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Stressenergy tensor of a wire under stress 
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#73
May1207, 05:43 PM

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The first part of the answer is that in the absolutely pure hyperelastic model, the force that holds the material together is an infiniterange 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 zerowidth 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 centroidframe 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 xaxis 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 centroidframe energy to fall below the rest mass. As soon as the tension starts to drop, there will be a barrier rising up in the centroidframe 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 centroidframe energy barrier steeper. 


#74
May1307, 02:11 PM

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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)(s1)(s4) 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. 


#75
May1407, 12:23 AM

P: 59

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. 


#76
May1407, 01:52 AM

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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 strangelooking 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. 


#77
May1407, 02:54 AM

P: 59

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. 


#78
May1407, 04:15 PM

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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.
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, massenergy, 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 LangevinLandauLifschitz metric applies to this Riemannian threemanifold), * 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 welldefined (e.g. Born rigid, radar distance) versus things which so far have not been welldefined (pervect's alleged Born rigid spinup procedure, which may be related to an alleged notion of "rigid spinup" 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 weakfield 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. 


#79
May1407, 06:17 PM

P: 59

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 symmetrypreserving perturbations to r around the stationary solutions, these arbitrary small perturbations to the shape seem always to be of higher energy. 


#80
May1507, 12:48 AM

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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(s1)(s4) 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. 


#81
May1507, 01:14 AM

P: 59

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 runaway 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. 


#82
May1507, 01:27 AM

P: 59

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. 


#83
May1507, 03:07 AM

P: 59

What is it I'm missing here? 


#84
May1507, 03:27 AM

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#85
May1507, 03:50 AM

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I see at least one very fundamental point that probably needs to be addressed in the Gron paper, conveniently online 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 spinup. Gron writes: 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 spinup the disk) one can also make the change in the proper proper distance ds between two nearby points during the spinup 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 spinup process is not (cannot) be perfectly Born rigid, it can approach the ideal as closely as desired. 


#86
May1507, 04:32 AM

P: 59

I'm having a lot of trouble following what you've done in your dynamics calculations. Before taking the divergence of T, what quantities do you allow to become functions of t? In principle everything is now a function of t: r, omega, rho, P (though of course the material model will link those latter two to the behaviour of r). Are you holding some of these constant as an approximation, in order to look at perturbations around the equilibrium solutions, as opposed to computing whole dynamic trajectories? 


#87
May1507, 06:00 AM

P: 59

As we decrease n from 1, i.e. increase the stretching of the hoop, if you look at the second derivative of E(r) for L held constant: (1) Initially, E''(r) is +ve, i.e. the stationary solutions are stable. (2) Before we hit E_max, E''(r) blows up to +ve infinity. This need not be anything pathological in the physics; it just means that at this particular stationary solution, there is literally no perturbation in r possible that will keep the angular momentum unchanged. (3) E''(r) is then ve (coming back towards zero from ve infinity); in this region the stationary solution is unstable, sitting on an energy ridge. However it is contained on the lowerr side by an energy trough, and on the higherr side by conservation of angular momentum, because there is a maximum r beyond which no more states exist with this L. (4) Exactly at E_max, E''(r) is zero. The lowerr energy trough and the ridge merge into a "shelf", a point of inflection in the curve (just like the graph of y=x^3), which resists movement to lower r but is unstable under perturbations that increase r. However, once again conservation of angular momentum seems to contain the hoop from exploding, because the curve of constant L stops at a finite r. (5) Immediately after this, E''(r) is +ve again, and the stationary solutions are stable again. 


#88
May1507, 02:53 PM

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OK, besides being terse, I was in somewhat of a hurry, so let's take a closer look at this. I'm definitely not infallible. What I did was first to approximate T^ab in a cylindrical coordinate chart (t,r,theta,z) by assuming that only the following terms were nonzero. T^{rr}, T^{rt}, T^{r theta}, T^{theta theta} This is all being done in a coordinate basis, so the basis vectors are not normalized. The second term was added by the assumption that I had a radial velocity, and I argued that T^{rr} could be excluded as it was of second order in radial velocity. All of these terms were assumed to be functions of r and t, and independent of theta and z. I did not substitute for the actual functions yet, I kept things symbolic. [add]You may have already noticed that if we don't refrain from substituting, we get an awful mess that's hard to interpret. The continuity equation [tex]T^{ab}{}_{;b}[/tex] or [tex] \nabla_b T^{ab}[/tex] (using Grtensor to calculate the covariant derivatives) then gave me: [tex]\frac{\partial T01(r,t)}{\partial t} + r T22(r,t) = 0[/tex] So the only thing being differentiated with t is T01. I didn't substitute in the terms for the variation of T^01 with w, etc. as I probably should have  but if the outgoing radial momentum T01 is positive and accelerating, or negative and deaccelerating, we know the system is unstable. We are looking for feedback that drives T01 to zero, and the equilibrium case has T01 = 0 and not changing with time. I was rather pleased that there were no obvious "deltafunction" type terms resulting from the differentiation of a step function in this particular component of the continuity equation  if you include T^{22}, this is not the case, for instance, you'll see [itex]\frac{\partial}{\partial r}[/itex] of a nonzero function. I haven't been keeping really close tract of the delta functions though, I've been assuming that if the continuous part works out correctly, the delta functions will too. There are some less obvious deltafunction type terms in the first term, though. As the hoop expands, T01 evaluated at a specific (r,t) will jump suddenly from 0 to a nonzero value, implying that its time derivative has a delta function. 


#89
May1507, 06:44 PM

P: 59

Thanks very much for the explanation, I was confused about your coordinate system but it's all clear now. (I also had a weird mental block and failed to recognise why T^22 will be zero in equilibrium solutions ... )
On the stability front ... at least with the axially symmetric constraint, I find even the breakable hoop to be stable away from its energy extrema. I expect its first energy peak to be, er, quasistable in the same odd way that the hyperelastic hoop is, but I can't determine anything yet about the energy valley or the second, higher peak; the algebra is so horrendous that I'm running out of memory in Mathematica. 


#90
May1507, 09:57 PM

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Not everyone knows that the distinguished mathematician Fermat was also a judge and a sometime ambassador who handled important and sensitive negotiations, apparently with great success. Alas, it seems that they no longer make such mathematical diplomats! Certainly I seem to be formed from a different die
All I was really trying to say was: please study the WP articles in the versions I cited, Greg's web pages, Gron's review paper, and ideally at least some of the papers cited therein, plus this thread to date, before speaking up, because this is subtle stuff so we don't want to encourage people to reinvent hexagonal wheels. OK, getting back on track: a quick comment: Just to be clear, I don't consider Gron's paper definitive at all, just the best review currently available. (To be truthful, it might be "the best out of a field of one", but he did a lot of work and it really is a pretty good review.) As I mentioned, Gron completely overlooks the crucial issue of multiple operationally significant notions of distance in the large for accelerating observers, and he appears to confuse quotient manifolds and submanifolds when he discusses the LangevinLandauLifschitz metric (which describes "the geometry of a rigidly rotating constant omega disk" as a Riemannian manifold, but this is the quotient of Minkowski spacetime by the Langevin congruence, not any submanifold!). 


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