpha U^t=\gamma \rho U^\alpha \ne p^\alpha$. The extra gamma factor gives additional weight to the parts that are moving faster in your frame (the frame where $t$ is constant on your hypersurface of integration).

By the way, i think it's important to spell out that $\rho$ here refers to the local rest mass density scalar, $\rho=T^{\alpha \beta}U_\alpha U_\beta$ rather than the coordinate dependent energy density $T^{tt}$, which I think is also sometimes denoted by $\rho$. The scalar is useful for defining local momentum and force densities as tensors, but the fact the relationship between $\rho$ and $T^{tt}$ varies with $U$ means that these tensors are not good for hypersurface integrals. I also am not sure how to write the conservation laws using such local tensors instead of the density tensors, where you take the 4-divergence using the additional index.

 

[PeterDonis]

To integrate over constant $t$, what you need is the $t$ components of the rank-3 angular momentum density tensor, $M^{\alpha \beta t}=X^\alpha T^{\beta t}-X^\beta T^{\alpha t}$. But the "kinetic" part of $T^{\alpha \beta}$s given by $\rho U^\alpha U^\beta$ so the $t$ components $T^{\alpha t}=\rho U^\alpha U^t=\gamma \rho U^\alpha \ne p^\alpha$.

You're assuming that the stress-energy tensor is that of a perfect fluid, since that's the one for which the "kinetic" part is $\rho U^a U^b$. I'm not sure that approach works if we use delta functions; we would have to model the cylinder and the helix as continuous substances occupying an open region of spacetime. I'm trying to avoid doing that if at all possible since it greatly complicates the math.

Heuristically, though, I think you're correct that there should be an extra factor of $\gamma$ in the 4-momentum $P^a$, since the individual pieces of matter whose worldlines are given by $U^a$ are not at rest in the frame we are using. That doesn't change which components are nonzero, since there are no derivatives in the angular momentum or torque tensors; it just adds an extra factor of $\gamma$ to those tensors.

i think it's important to spell out that $\rho$ here refers to the local rest mass density scalar, $\rho=T^{\alpha \beta}U_\alpha U_\beta$ rather than the coordinate dependent energy density $T^{tt}$

Yes, I think that's correct--and that's another way of seeing why the formulas should have a factor of $\gamma$ added, as above. If $\rho$ is the mass/energy density measured by an observer moving with the matter (in this case the rotating cylinder/helix), then the mass/energy density measured by an observer at rest in the frame we are using will be $\gamma \rho$.

the fact the relationship between $\rho$ and $T^{tt}$ varies with $U$ means that these tensors are not good for hypersurface integrals

You can still do the integrals, you just have to be careful to specify the integrands taking into account the effects discussed above.

I also am not sure how to write the conservation laws using such local tensors instead of the density tensors, where you take the 4-divergence using the additional index

I'm not sure what you mean. The fundamental conservation law is $T^{ab}{}_{; b} = 0$, which is local (valid at every event, in any frame), and involves a "density tensor" (the SET). Any other conservation law, using a tensor derived from $T^{ab}$, can be derived from this one. But conservation laws that are not local, i.e., that hold on some foliation of the spacetime into spacelike hypersurfaces, will involve integrals over those hypersurfaces, so you have to be careful to specify the integrands, as above.

The case we're discussing is actually pretty simple, since we are in flat spacetime and are using inertial coordinate charts, which means there are no connection coefficients and the local conservation law is just $T^{ab}{}_{, b} = 0$. We also have obvious foliations into spacelike hypersurfaces, namely those of constant coordinate time in whatever frame we choose. So the issues that arise in curved spacetimes or non-inertial charts don't arise here.

 

[PeterDonis]

I think you're correct that there should be an extra factor of γγ\gamma

Yes, I think that's correct

Actually, on looking back at the formulas I gave for $\rho$, I'm not sure the extra factor of $\gamma$ should be there, because I defined $\rho$ using the rest mass $m$ of the object as a whole. That rest mass already includes the internal kinetic energy due to the motion of parts of the object, as long as those motions average out to zero, i.e., as long as we are in the center of mass frame. So actually I was defining $\rho$ as $T^{tt}$ (actually, the mixed component $T^t{}_t$, strictly speaking, since that's the one that doesn't include any effects of the metric or the metric signature). The mass/energy density measured by an observer rotating with the internal parts, by my definition, would be $\rho / gamma$.

As I said before, none of this affects which components of the various tensors are nonzero, so it doesn't affect the key question of what is necessary, dynamically, to make the motion in question a free motion.

 

[TSny]

Again, $\lambda \omega p \theta \sin \theta$ is the $y$ component of the term $\rho (\vec \omega \cdot \vec r) \vec r$. But the angular momentum density is given by $d \vec J = \rho [r^2 \vec \omega - (\vec \omega \cdot \vec r)\vec r]dV$, so you get the opposite sign.

Ok, I've got the signs settled now and your final answer for $J_y$ in #61 looks good.
I will now move on to Peter's more general modeling.

 

[maline]

So actually I was defining $ρ$ as $T^{tt}$

Okay, glad we got that clarified. But note that with that definition, multiplying $\rho$ by a tensor such as $U$ does not give a tensor quantity.

The mass/energy density measured by an observer rotating with the internal parts, by my definition, would be $ρ/\gamma$.

Actually I think the ratio should be $\gamma^2$, one gamma factor for the length contraction and another for the kinetic energy. This is consistent with $T^{\alpha \beta}=\rho U^\alpha U^\beta$ for the case of dust (using the scalar $\rho$).

As I said before, none of this affects which components of the various tensors are nonzero, so it doesn't affect the key question of what is necessary, dynamically, to make the motion in question a free motion.

This is true so long as those integrals that give zero do so for each cylindrical shell separately. Until now our distributions have been confined to a single cylindrical shell, so these details were not important, but sooner or later we will have to move on to a 3D model with a stress tensor, and we probably will not be able to deal only with particular shells.

Ok, I've got the signs settled now and your final answer for $J_y$ in #61 looks good.
I will now move on to Peter's more general modeling.

Don't forget the elephant in the room: these analyses, both for the center of mass and for the angular momentum, are dealing only with the movement of the mass. But the helix, even in the CoM frame, is a moving body under stresses that are not perpendicular to the motion. That gives a whole other sector, of both energy and momentum, where has of yet we have no idea what happens. It may well be that after taking these into account, there is no possible mass and stress distribution that will allow a free rotation!

 

[PeterDonis]

note that with that definition, multiplying $\rho$ by a tensor such as $U$ does not give a tensor quantity

Yes, but we can fix that by getting the factors of $\gamma$ right. To put it another way, we can view $\rho$ as I wrote it as $T^{ab} (e_0)_a (e_0)_b$, where $(e_0)$ is the timelike basis vector of our coordinate chart; physically this would be the mass/energy density measured by an observer at rest in the chart (i.e., relative to the center of mass of the object). To correct it to $T^{ab} U_a U_b$, which is what we want if we are going to use $\rho U^a$ as the momentum density $P^a$, just means multiplying or dividing by appropriate factors of $\gamma$.

Actually I think the ratio should be γ2γ2\gamma^2, one gamma factor for the length contraction and another for the kinetic energy. This is consistent with $T^{\alpha \beta}=\rho U^\alpha U^\beta$ for the case of dust (using the scalar $\rho$).

Yes, I think you're right.

sooner or later we will have to move on to a 3D model with a stress tensor

I'm still hoping we can avoid that level of complication. But see below.

the helix, even in the CoM frame, is a moving body under stresses that are not perpendicular to the motion

This is the part I'm not sure how to capture. I have been hoping that we can somehow capture it using the distributions I gave, limited to the cylindrical shell. But it is not captured in any of the tensors I have computed so far; neither the kinematic decomposition, nor the angular momentum as I've computed it, nor the torque, depend on purely space-space components of the stress-energy tensor (pressure and shear stress), which is what you are talking about.

The only other possibility I can see is to move to the full 3-index angular momentum tensor, which uses the SET directly, and see if any of its components differ between the cylinder and the helix.

 

[maline]

Ok, I want to take a crack at the full stress tensor analysis. I think the cleanest way to go about this is to work in Born coordinates, that is, cylindrical polar coordinates that rotate with the body. In these coordinates the body is stationary. That helps us in several ways. The most important advantage I see is that the stress-related components of the SET take their "natural form" in this chart, which is that of the the purely spatial 3-tensor $-\sigma^{ij}$. In fact, the whole SET (assuming no distortion and no EM fields) is just $$T^{\alpha \beta} =\begin{pmatrix} \rho & 0 \\ 0 & -\sigma^{ij} \end{pmatrix}$$
So, here goes. The transformation from Born coordinates $(t,r,\phi,z)$ to rectangular $(t',x',y',z')$ is: $$ t'=t,~x'=r \cos(\phi+\omega t) ,~y'=r \sin(\phi+\omega t), ~z'=z$$
The partial derivatives are: $${J_\alpha}^\beta =\frac {\partial X'^{~~\beta}}{\partial X^\alpha}=\left( \begin{array}{cccc} 1 & -\omega r \sin(\phi+\omega t) & \omega r \cos(\phi+\omega t) & 0 \\ 0 & \cos(\phi+\omega t) & \sin(\phi+\omega t) & 0 \\ 0 & -r \sin(\phi+\omega t) & r \cos(\phi+\omega t) & 0 \\ 0 & 0 & 0 & 1 \end{array} \right) $$ Therefore the metric tensor is: $$g_{\alpha \beta}={J_\alpha}^\mu \,\eta'_{\mu \nu} \,{J_\beta}^\nu= \begin{pmatrix} -1+\omega^2 r^2 & 0 & \omega r^2 & 0 \\ 0 & 1 & 0 & 0 \\ \omega r^2 & 0 & r^2 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}$$ I plan to work on this a bit at a time. The next step is to write the formula for the covariant 4-divergence of a rank 2 contravariant tensor in these coordinates. We can apply this to the SET and get 4 equations (PDE's) we will need to solve.

If anyone else wants to continue where I leave off, please go right ahead.

 

[maline]

All right, so it seems the covariant 4-divergence of a rank 2 tensor does not have a simple form (using $\sqrt {-g}$, for instance) except where the tensor is antisymmetric, which the SET obviously is not. We will have to use the Christoffel symbols explicitly.

The nonzero partial derivatives of the metric are: $$g_{tt,r}=2r\omega^2~~~g_{t\phi,r}=g_{\phi t,r}=2r\omega~~~g_{\phi \phi ,r}=2r$$
We wish to calculate the Christoffel symbols, using the explicit formula $\Gamma^\alpha_{\beta \gamma}=\frac 1 2 g^{\alpha \mu}(g_{\mu \beta,\gamma}+g_{\gamma \mu,\beta}-g_{\beta \gamma, \mu})$. For this we also need the contravariant metric tensor. Matrix inversion gives: $$g^{\alpha \beta}=\begin{pmatrix} -1 & 0 & \omega & 0 \\ 0 & 1 & 0 & 0 \\ \omega & 0 & r^{-2}-\omega^2 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}$$
Now for the Christoffel symbols: $$\Gamma^r_{tt}=-\frac 1 2 g^{rr}g_{tt,r}=-r\omega^2~~~\Gamma^r_{\phi\phi}=-\frac 1 2 g^{rr}g_{\phi\phi,r}=-r~~~\Gamma^r_{t \phi}=\Gamma^r_{\phi t}=-\frac 1 2 g^{rr}g_{t \phi,r}=-r\omega \\ \Gamma^t_{tr}=\Gamma^t_{rt}=\frac 1 2 g^{tt}g_{tt,r}+\frac 1 2 g^{t \phi}g_{\phi t,r}=-r\omega^2+r\omega^2=0 \\\Gamma^t_{\phi r}=\Gamma^t_{r \phi}=\frac 1 2 g^{tt}g_{t \phi,r}+\frac 1 2 g^{t \phi}g_{\phi \phi,r}=-r\omega+r\omega=0 \\ \Gamma^\phi_{tr}=\Gamma^\phi_{rt}=\frac 1 2 g^{\phi t}g_{tt,r}+\frac 1 2 g^{\phi \phi}g_{t \phi,r}=r\omega^3+\frac \omega r-r\omega^3=\frac \omega r\\ \Gamma^\phi_{\phi r}=\Gamma^\phi_{r \phi}=\frac 1 2 g^{\phi t}g_{t \phi,r}=\frac 1 2 g^{\phi \phi}g_{\phi \phi,r}=r\omega^2+\frac 1 r -r \omega^2=\frac 1 r$$
All the others are clearly zero because all the metric derivatives involved are zero.

Now we can work out the covariant divergence equations, representing local energy/momentum conservation. We assume all derivatives by $t$ are zero, and $T^{ti}=T^{it}=0$ everywhere for each spatial index $i$. For the "energy" equation, we have $${T^{t \alpha}}_{;\alpha}={T^{t \alpha}}_{,\alpha}+\Gamma^t_{\alpha \mu}T^{\mu \alpha}+\Gamma^\alpha_{\alpha \mu}T^{t \mu}=0$$All of these terms vanish independently, which is intuitive. Our body is "stationary" in these coordinates and no "energy" is "flowing" anywhere.

For the "$z$ momentum", we get $${T^{z \alpha}}_{;\alpha}={T^{z \alpha}}_{,\alpha}+\Gamma^z_{\alpha \mu}T^{\mu \alpha}+\Gamma^\alpha_{\alpha \mu}T^{z \mu}={T^{zi}}_{,i}+\Gamma^\phi_{\phi r}T^{zr}=\\={T^{zi}}_{,i}+\frac 1 r T^{zr}=0$$This is the same as with a truly stationary body in ordinary polar coordinates: the "flow" in the $r$ direction has a change in "density" as the $\hat r$ vectors spread apart.

For the "$\phi$ momentum": $$ {T^{\phi \alpha}}_{;\alpha}={T^{\phi \alpha}}_{,\alpha}+\Gamma^\phi_{\alpha \mu}T^{\mu \alpha}+\Gamma^\alpha_{\alpha \mu}T^{\phi \mu}=\\={T^{\phi i}}_{,i}+\Gamma^\phi_{\phi r}T^{r \phi}+\Gamma^\phi_{r \phi}T^{\phi r}+\Gamma^\phi_{\phi r}T^{\phi r}=\\= {T^{\phi i}}_{,i}+\frac 3 r T^{\phi r}=0$$The additional two $T^{\phi r}=T^{r \phi}$ terms are because "$r$ momentum" needs to shift its direction as it "flows" along the curve in the $\phi$ direction, and because the magnitude of a unit of "$\phi$ momentum" changes with $r$.

For the "$r$ momentum": $$ {T^{r \alpha}}_{;\alpha}={T^{r \alpha}}_{,\alpha}+\Gamma^r_{\alpha \mu}T^{\mu \alpha}+\Gamma^\alpha_{\alpha \mu}T^{r \mu}=\\={T^{r i}}_{,i}+\Gamma^r_{tt}T^{tt}+\Gamma^r_{\phi \phi}T^{\phi \phi}+\Gamma^\phi_{\phi r}T^{r r}=\\={T^{r i}}_{,i}-r\omega^2 \rho-r T^{\phi \phi}+\frac 1 r T^{rr}=0$$The $r\omega^2 \rho$ term is, of course, the centrifugal force- the source of all our stress. It is the only place that the mass distribution enters the equations. The T^{\phi \phi} term is because of the "$\phi$ momentum" changing direction as it flows, and it expresses the fact that the body, in resisting centrifugal expansion, will be under tension in the $\phi$ direction. In the case of a rotating ring, for instance, this tension is the only stress needed to maintain rigidity. That's why I claimed a few posts ago that the rotating ring can just as well consist of a flexible rope and still be a "rigid body".

We have arrived at a very surprising result: the form of these equations is exactly the same whether the speed $\omega r$ is tiny, or highly relativistic! No $\gamma$ factors appear. In fact $\omega$ itself does not appear except in the centrifugal force term. This implies that any solution for the stresses in a slow rotation- a purely classical problem- can simply be multiplied by a constant $(\omega_2^2/\omega_1^2)$ to be valid for any value of $\omega$!

In particular, Peter's solution for a freely rotating helix, in post #118, should certainly work for slow rotations (if it is modified to make the helix 3D, which is simple). Whatever the stresses are in that case, they satisfy local energy-momentum conservation. So we can always transfer that solution to Born coordinates, multiply by the appropriate factor, and get a fully relativistic solution. If it satisfies the local conservation laws then it will automatically be a free motion as well. So freely rotating relativistic helices definitely can be described! We are still stuck with our paradox!

 

[TSny]

I’m still working on this. I agree that there are probably many possible choices for variable mass densities that would produce freely rotating helices. But the nature of the stresses in the helix are causing me problems. There will be compression, tension, shear, and bending moments that vary in a complicated way along the helix.

I did run across some notes that I made years ago on a helical system with simple stress, but with the disadvantage of requiring external forces at the ends of the helix.

https://www.dropbox.com/s/20kfa57au...ting Helix Paradox edited 5_30_2017.docx?dl=0

The helix consists of a flexible cable of uniform mass. The only stress is tension in the cable. As the helix rotates, it maintains its shape with just the tension and the external forces at the ends. The model assumes the rotation speed of the helix is nonrelativistic.

In the frame of reference where the helix is rotating but not translating along its axis, the forces on the ends have magnitudes equal to the tension and tangent to the helix. In the frame in which the helix is unwound into a straight line, the forces on the end are equal in magnitude, opposite in direction, but are not parallel to the cable. The result is that the net force on the cable is zero but there is a nonzero torque (couple). The paradox is double: (1) How can the cable rotate around the coordinate axis if there is zero net external force? (2) How does the cable maintain its orientation parallel to the coordinate axis despite the external torque which appears to be in a direction that would change the orientation?

The notes show, I hope, how the contribution of the tension to the momentum and angular momentum in the unwound coordinate system resolves the paradox. Even though the cable revolves around the coordinate axis, there is no linear momentum associated with this rotatory motion. The system only has a constant linear momentum parallel to the cable. So, no external force is required. However, in general there is a changing angular momentum which is shown to equal the applied torque. So, it appears to all work out.

The obvious shortcoming of this model is the need for the external forces at the ends of the helix. Also, in the notes, I assume that the rotational speed of the helix is nonrelativistic to simplify the calculations.

I don’t know if anyone will want to wade through the details of the notes. I tried to keep it very elementary. But the calculations are inelegant and tedious. I used several different coordinate systems, including some that are comoving with an element of the helix. Sorry for not including any diagrams in the notes.

 

[pervect]

I’m still working on this. I agree that there are probably many possible choices for variable mass densities that would produce freely rotating helices. But the nature of the stresses in the helix are causing me problems. There will be compression, tension, shear, and bending moments that vary in a complicated way along the helix.

This thread has gotten too long for me to follow, but I thought your simplified version of the helix which was a loaded beam was a promising approach and simpler than the other approaches I'd seen. Unfortunately, I'm not familiar with the stresses in a loaded beam in any detail, so I wasn't able to proceed further.

This simple straight loaded beam will be twisted into a helix when one does the appropriate Lorentz boost, so I assume that some of the results for helices (which I haven't had the time or inclination to look at in detail) will still be applicable using the simpler "loaded beam" variant of the problem. I believe the beam must have a finite width to support the loads with finite stresses. I believe that an analysis based on the stress-energy tensor will show that sections of the beam under tension (in the direction of the boost) will have a negative energy density in the boosted frame, and sections of the beam under compression (in the direction of the boost) will have a positive energy density. I don't think the stresses in the directions transverse to the boost should matter. Because the beam will no longer be massless, in the boosted frame, the beam's contribution to the angular momentum must be evaluated, and I expect it will explain the "paradox".

 

[PeterDonis]

I believe that an analysis based on the stress-energy tensor will show that sections of the beam under tension (in the direction of the boost) will have a negative energy density in the boosted frame

This is not correct if the beams are made of ordinary matter, since ordinary matter satisfies all of the energy conditions, one of which is that the energy density is positive in any frame. Another is that the stress is less than or equal to the energy density in any frame.

Assuming a "massless" beam should not change the above, because "massless" does not mean "zero energy density in the rest frame", and objects made of massless particles still obey the energy conditions. "Massless", in terms of the stress-energy tensor, means that the SET is something like that of the electromagnetic field, as given, for example, here:

https://en.wikipedia.org/wiki/Electromagnetic_stress–energy_tensor

Note that the key property due to masslessness is that the SET is traceless, not that the energy density is zero.

In order to violate the energy conditions and have negative energy density in some frame, the beam would have to be made of exotic matter, but I don't think that was the intent of the model, and I would not recommend making such an assumption anyway.

 

[PeterDonis]

"massless" does not mean "zero energy density in the rest frame",

There might be a terminology issue here; what I said in the above quote is the standard definition of "massless", but it might not be the one being used in this thread.

However, the more important point is that, if the beam has zero energy density but nonzero stress in any frame, then it violates the energy conditions, and I don't think we should be basing any possible resolution of the paradox in this thread on a model that violates the energy conditions. We want, if possible, to explain how a helix made of ordinary matter either can or can't rotate freely about its axis. There should be some way to resolve that without having to bring in exotic matter.

 

[TSny]

We want, if possible, to explain how a helix made of ordinary matter either can or can't rotate freely about its axis. There should be some way to resolve that without having to bring in exotic matter.

Doesn't your sectional model show that this is possible?

I think the only conditions that must be met for a helix to be able to rotate freely at nonrelativistic speed is

$\int{\rho(\theta) \cos(\theta) d \theta = 0}$ and $\int{\rho(\theta) \theta \sin(\theta) d \theta = 0}$

These conditions were mentioned earlier in the thread.

There are many ways to choose the variable mass density $\rho(\theta)$ to satisfy these conditions.

It’s OK to consider only nonrelativistic speeds since we can still unwrap such a helix with a boost, as long as the pitch of the helix is great enough.

 

[maline]

Note that the key property due to masslessness is that the SET is traceless, not that the energy density is zero.

But if the body is perfectly rigid, the only way stress (other than in the direction of motion) will contribute to the energy density is if you put it in by hand, to enforce the energy condition. With real matter, stress causes strain (deformation) which stores energy and automatically enforces the condition, but we don't want to work with that, and anyway it won't work for a massless body because the elastic modulus needs to be less than the rest mass density.

 

[PeterDonis]

Doesn't your sectional model show that this is possible?

As far as I can tell, it does, but we still haven't resolved the apparent paradox that in the frame that "unrolls" the helix, we have a straight rod apparently rotating about an axis off of the rod.

It’s OK to consider only nonrelativistic speeds since we can still unwrap such a helix with a boost, as long as the pitch of the helix is great enough.

This doesn't resolve the apparent paradox, it just makes the math somewhat easier since some approximations can be made.

 

[PeterDonis]

if the body is perfectly rigid

Which it can't be in relativity.

the only way stress (other than in the direction of motion) will contribute to the energy density

Stress doesn't contribute to the energy density. They are different components of the stress-energy tensor.

With real matter, stress causes strain (deformation) which stores energy and automatically enforces the condition, but we don't want to work with that

Why not?

it won't work for a massless body

Yes, and I'm saying that IMO that means we should not use a "massless body" (by which you appear to mean a body with zero energy density in its rest frame) in our model. Anyway, a real helix won't have a massless rod and strings. We ought to be able to resolve the apparent paradox we've been discussing without putting things into the model that aren't there in a real helix.

 

[TSny]

This simple straight loaded beam will be twisted into a helix when one does the appropriate Lorentz boost, so I assume that some of the results for helices (which I haven't had the time or inclination to look at in detail) will still be applicable using the simpler "loaded beam" variant of the problem.

I'm not following you here. Maybe you're thinking of a different "loaded beam" scenario. In the three-particle model of post 147, the beam lies along the x-axis (boost axis). The beam cannot be twisted into a helix by a boost along the axis.

I believe the beam must have a finite width to support the loads with finite stresses.

Yes, I agree.

I believe that an analysis based on the stress-energy tensor will show that sections of the beam under tension (in the direction of the boost) will have a negative energy density in the boosted frame, and sections of the beam under compression (in the direction of the boost) will have a positive energy density.

Yes, I think that's true. As Peter has pointed out, this would violate a postulate that the energy density must be positive in all frames of reference. In the case of the three-particle model, we can avoid the problem by simply allowing the rod to have mass. I initially chose a massless rod in order to make the "paradox" more dramatic. But we can add mass to the rod and the motion of the system will still appear paradoxical in the boosted frame in which the three masses are all lined up on the same side of the rod.

I don't think the stresses in the directions transverse to the boost should matter.

If the boost is along the x direction, then the stresses T_yy and T_zz in the rest frame will not matter. But the shear stresses, such as T_zx are important in resolving the paradox.

Because the beam will no longer be massless, in the boosted frame, the beam's contribution to the angular momentum must be evaluated, and I expect it will explain the "paradox".

In the three-particle system, the rod is along the x-axis. So, at first, you would think that it cannot contribute to angular momentum of the system. However, calculations show that the shear stress in the rod actually does contribute to the angular momentum of the system in the frame in which the particles are lined up. These contributions combine with the angular momenta of the particles to produce a total angular momentum of the system that has only an x-component and remains constant in time. This is consistent with zero torque acting on the system.

 

[TSny]

... ordinary matter satisfies all of the energy conditions, one of which is that the energy density is positive in any frame. Another is that the stress is less than or equal to the energy density in any frame.

Yes, this is a key point that I'm just now appreciating. For the flexible cable (or string) model of post #173, it is easy to check that the tension in the helical string must violate the stress condition if the helix can be unwound by a boost. In the notes that I link to in that post, equations (12) and (14) combine to show that the tension $\tau_0$ is given by

$\tau_0 = \lambda_0 \frac{1+u^2 v^2}{v^2} > \lambda_0 \,\,\,\,$ (in units where $c = 1$).

Here, $\lambda_0$ is the linear mass density of the string in the rest frame, $u$ is the tangential speed of rotation of an element of the string, and $v$ is the boost speed needed to unwind the helix.

The tension stress $\sigma$ is given by the tension divided by the cross-sectional area of the string. So, we get $\sigma > \rho _0$ where $\rho_0$ is the volume mass density.

I'm very inclined to think that the stress condition is going to be violated in any freely rotating helix that can be unwound by a boost.

 

[PeterDonis]

I'm very inclined to think that the stress condition is going to be violated in any rotating helix that can be unwound by a boost.

I think this is a possible resolution of the apparent paradox, yes.

 

[maline]

I'm very inclined to think that the stress condition is going to be violated in any freely rotating helix that can be unwound by a boost.

I think this is a possible resolution of the apparent paradox, yes.

Bravo! @AVentura, it looks like the answer to your paradox has been found!

I want to point out that violation of the weak energy condition can be thought of as an answer in two different ways. From the physical perspective, if it's true that a helix that is rotating fast enough must violate the energy conditions, then the "paradoxical" situation is just impossible. What will happen if you try to get the helix to rotate is that as the tension of fighting the centrifugal force builds up, the material will start to deform, generating elastic energy, which in turn contributes to the rest mass and the centrifugal force, requiring even more tension. No material will be able to supply the necessary tension, and the body will be destroyed.

From a more mathematical perspective, if we consider solutions that violate the energy conditions as valid (say using exotic matter), we get the result that the energy density will be negative in some parts of the body in some frames. In particular, if we can show that it is negative in the outer edge of the unwound helix in the boosted frame, that can answer the paradox as follows: The problem with a straight rod revolving around an axis parallel to itself is that the center of rotation, which must be the center of mass, is outside the (convex hull of) the body. But the fact that the center of a mass distribution must lie within the distribution depends critically on mass being nonnegative! So in "exotic matter" situations, we can indeed have a center of mass situated well outside the body, and revolve around it with zero linear momentum.

 

[AVentura]

What will happen if you try to get the helix to rotate is that as the tension of fighting the centrifugal force builds up, the material will start to deform, generating elastic energy, which in turn contributes to the rest mass and the centrifugal force, requiring even more tension. No material will be able to supply the necessary tension, and the body will be destroyed.

Being a closed system, the increase in invariant mass could only come from the kinetic energy, right? That's possible if the radius increases, like a spinning figure skater opening her arms. A spinning ring could prevent this (right?), but why not a helix?

Since we are verbalizing theories here's mine: after setting any helix in motion elastic energy will move, putting some energy in a new place along the helix. This unbalances it. Any attempt to add mass to balance it back exacerbates the situation.

Stress doesn't contribute to the energy density. They are different components of the stress-energy tensor.

So elastic energy is different than stress? Sorry for the elementary question. I just know a compressed spring has more invariant mass than an otherwise relaxed one.

 

[PeterDonis]

the increase in invariant mass could only come from the kinetic energy, right?

Not necessarily, no. The bonds between neighboring atoms basically act like springs--putting them under tension increases the stored energy, even if the atoms aren't moving relative to each other (after some small adjustment due to the tension).

So elastic energy is different than stress?

Yes. The stress is a space-space component of the stress-energy tensor. But a body under stress also has a greater energy density, i.e., "elastic energy", so the time-time component of the stress-energy tensor also increases.

 

[TSny]

Here’s an attempt to show, or at least suggest, that the stress at certain points in the helix must be of order $\rho c^2$.

Consider the last half turn of the rotating helix and look at the angular momentum of this section. We choose a coordinate system such that this section is described at the present moment by $(x, y, z) = \left( a \theta, R \cos \theta, R \sin \theta \right)$, $0 < \theta < \pi$. So, the origin of this system is not the same as the origin that was used for the entire helix. Our new $y$ axis passes through one end of this last half turn of the helix. Assume the helix is rotating in this coordinate system at non-relativistic speed with angular speed $\omega$. This section of the helix has a linear mass density $\lambda$ which we take to be constant for this section.

The $z$ component of angular momentum of the section relative to the point (x, y, z) = (0, 0, 0) at this moment is found to be

$L_z = -\lambda \pi \sqrt{1+\frac{R^2}{a^2}} a^2 u \,\,\,\,$ where $u = \omega R$

The rate of change of this component of angular momentum is due to the rotation of the helix and has a magnitude given by

$|\frac{dL_z}{dt}| = \omega |L_z| =\lambda \pi \sqrt{1+\frac{R^2} {a^2}} a^2 u^2 / R$

The direction of $\frac{dL_z}{dt}$ is the positive $y$ direction. So, there must be a positive $y$ component of torque acting on this section of the helix which has a magnitude

$\tau_y =\lambda \pi \sqrt{1+\frac{R^2}{a^2}} a^2 u^2 /R$

For it to be possible to unwind the helix with a boost, we must have $a = \frac{c^2 R}{uv}$, where $v$ is the boost speed. So,

$\tau_y = \left(\lambda c^2 \right) \pi \sqrt{1+\frac{R^2}{a^2}} \frac{c^2}{v^2} R$

The forces acting on this section of the helix are due to the stresses at the end of the section that is connected to the rest of the helix. It is not hard to see that the only form of stress that can produce torque in the y-direction relative to the origin $(0, 0, 0)$ is a “bending moment”. This is because the end of the section where the stresses are acting is located on the $y$ axis. (I can try to add more explanation of this point, if necessary.)

The bending moment is given by $M_B = \int{T_{\tilde{x} \tilde{x}}\tilde {z} d\tilde{y} d\tilde{z}}$. The tildes refer to a coordinate system with origin at the center of the cross section of the rod with the $\tilde{x}$ axis perpendicular to the cross section of the rod and the $\tilde{y}$ axis parallel to the $y$ axis. The integration is over the cross section of the rod at $\theta = 0$. In order of magnitude, we can write this as $M_B \approx \left<|T_{\tilde{x} \tilde{x}}| \right> r A$ where $r$ is the radius of the rod, $A$ is the cross sectional area of the rod, and $\left< |T_{\tilde{x} \tilde{x}}| \right>$ represents an average of the absolute value of $T_{\tilde{x} \tilde{x}}$ over the cross section.

Thus, we must have $\left<|T_{\tilde{x} \tilde{x}}| \right> r A \approx \left(\lambda c^2 \right) \pi \sqrt{1+\frac{R^2}{a^2}} \frac{c^2}{v^2} R$.

Then, $\left< |T_{\tilde{x} \tilde{x}}| \right> \approx \left(\rho c^2 \right) \pi \sqrt{1+\frac{R^2}{a^2}} \frac{c^2}{v^2} \frac{R}{r}$, where $\rho = \lambda/A$ is the volume mass density of the section.

Since $r < R$ and $v < c$ we find

$\left<|T_{\tilde{x} \tilde{x}}| \right> \sim \left(\rho c^2 \right)$

 

[maline]

So elastic energy is different than stress? Sorry for the elementary question. I just know a compressed spring has more invariant mass than an otherwise relaxed one.

As you say, a deformed body has more invariant mass. From a microscopic perspective, this is because the atoms are moved away from their equilibrium positions, forcing the electrons into wavefunctions that do not do as well at minimizing the electromagnetic energy. So you can say that elastic energy is really EM field energy.

Stress is the force derived from the gradients of this energy. As long as the deformations are small, the energy is approximately quadratic in the deformation, so the stress is linear in the deformation (strain). This is "Hooke's law" familiar from discussions of springs. In more detail, typical materials have three elastic moduli, relating the stress linearly to the tensile, shear, and bulk strains respectively. For simplicity, let's think of the strain as one dimensional (stretching or compressing in the $x$ direction with no change in the perpendicular directions). Then $$\frac {\Delta V} V = \frac {A\Delta L_x} {AL_x}= (1/K) \sigma_{xx}$$ where $\Delta V$ and $\Delta L_x$ are the changes in volume and in length in the $x$ direction, respectively, $A$ is the cross sectional area in the $y,z$ directions, $K$ in the bulk modulus, and $\sigma_{xx}$ is the tension or stress resisting further deformation.

For idealized "perfectly rigid" materials, $K\rightarrow\infty$, so there can be unlimited stress with no (or infinitesimal) strain, and therefore no elastic energy. But when $K$ is finite, the elastic energy is given by $$\Delta E= \int_{L_x}^{L_x+\Delta L_x}\,dl\,A\sigma_{xx}=A\int_{L_x}^{L_x+\Delta L_x}\,dl\,K\frac{l-L_x} {L_x}=\frac{AK}{2L_x}(\Delta L_x)^2=\frac {KV} 2 {\left(\frac {\Delta L_x}{L_x}\right)}^2=\\=\frac {KV} 2 {\left(\frac {\sigma_{xx}}K\right)}^2=\frac V {2K} \sigma_{xx}^2$$ Thus the added energy density is proportional to the stress squared, and inversely proportional to the bulk modulus.

However, the speed of sound in a solid is given by $\sqrt {\frac K \rho}$, where $\rho$ is the rest mass density. In relativity this must be less than $c$, so we must have $K< \rho c^2=E/V$. Therefore $$2 \frac E V \frac{\Delta E} V =\frac{\Delta(E^2)}{V^2}> \sigma_{xx}^2$$ and in particular, the total energy density will always remain more than the stress. This is how the weak energy condition is "enforced".

 

[maline]

However, the speed of sound in a solid is given by $\sqrt {\frac K \rho}$, where $ρ$ is the rest mass density. In relativity this must be less than $c$, so we must have $K< \rho c^2=E/V$.

I should add that this bound is purely theoretical. Real materials come nowhere close to this value. For instance, diamond has a very high value of $K$, given as $443~ [GPa]$. But $\rho c^2$ would be something like $3.17 \times 10^{11} [GPa]$.

 

[AVentura]

I never dreamed the answer would be this complicated. Even though the gist of the answer seems simple, I think. Thank you for the explanation @maline. Thank you @TSny, @PeterDonis. Thank you everyone.
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[PeterDonis]

I never dreamed the answer would be this complicated.

I think that's a sign that it was a good question.

 

[AVentura]

Thank you Peter.

Side note about this "relativistic twisting", I came across this while thinking about how it transforms wave functions. If you treat the real and imaginary components of the wave function as space, it ensures all observers see a wavelength that corresponds to the momentum that they would observe. At least for a non-localized free particle. The wave function is just a helix. Is this surprising? If not, is it already theorized that wave functions live on the Kaluza-Klein "cylinder"?

 

[PeterDonis]

The wave function is just a helix

A helix if you equate the axis of the helix with a single space dimension (note that this is based on an idealized "particle" that can only move in one space dimension), and the two dimensions transverse to the helix with the complex plane, yes. But that's very different from an actual helix in actual 3-dimensional space; the "space" in question is the abstract Hilbert space of complex functions of a single real variable.

is it already theorized that wave functions live on the Kaluza-Klein "cylinder"?

Not that I'm aware of; I don't know of any useful analogy between Kaluza-Klein type geometric spaces and the Hilbert spaces used in quantum mechanics.