Stress-energy tensor of a wire under stress

Ich

Glad you feel uneasy about this limit too - I really don't trust my intuition when it comes to GR.
You're right, I missed that point. But still: You identify increasing tension with the release of binding energy - another point where my intuition fails. It should be the other way round.
So somehow this whole scenario could be really consistent. I'm curious what will follow.

Thanks for the answers.

gregegan

In the Newtonian world it always is, but stretching a material in SR has two opposing effects.

In the material's local frame, it adds to the energy density, and it produces a tension (i.e. a negative pressure). However, in the centroid frame those two things get mixed by the Lorentz transformation, and what someone in that frame measures as energy density is a combination of the material frame's energy density and its pressure. Since the pressure in this case is -ve, that drives down the energy density measured in the centroid frame.

Since there are two competing contributions to the centroid-frame energy density as the material becomes more stretched, it makes perfect sense that this quantity should reach a peak and then decline. The fact that it declines below the rest mass energy is a bit startling, but given that the only limit we're imposing on the elasticity of our material is that no energy density ever gets measured to be literally zero, we shouldn't be surprised that it can get close to zero.

gregegan

Sorry, this statement was a bit careless. The energy density in the centroid frame actually falls with increasing omega, reaches a minimum, then starts to rise. The initial fall is simply because the hoop is being stretched, spreading its rest mass more thinly, an effect which at first will completely dominate over anything related to tension or potential energy.

But the behaviour of the radius, which first increases and then decreases, means that the total energy in the centroid frame first rises, then falls. The radius must fall eventually, because relativistic length contraction eventually overwhelms the expansion of the hoop material in its rest frame.

The gist of it is that there are so many competing terms and factors pulling in different directions that it's not all that surprising that the total energy is not monotonic in omega.

pervect

In combination with taking the total angular momentum of the hoop, what the math is telling me that at some value of omega, a spinning circular hoop will have less energy AND less angular momentum when we increase the angular velocity omega. This appears to be true for both Greg Egan's hyperelastic hooops, and the Born rigid hoops I analyzed.

What these equations would be telling us, if we interpret them literally and assume stability, is that a circular hoop spun up at some velocity would spin up even faster if we applied a negative torque to it, a torque that would reduce it's angular momentum and reduce its energy. I.e., if we applied some brakes to the hoop, the hoop would actually spin faster, transferring energy from the hoop to the brake pads. Energy which we could use to say, run a steam engine (or whatever). Eventually, the hoop would transfer a good proportion of its rest energy to the brake pads.

This is a pretty bizarre result. I have a strong suspicion, though, that an actual hoop would not so nicely provide us with a source of energy while shrinking its radius and increasing omega in a symmetrical manner. I have a strong suspicion that the assusmption of circular symmetry will fail, that the hoop will behave in a rather unstable manner under these conditions.

Unfortunately, I'm not quite sure how to go about analyzing the stability analytically. I think this boils down to evaluating the stability of [tex]\nabla_b T^{ab} = 0[/tex] around the operating point of the [tex]T^{ab}[/itex] for some particular rotating hoop.

gregegan

That's a really good question!

The first part of the answer is that in the absolutely pure hyperelastic model, the force that holds the material together is an infinite-range force. You really can't break the hoop, no matter what you do to it, because you can never get one part of it out of the range of attraction of the rest. And the way the physics works out, at least for the case of zero-width hoops, this never actually clashes with the weak energy condition. It gets asymptotically close to violating it, but it never quite does so. Even as omega goes to infinity we always have n>n_min, where n_min is set by the weak energy condition rather than any property of the material.

OK, but what about something a bit closer to reality? What about a force that mimics the hyperelastic force out to a certain threshold, and then dies away? So long as the threshold is high enough, that ought to be able to take our hoop into the strange zone where its centroid-frame energy is less than its rest mass, but it doesn't seem to offer any guarantee that the hoop won't come apart after we get there, leading to the apparent paradox of explaining how the fragments can separate when they don't have enough energy.

I think the answer is that you have to consider the potential energy diagram for this force, and its derivative, which gives the tension. Along the x-axis is 1/n, the expansion factor for the material. Starting out at 1/n=1, the PE is 0, and flat, so the tension is zero. Initially the PE rises quadratically, with the tension increasing linearly. So far, this is just like a spring. If it went on like this forever, we'd have the idealisation of hyperelasticity.

But we want our new force to lose its grip eventually, which means we want the PE to stop climbing, hit a maximum, and come down to zero. To do that continuously, though, will have consequences for the tension. As the PE stops increasing and reaches its maximum, the tension will come back to zero.

But it's only the tension being so high that allowed the centroid-frame energy to fall below the rest mass. As soon as the tension starts to drop, there will be a barrier rising up in the centroid-frame energy that needs to be overcome in order for the material to expand any further. The material can break eventually, it can get free of the force holding it together, but only if energy is supplied to get it over this barrier.

The material can't "just break", as if someone instantly pulled the plug on the force. Of course the potential energy can change as rapidly as we like, but that will only make the wall of the centroid-frame energy barrier steeper.

pervect

I've had a couple of more thoughts:

The first thought is to model a "breakable" hoop that has a limit on pressure. I eventually came up with

for 0<s<4
P = (1/4)(s-1)(s-4)

for s>4
P=0

where s = 1/n is the "stretch factor"

Peak pressure occurs at s=2.5. The speed of sound should be less than 1 at s=1. (k should be .75 if you take the slope). The weak energy condition should be imposed naturally by the hoop elongating indefinitely and not have to be added artificially.

I won't go into the details of the calculation, but if I'm doing it correctly it appears that this hoop still has a maximum in the E curve, at s=1.62.

The second thought I had is to model the dynamics of the hoop. As a first step, we can keep circular symmetry (which makes the calculations much simpler) but simply allow the radius of the hoop to be a function of time. I haven't looked at the details yet as to how to do this, but I think it should allow us to create hoops with over the statically allowed maximum energy E, and study their evolution.

While E_max appears to be a limit on the amount of energy a static hoop can hold, it should be possible to create a dynamic hoop with more energy than that.

gregegan

That's a nice model! I'd been doing something with an exponential drop-off, but it was analytically intractable. I didn't want to use a piecewise-defined function, because I thought that would be a pain in the neck, but on reflection it's almost irrelevant, because for s>4 in your model we simply know that there are no stable solutions. This model's so much easier to work with; I managed to get quite simple analytic expressions for r, omega and E in terms of s.

I assume you're using rho_0 = 1, in which case I get the same result. What's more, after reaching the maximum at s=1.62, I think E crosses the rest mass at s=2.19, hits a minimum of about 94% of the rest mass at s=2.45, then climbs up above the rest mass, long before breaking at s=4. (There's a second peak and fall just before the end, but it all takes place at much higher energy.) So as I guessed, even when the material model allows breakage, it can't happen until you get back above the rest mass.

Interestingly, at the same value for s as the energy minimum, r also has a minimum, and omega has a maximum. So as you drive the hoop along the path of monotonically increasing s, its radius first increases, then falls, then increases again; whereas its angular velocity first increases, then decreases. For a given omega less than the maximum possible value, there will always be two solutions with different radii, one much more stretched and energetic than the other.

Good luck! I might try something similar if I have time. One thing this ought to help reveal is whether even these symmetrical solutions are stable to small perturbations in r. In planetary dynamics, the nice thing to do is to fix the angular momentum L, turn omega into a function of L and r, and then use that to plot E as a function of r. Circular orbits then show up as the flat bottom of an energy trough. But in our context, we don't know yet if all our symmetrical solutions are in troughs, or if some of them sit on the top of ridges.

pervect

Yes, rho(0) was 1, and after the maximum at 1.62, E has a minimum below the rest mass at s=2.44, after which it starts rising. A rather strange-looking curve, especially considering how simple the defining function was.

I expect that at least some of these solutions represent unstable equilibrium solutions though, rather than a state that would actually represent a stable configuration of the hoop.

Certainly a Newtonian hoop would not be stable with a material that weakened as it stretched (say from s=2.5 to s=4).

Find out for sure which ones are stable in the relativistic case is the next big task (even the simpler subtask of considering only the stability of radially symmetrical hoops looks pretty involved.) While I think the equations needed should be generated from the divergence relations T^ab;b=0, T^ab won't be nearly as simple as it was before.

gregegan

I think the stability question for axially symmetric hoops can be answered without computing the full dynamic equations that allow r to be a function of t.

I computed angular momentum L as a function of v^2 and r, solved the cubic in v^2 for some fixed L, and fed that v^2 into E, to get E as a function of r, for some fixed L. Note that this is all done without imposing div T=0, because the point is to look at adjacent states which are not stationary solutions (also, this is back in the hyperelastic model, not the breakage model).

The plots I get show the stationary solutions lying at the bottom of troughs for E, even when E is falling with increasing omega, and even when E < rest mass. In other words, perturbing r in either direction always means adding energy to the stationary solutions, so they ought to be stable, at least under perturbations that respect the axial symmetry.

If there's any instability it must involve the hoop losing its circular shape. It's conceivable to me that the stretched hoop might be vulnerable to crinkling at some point where expanding its length (and hence increasing the tension) lowers the overall energy, but I'm still trying to think of a reliable way to check this without a week's worth of algebra and/or numeric computations.

Chris Hillman

I am glad to see Greg Egan has taken up my discussion with pervect, and I hope that some of the many interesting and physically/philosophically/mathematically interesting issues related to rotating matter will be fruitfully discussed by them in this thread.

Ich, FYI, I feel more comfortable limiting my conversations here to individuals who have divulged their identity to me (perhaps by PM).

I don't know anything about your background or your motivations for commenting in this thread, but FYI the reason I feel that it would be best for posters other than Greg and pervect to keep silent (unlike middle Egyptian, English lacks a verb for keeping quiet--- is this why American tourists are so loud?) is that there are many subtle issues here which experience shows are difficult to explain to persons lacking a strong background in math, physics and even a philosophical bent. I don't wish to see either of them distracted by naive questions--- or even worse, foolish statements based on neglecting known technical, physical, or philosophical issues, particularly if these have already been mentioned earlier in the thread. I also feel that those who have made no attempt to get some sense of the vast literature on rotating relativistic matter are unlikely to play a helpful role here.

In recent days, I have been working on some related issues but am unhappy with the fruit of my labors (things less thoughtful investigators would sloppily label "exact solutions", but which I currently suspect are physically misleading), so I'll bow out of this thread now, although I hope everyone else will let pervect and Greg continue their discussion. But I'd like to leave this thread with one last attempt to stress that there are many subtle issues here, and failure to bear them all in mind will certainly result in conceptual errors, uneccessary confusion, physical absurdities, and nonsense generally.
So here are a few final hints for getting started on thinking about this stuff, mostly addressed to hypothetical intelligent lurkers who are intellectually capable of appreciating subtleties and of bearing multiple issues in mind:

Some important distinctions:

* density and other variables in strained versus unstrained material,

* frame fields (AKA anholonomic bases) versus coordinate bases in a given
chart,

* frame field versus corresponding congruence,

* Langevin frame proper (constant omega, observers move in circular orbits
with constant radius) versus the variable omega generalization (observers
move in constant radius circular orbits but their speed varies),

* congruence (fills up region of spacetime) versus the world sheet of a hoop (doesn't fill up a region of spacetime),

* Born chart ("rotating cyl. chart") versus cyl chart (used in this thread),

* Axel, the inertial observer stationary wrt centroid of disk/hoop, versus Barbarella, a hoop/disk riding observer (if constant omega, she is one of the observers whose world lines are given by Langevin congruence for that omega),

* radius, mass-energy, angular momentum of hoop measured by Axel (makes sense), versus the same as measured/computed by Barbarella (won't make sense, at least not without very careful qualification),

* clock synchronization by Axel and friends (makes sense) versus by Barbarella and friends (impossible even for a hoop--- c.f. Sagnac effect),

* Born rigid congruence (vanishing expansion tensor) versus other notions of "rigidity",

* pervect's position (no problem) versus my position (nothing shown either way) on pervect's claim that it is possible to define a notion of spinup of an elastic hoop (with radius expanding or contracting as described by Axel) which remains rigid throughout the evolution (in the sense that the world lines in the world sheet of the hoop can be enlarged to a Born rigid congruence),

* Alleged orthogonal spatial hypersurfaces for Langevin observers (doesn't exist, since Langevin congruence has nonzero vorticity) versus the quotient manifold (quotient of Minkowski spacetime by the Langevin congruence) (which does exist; indeed the Langevin-Landau-Lifschitz metric applies to this Riemannian three-manifold),

* multiple operationally distinct notions of "distance in the large" for accelerating observers even in flat spacetime--- c.f. problems with speaking carelessly about "the circumference of the hoop measured by Barbarella and friends",

* constant omega versus nonconstant omega (I discussed a generalization of Langevin congruence to variable omega, but these observers maintain constant radius as measured by Axel, so aren't suitable for discussing pervect's alleged "Born rigid" spinup of a hoop,

* crude conditions on "physical acceptability" like energy conditions, speed of sound, versus "physically realistic" models,

* making a computation versus interpreting it; a good physicist never omits the latter and in fact may spend most of his effort on this task,

* conclusions which depend upon choice of a physical model and those which do not; I feel that some important points require studying specific physical models and considering limits in order to have confidence that "any
reasonable model" would have such and such qualitative behavior.

* things which have been well-defined (e.g. Born rigid, radar distance) versus things which so far have not been well-defined (pervect's alleged Born rigid spinup procedure, which may be related to an alleged notion of "rigid spin-up" suggested by Grunbaum and Janis, which I also currently consider unconvincing).

* exact solutions of ODEs mentioned by Greg, pervect and myself (typically hard to obtain) versus approximations via perturbation theory (which can also yield valuable physical insight),

* attempting str treatments (pervect and Greg) versus exploring gtr treatments (me only),

* Newtonian limit (str or gtr) versus weak-field limit (gtr); I advocated latter as a stepping stone to exact solutions in gtr. I expect to expend more work laying the foundation to interpret such solutions than in actually finding them.

Further general issues:

* what can be neglected? e.g a nonspinning inertial frame for Langevin observers will appear to spin wrt Axel as per Thomas precession.

* which idealizations are "physically acceptable"? "Physically reasonable?"

* what are the criteria for "physical acceptability", anyway?

* perturbation analysis is usually very helpful when things get confusing and formulas get messy, but choice of variables is critical, i.e. this is a delicate art.

And a general reminder:

The literature on rotating disks and hoops is large and spread over many decades, journals, and several languages. None of these authors have taken all relevant considerations into account, so none of them have provided fully correct treatments. Some have come much closer than others, however, in fact much of the literature consists of independently recommiting old errors.

All parties should bear in mind the advice of George Santayana, which I'll paraphrase as the warning that "those who [fail to study past errors] are condemned to repeat [them]." Study the literature, or else forfeit the honorable title of scholar! Grrr!

A good place to begin is the review paper by Gron and papers cited therein:
http://freeweb.supereva.com/solciclos/gron_d.pdf

Last but not least, this list is incomplete.

gregegan

For what it's worth, I've now done some calculations which make me suspect that the hoop probably won't be vulnerable to crinkling, at least from small perturbations.

I computed L and E for a state where some small, arbitrary function delta*f(theta) is added to the radius. I found series expansions for both L and E to second order in delta; the coefficients for both included integrals over theta of f, f^2, and (f')^2. Numerically, I found that requiring L to be constant to second order always resulted in E being constant to 1st order, and with a +ve coefficient for the delta^2 term. In other words, just as with symmetry-preserving perturbations to r around the stationary solutions, these arbitrary small perturbations to the shape seem always to be of higher energy.

pervect

Are you at a point yet where you can make some predictions for the stability of the hoop, in either sense, near (after) the point where E reaches a maximum?

I think I'm getting the rather interesting result that for a sufficiently strong hoop, there is a limit to how much energy it can hold (the first peak of E on the curve) but this limit is governed by an implosion failure rather than an explosion failure.

As far as the dynamics go:

If I drop terms of order vr^2, where r is the radial velocity, I find that we only add T^{tr} to the stress energy tensor. This comes from the equation

[tex]
T = \rho \vec{u} \times \vec{u} + P \vec{w} \times \vec{w}
[/tex]

and adding a radial component to v while letting w remain unchanged.

There is a second order term T^{rr}, also second order corrections to other terms, which I ignore.

Keeping only this linear terms the continuity equation says simply

[tex]\frac{\partial T^{01}}{\partial t} = r T^{22}[/tex]

so all we need to do is look at the value of T^22 (in the lab frame) to determine whether or not the radial velocity v, which is proportional to T^{01}, accelerates or deaccelerates.

This gives us a rather physical interpretation of T^22 in the lab frame by the way - we previously calculated that this was zero, now we see again that the continuity equation requires this for an equilibrium hoop. Furthermore, its sign controls expansion or contraction - a positive sign means expansion (more precisely, a positive acceleration of rate of expansion dvr/dt, assuming vr is small).

T^22 ignoring second order corrections is just
[tex]
\frac{\rho \omega^2 r^2+P}{r^2\left( 1-\omega^2 r^2 \right)}
[/tex]

so basically it's just the sign of [tex]\rho \omega^2 r^2+P[/tex] which should determine the direction of acceleration or deacceleration of expansion.

I am not terribly confident that radial symmetry would be maintained during the implosion process - I just don't see any stable state that a hoop with "too much energy" could reach. For instance, in the "breakable hoop" with P = .25(s-1)(s-4) we see that E eventually climbs up above the first peak, but IMO this portion of the curve is radially unstable.

I'm not too sure about what happens near the first valley of the E curve yet.

gregegan

I just found something delightfully weird! It was premature of me to jump to the conclusion (on the basis of some numerically derived plots) that all the axially symmetric solutions were stable. There turns out to be a narrow range of parameters where they aren't.

To make the calculations more tractable, instead of trying to express E(r) for constant L explicitly, I used L=constant to implicitly define a relationship between v^2 and r, and then took derivatives of that equation to evaluate the first and second derivatives of E(r). (Conceptually this is all the same as using omega and r, rather than v^2 and r, but the algebra is much simpler using v^2.)

At the solution points, E'(r) was zero, which was reassuring, but the second derivative E''(r) was an expression not guaranteed to be positive. Close to the point where r reaches its maximum, there is a range of values where E''(r) at the solution point is negative, i.e. the solution lies on an energy ridge.

Here's the weird part, though: if you fall off the ridge in the direction of increasing r, you approach an edge to the energy curve, beyond which there are no states which have this value of L. Also, there's an energy trough just inside, so if a hoop started out on the ridge and fell inwards, it would get caught in that trough and presumably oscillate radially between two (quite close) r values. I don't think there's any run-away behaviour, leading to hoops either exploding or imploding, but at this point I wouldn't bet my life on it. Maybe it will take the complete dynamic equations for axially symmetric states to fully understand this, after all.

If anyone feels like checking this out, an example is k=0.32, rho_0=1 (this corresponds to K=0.4 for the K used on my web page), and this phenomenon occurs between n=0.510 and n=0.523, where n is the compression factor, which pervect showed how to use to parameterise the (r,omega) solutions.

gregegan

I think my last post partly answers that; in the region where r, L and E reach maxima (and of course they don't all reach maxima at exactly the same point), I see some definite instability even for axially symmetric states, but it looks to me as if it's probably contained on both sides.

I'm not studying the breakable hoop at this point, though; it sounds like you are?

I think I need to look more closely at the dynamics.

gregegan

Sorry to be thick, but I'm completely lost here! The numerator of the formula above will rarely, if ever, be zero. Are you claiming that whenever this expression is non-zero, the hoop is undergoing radial acceleration in one direction or another? That can't be true, or there'd be no constant-radius solutions at all; v_r would always be changing, even if it was initially zero.

What is it I'm missing here?

pervect

Typically r reaches a maximum first - I've always found L and E reach a maximum at the same time, however. I suspect that the later (L&E reaching a maximum at the same time) is required for consistency of the physics, that any torque that increases L must be in the same direction as omega and hence increase E, and that any torque that decreases L must decrease E, i.e. that the slopes of L and E must have the same sign for physical reasons. If they must have the same sign, they must switch signs at the same time.

pervect

I see at least one very fundamental point that probably needs to be addressed in the Gron paper, conveniently on-line in draft (?) form at

http://freeweb.supereva.com/solciclos/gron_d.pdf

This is a true, but I think potentially misleading remark made by Gron that I think fuelled some of the earlier rather long discussion of the spin-up.

Gron writes:

Going into more detail, one can see that it is indeed not possible to EXACTLY accelerate a ring in a Born rigid manner as Gron argues in the section on "Kinematical solution of Ehrenfest’s paradox".

Note that this also applies to the thin ring, not the disk.

What I believe *IS* possible, however, is that by taking the limit as w(t) increases very slowly (the limit in which one takes an infinite amount of time to spin-up the disk) one can also make the change in the proper proper distance ds between two nearby points during the spin-up process less than [itex]r_ch \, ds[/itex], where [itex]r_ch[/itex] is an arbitrarily small number. Note that any pun about r_ch being a small number might well be intentional.

So while the spin-up process is not (cannot) be perfectly Born rigid, it can approach the ideal as closely as desired.