Stress-energy tensor of a wire under stress

[pervect]

OK, it's been awhile, but I think I'm getting the same answer.

If v_c is the speed of sound in the unstretched rod, the maximum value of the stretch factor s where the speed of sound becomes equal to c is

$${\frac {\sqrt {6+3\,{{\it v_c}}^{2}}}{{3 \it v_c}}}$$

For a Newtonian hoop, the speed of sound would increase simply by the stretch factor s - the Lagrangian is the same, but one has to multiply the wave propagation speed for phi by the stretch factor s to get the actual physical wave speed through the medium.

For the relativistic hoop, we'd have something like:

eta(t,phi) = s0*phi + f(phi - beta*t)

The Lagrangian density of the hoop is
$$\mathcal{L} = -\rho(s) \, s \, \sqrt{1 - \eta_0^2} d \phi$$

where
$$s = \frac{\eta_1}{\sqrt{1-\eta_0^2}}$$

which sensibly reduces to $\eta_1$ in the non-relativistic limit, and

$$\rho(s) = \frac{2 + v_c^2 \, (s-1)^2}{2 \, s}$$

Approximating the Lagrangian as a quadratic gives:

Lapprox = (-1/2*v_c^2*eta_1^2+1+1/2*v_c^2) eta_0^2 - v_c^2 eta_1^2

[add]Question - Can I really justify this approximation, though - eta_0 may not be small.We can divide this by the coefficient of eta_0^2 to find the effective velocity v_y^2, and set v_y * eta_1 = 1 to solve for the maximum stretch factor eta_1.

For v_c^2 = 1/2, this puts the maximum stretch factor at around 1.225, well before the energy peak.

Plugging the critical value of s into the equation for dr/ds and dE/ds seems to indicate that both the radius and the energy are increasing when s reaches its maximum allowable value.

 

[gregegan]

OK, it's been awhile, but I think I'm getting the same answer.

We seem to agree 100% on the speed of sound in a straight rod under tension. I'm not sure that I fully understand your other calculations, but we seem to get roughly the same result, in as much as I find the phase velocity of one mode of vibrations in the hoop to hit c when s is approximately 1.29, at least for short wavelengths (which is just the same point as where it happens for a rod).

I've been agonising a lot about trying to compute the rate of information flow -- the front velocity rather than the phase velocity -- but when I look at this plot:

http://www.gregegan.net/SCIENCE/Rings/RelativisticVibrationSpeed.gif

(explained a bit more on the web page
http://www.gregegan.net/SCIENCE/Rings/Rings.html
towards the end, in the section on relativistic vibrations)

where the curves of different colour represent different values of m, i.e. different wavelengths, it looks as if the phase velocity for the relevant modes is becoming almost independent of m by the time you get to m=5 (the magenta curves), so there really shouldn't be a huge amount of dispersion. There are a lot of quibbles here; we can't really take the linearised PDE seriously when m goes to infinity, as it needs to when we analyse a sharp-edged pulse. But it does seem quite plausible that the rate of information flow demanded by the model becomes superluminal around s=1.29, not only well before the energy peak (s=5/3, or 1.6667), but also well before the crisis point where the rate of change of angular momentum wrt omega is zero, which intersects the equilibrium states at s=1.617.

If this is true, it's interesting that things don't go haywire with the equations of motion as soon as the velocity of sound becomes superluminal! It's only when you stretch the hoop quite a bit beyond that point that the dynamics become completely absurd.

 

[pervect]

What I think is interesting is that from a purely Newtonian POV, that the speed of sound in the rod should increase with the stretch factor. Thus if you stretch a wire so it is 10% longer, the speed of sound in that wire should also increase by 10%. I've been meaning to redo this analysis in terms of force/body diagrams rather than a Lagrangian, but I haven't gotten around to it.

There are relativistic effects as well.

As far as the pertuabations go, I think that ideally they should have bounded first and second derivatives, though perhaps a step function in velocity (a bounded and low first derivative of position, but no bound on acceleration) might be acceptable.

 

[gregegan]

I've analysed a pulse with a bounded and continuous second derivative propagating through a relativistic hoop; movie at:

http://www.gregegan.net/SCIENCE/Rings/SmoothPulseInHoop.gif

and more details at:

http://www.gregegan.net/SCIENCE/Rings/Rings.html

While I can't pinpoint exactly where the hyperelastic model begins to imply superluminal signalling, it's clear from the behaviour of a pulse like this that the transmission of information is not significantly slower than the short wavelength limit of the linearised PDE, which becomes superluminal at exactly the same stretch factor where a straight string's speed of sound becomes superluminal.