Stress-energy tensor of a wire under stress

[pervect]

In this case, though, I don't really have a vector field. I have a one-parameter subfamily of curves (the parameter is theta_0, the value of theta at t=0) that do not fill all of space-time as a congruence should, but which do trace out the worldline of any point initally on the hoop.

 

[Chris Hillman]

In this case, though, I don't really have a vector field. I have a one-parameter subfamily of curves (the parameter is theta_0, the value of theta at t=0) that do not fill all of space-time as a congruence should, but which do trace out the worldline of any point initally on the hoop.

I guess you are worried that if you let $r_0, \, \theta_0$ vary (to make a three parameter family of curves which on dimensional grounds we hope will fill up some neighborhood without intersections), your curves might intersect?

 

[pervect]

While I think it is probably possible to define the "circumference" of the hoop by performing a branch cut along a worldline of the congruence, I think that it's also possible to avoid talking about the circumference simply by insisting that the worldlines maintain a constant distance (aka a constant "orthogonal deviation vector").

You keep bringing up the topic of the circumference, and I keep attempting to avoid it. If you really want to define a circumference, go ahead and define the circumference via a branch cut. The important point is that the branch cut be one of the worldlines of the congruence. Then we can talk about 'circumference vs time' if we really want to. (And I think we can obsserve that the circumference, in this sense, is constant, though I'd have to double-check this.) But I don't particular want to, I'd just as soon talk about the separation between close world-lines being constant.

To avoid some of the acceleration issues, we can imagine stopping the spinup process to perform our measurement of the separation of the worldlines, spin it up a little more, stop it again and check that our wordlines are still a constant distance apart, etc, though the only cure for dealing with the centripetial accleration is to chose worldlines close enough to each other initially.

This Born-rigid motion allows us to use the simplest possible model - a constant density wire that does not elongate at all with stress. This sort of motion is not possible for anything but a hoop of zero thickness, however, as has been previously noted.

The approach can be adjusted as needed to use a more elaborate material model if desired.

 

[pervect]

I think I may have a better handle on what you're looking for - a velocity field .

To fill all of space-time, we need to consider multiple hoops. Let's consider how r(t) varies, given that the initial radius was r0

We need
$$r(t) = \frac{r0}{\sqrt{1+r0^2 w(t)^2}}$$

By definition the theta component of the coordinate velocity is given by
$$\frac{d \theta}{d t} = w(t)$$we need to know what $$\frac{d r}{d t}$$ is.

This is
$$\frac{d r}{d t} = \frac{dr}{dw} \frac{dw}{dt}$$

It turns out, unless I'm making an error, that
$$\frac{dr}{dw} = -\frac{r0^3 w}{(1+r0^2 w^2)^{3/2}} = -r^3 w$$

so

$$\frac{dr}{dt} = -r^3 w \frac{dw}{dt}$$

Exactly what to do with this, I'm not sure :-). I suppose we need to convert this to a 4-velocity from an ordinary velocity for starters.

[add]
If we don't normalize v first, and just set dt/dt = 1, v(bup, cbdn) has a lot of zero components in the Langevian basis (with w replaced by w(t)).

with t=1, r=2, theta=3, and z=4

v^(1)_(3), v^(3)_(3) and v^(4)_(3) are all zero
v^(2)_(3) is nonzero however.

If we normalize v first, v^(1)_(3) no longer vanishes. Still a lot of zero components, but not as many.[add^2]
Well, this is getting nowhere fast. Basically I expect members of the the congruence to maintain the same separation (measured orthogonal to the congruence) from each other if they start on the same radius.

I do not expect the expansion tensor to vanish.

I expect the separation vector to rotate, too.

 

[Chris Hillman]

Still trying to clarify the nature of our current disagreement

While I think it is probably possible to define the "circumference" of the hoop by performing a branch cut along a worldline of the congruence, I think that it's also possible to avoid talking about the circumference simply by insisting that the worldlines maintain a constant distance (aka a constant "orthogonal deviation vector").

When you say "constant distance" here, clearly you still mean vanishing expansion tensor.

You keep bringing up the topic of the circumference
[of a rotating hoop, as described in some specified fashion by comoving observers riding on the hoop], and I keep attempting to avoid it.

I'm not trying to be difficult --- I'm trying to get you (and lurkers) to see that there is a serious and challenging issue here. Just one of many, unfortunately.

I only act like a philosopher exposing hidden assumptions when its really neccessary, i.e. when people are in fact arriving at incorrect results, or as in this case, when everyone seems to arrive at a different result, and to be convinced, through a failure of imagination, that only his own result can be correct. When a whole bunch of smart people have said "it's perfectly simple, and the result is $R_j$, where $R_1, \, R_2, \dots$ are all different, this can be an indication that they are all confused. I think you know that I don't often say "there are serious issues here"--- most of the stuff which comes up in public forums really is pretty trivial once you've mastered the elementary stuff. This problem isn't like that--- it bites!

If you really want to define a circumference,

My point is that depending upon what you mean (see "distance in the large" again) by "define a circumference", this may be impossible.

To avoid some of the acceleration issues, we can imagine stopping the spinup process to perform our measurement of the separation of the worldlines, spin it up a little more, stop it again and check that our wordlines are still a constant distance apart, etc, though the only cure for dealing with the centripetial accleration is to chose worldlines close enough to each other initially.

This answers none of my objections. You haven't clarified, much less justified, how your alleged "equivalence" should work. Again, one of my points is that you want to perform a thought experiment in which you compare the mass-energy of a hoop, as computed by an inertial observer, "before" and "after" being spun-up to a constant angular velocity. You are claiming that the "after" hoop has smaller radius. To justify that you need to explain how to put an equivalence relation on rotating disks characterized by $R, \omega$, which declares certain disks to be equivalent via a particular kind of spin-up. You can certainly stipulate that the spin-up be very slow, or that it maintains vanishing expansion at all times, as long as you show your stipulations (1) yield realizable congruences (2) establish the necessary equivalence relation.

This Born-rigid motion allows us to use the simplest possible model - a constant density wire that does not elongate at all with stress. This sort of motion is not possible for anything but a hoop of zero thickness, however, as has been previously noted.

Are you still claiming that rigid acceleration acceleration is possible for a "thin" hoop (infinitesimal cross section)? I have provided (weak) mathematical evidence that this might not be true.

The approach can be adjusted as needed to use a more elaborate material model if desired.

I think our core disagreement concerns how to establish the necessary one-parameter equivalence relation on rotating hoops (constant omega). I think we agree that once an equivalence is established, we know how to compute the energy and angular momentum, and then using our equivalence we can compare the energy/ang.mom. of a hoop rotating with constant rate $\omega$ with "an identical nonrotating hoop".

You wish to find some way of doing this which avoids any neccessity of postulating a material model; I doubt this can be done but am prepared to learn otherwise if I am wrong about that. My intuition is that postulating a material model may be unavoidable, but that a fairly simple elastic material will suffice to establish an equivalence.

There is another reason for my interest in material models: to (1) and (2) above I'd like to add (3) the equivalence should be "physically reasonable". Let me explain what I mean by that.

I feel that it is important not only to concoct a mathematically well-defined equivalence (and it is clear to me that there are many ways of doing this), but to justify one as being preferred on the grounds that it is the simplest physically reasonable method.

How do we assess whether an equivalence is physically reasonable? Very simple: the slow rotation limit should be an excellent approximation to a reasonable Newtonian model. Any equivalence which claims that even a tiny increase in rotation rate from zero will decrease the radius strikes me as physically objectionable, because no physically reasonable hoop would behave that way.

 

[pervect]

First off, let me say that I appreciate your time and help, and if I sounded a little frustrated, it was probably because I was getting a little frustrated. But the problem is not your help, which I do appreciate, the problem is the problem itself.

I got some more sleep and sent a PM, and also came up with some new ideas. For the benefit of any hardy souls still with us (if any), I'm saying that the expansion tensor won't vanish on the entire 4-d manifold, it will only vanish on the submanifold generated by the congruence of paths passing through a circle. The disk must be of zero thickness. The expansion tensor of an actual disk of nonzero thickness does not vanish, this is why the expansion tensor of the manifold does not vanish, it only vanishes on the submanifold.

At first this didn't seem all that helpful, but then I realized we could always find the induced metric on the submanifold and calculate the tensor there.

Hopefully this will be a little clearer. Anyway, time for lunch.

 

[Chris Hillman]

Aha, a time zone clue!

First off, let me say that I appreciate your time and help, and if I sounded a little frustrated, it was probably because I was getting a little frustrated. But the problem is not your help, which I do appreciate, the problem is the problem itself.

You didn't sound frustrated. But discussing this problem is exhausting, because there are so many things to bear in mind.

I got some more sleep and sent a PM, and also came up with some new ideas. For the benefit of any hardy souls still with us (if any), I'm saying that the expansion tensor won't vanish on the entire 4-d manifold, it will only vanish on the submanifold generated by the congruence of paths passing through a circle. The disk must be of zero thickness. The expansion tensor of an actual disk of nonzero thickness does not vanish, this is why the expansion tensor of the manifold does not vanish, it only vanishes on the submanifold.

I agree that a "thin hoop" should be easier to study than a "thin disk", and I agree that in the former case, you have the option of defining a congruence in which the expansion tensor vanishes except for world lines which actually correspond to matter in the hoop. I still don't see how to obtain such a congruence however, subject to (1), (2).

If you (or I) can find one or otherwise show they exist, subject to establishing a unique equivalence, I'd want to move onto (3). That's why I've been studying Greg Egan's treatment of a linear accelerated elastic rod in flat spacetime.

At first this didn't seem all that helpful, but then I realized we could always find the induced metric on the submanifold and calculate the tensor there.

Whew!--- for a moment I thought you meant "the" (nonexistent) hyperslice orthogonal to the world lines in the variable omege type Langevin congruence.

(One of the elementary points in the full Ehrenfest paradox is that because we have a stationary timelike congruence we can form a quotient manifold, obtaining the Langevin-Landau-Lifschitz metric, but this is certainly not a hyperslice (submanifold)! And the LLL metric of course gives rise (as does any Riemannian metric) to too many notions of "distance in the large" (integrate length along multiple paths), none of which are particularly interesting physically. But in his PM, pervect says this submanifold is the "world sheet" of the hoop, which he says should gradually shrink as we spin it up. So, in the standard cylindrical chart, this world sheet would look like a cylinder in t,r,phi space--- we suppress the z coordinate--- whose radius is decreasing as we move upwards, i.e. increase time coordinate.)

OK, I agree that to realize your program, you need to set the expansion tensor to zero on this submanifold. And then show this gives a unique equivalence. Then given a spinning hoop, you'd know which nonspinning hoop (which radius) to compare it with. Then we could study slow rotation limit and compare with Newtonian analyses to argue over (3).

 

[pervect]

Here's what I'm getting.

the cylindrical metric is -dt^2 + dr^2 + r^2 dtheta^2 + dz^2

We define some function w(t) to represent spin-up, and we set

r(t) = 1/sqrt(1+w(t)^2)

We want to consider a submanifold having only (t,theta,z) and eliminating r. To calculate the induced metric on this submanifold, we just do some algebra.

The result we get is (for convenience I've removed the explicit dependence of w with t)

dt^2 = (-1 + (w dw/dt)^2/(1+w^2)^3) dt^2 + dtheta^2/(1+w^2) + dz^2

Now, we need to argue that we can make dw/dt as small as we like, and that in the limit where dw/dt is negligible, representing a very slow spin up, that our induced metric becomes

ds^2 = -dt^2 + dtheta^2/(1+w^2) + dz^2

I.e we get a series of different line elements, which converge to this unique induced metric when we take the limit dw/dt -> 0.

We need to do this _before_ we calculate the expansion scalar.

I'm not sure how mathematically rigorous this step is, but it seems at least plausible.

With this assumption, I get zero for the expansion scalar assuming I'm doing the calculation correctly.

The velocity field (unnormalized) I use is just (t,theta,z) = (1,w(t),0), which represents the curves in the congruence parameterized by time.

I pass this velocity vector to grnomralize first, then compute the expansion scalar.

 

[pervect]

Finally, if we make

r(t) = f(w(t))

and we repeat the above analysis, upon setting the expansion scalar to zero we get

f(w)^3*w + df/dw = 0, which has the solutions

f = 1/sqrt(w^2+ C)

Setting f=1 at w=0 yields the original expression. So if the original analysis holds up, it generates a unique answer for the radius of the disk vs w given the initial radius at w=0.

 

[Chris Hillman]

Time to study some previous work!

Hi, pervect,

I think it's time to visit the library, since we are approaching the point of reinventing the wheel, or more precisely, varieties of spinup procedures for attempted treatments of relativistic hoops. To mention just three relevant papers:

G. L. Clark, "The Problem of a Rotating Incompressible Disk", Proc. Camb. Phil. Soc. 45 (1949): 405.

For a small strain limit, compares a Newtonian with relativistic analysis of the spin-up of an elastic disk. Clark and other authors find that the radius increases, as described by inertial observer comoving with centroid, hereafter "Axel", for "axle observer". This is obviously relevant to our hoop discussion re (3) physical reasonableness of a proposed "equivalence" between rotating and unrotating disks.

W. H. McCrea, "Rotating Relativistic Ring", Nature 234 (1971): 399.

McCrea studies a hoop made of a material in which speed of sound (or better say the speed of p-waves?) equals speed of light (an elastic material variant of something often called a "stiff fluid" in the gtr literature), and finds $R=R_0/\sqrt{1-R_0^2 \, \omega^2} > R_0$ for radii measured by Axel "before" R_0 and "after" R the spinup. (The elastic deformation would be even larger for "softer" materials.)

A. Grunbaum and A. I. Janis, ''The Geometry of the Rotating Disk in the Special Theory of Relativity", Synthese 34 (1977): 281.

The authors introduce a spin-up condition ensuring that tangential stresses vanish and conclude a disk spun-up in this way has smaller radius (as described by an inertial observer comoving with the centroid). For our hoop problem, this would correspond to trying to spin up the hoop without introducing any tangential tensions, in fact I think it may correspond to the spinup procedure you are trying to formulate.

Another possible spin-up procedure we should both think about would be impulsive tangential blows delivered around the hoop, simultaneously and equal magnitude as described by Axel. As we know from twin paradox, impulsive blows can be more confusing than helpful, but we should consider this anyway.

Many of the other papers discussed in Gron's review are also relevant to our discussion, but I think these three might be particularly important for us. But bear in mind some generalizations about the literature on rotating disks and hoops:

1. None of the published papers are, in my view, fully correct,

2. The reason for this is that none of the authors have borne in mind all relevant considerations (for example, we haven't discussed the issue of whether Thomas precession mucks up our analysis--- I think not, but contrary views have been expressed!),

3. The worst papers come from authors who think "it's all perfectly simple" if you just think of it like they do.

So trust nothing, verify everything! Yep, whole lotta work for a simple seeming problem.

For example, while Gron's review is excellent, he fails to consistently distinguish between "congruence orthogonal hyperslice" and "quotient by congruence" manifolds (respectively impossible and possible for the constant omega Langevin congruence). He also fails to discuss the issue of multiple operationally significant notions of "distance in the large" for accelerating observers (and all rotating observers are accelerating, even if they have constant angular velocity).

In reading Gron, be careful to recall that most of his discussion doesn't involve spinup at all, but rather comparing a disk of radius R as described by Axel using the cylindrical chart with omega zero and nonzero, or the same as described by hoop riding observers using the Born chart. Note too the distinction between imagining small Born rigid measuring rods and the material of the disk itself; Einstein's analysis imagines rigid rods sliding on the disk, and would correspond roughly to a notion of circumference measured by the hoop riding observers which I called "pedometer" distance. But never forget that clocks can't be synchronized for these hoop riding observers (c.f. Sagnac effect).

In the paper by McGregor cited by Gron, M argues that the elastic potential energy should be fourth order in the rim velocity measured by Axel which we can try to verify and use in discussing physical reasonableness of proposed spinup procedures.

I should also say that I am still thinking about trying to adapt Greg Egan's analysis to spinning hoops.

 

[Chris Hillman]

Bulletin for lurkers: pervect and I (and Greg Egan!) are still working on this, but right now I at least am reviewing the literature, playing with some putative exact solutions I have found, with alternative congruences, charts, and so on.

I forgot to mention M"arzke-Wheeler coordinates for accelerated observers in Minkowski spacetime, which are an important addition to the roster. See gr-qc/0006095, which discusses MW coordinates for the Langevin observers.

Greg tells me he has found an exact solution (given in terms of an ODE) modeling a rotating hoop, using his stress-strain assumptions, which exhibits this behaviour under spinup: initially the hoop expands, as seen by Axel (the inertial observer comoving with the centroid) as per Newtonian theory, but eventually a relavistic correction counteracting the centrifugal force (roughly speaking) becomes significant. Thus, as described by Axel, the diameter of the hoop increases, but not as quickly as in Newtonian theory.

I rather easily found another exact solution (also given in terms of an ODE), using the stress-strain relation proposed by Clark for a "stiff elastic solid" (in which the propagation speed of p-waves is c, while the propagation speed of s-waves is less than c), which might model the interior of a linearly accelerated rod (I'm still thinking about whether or not I believe this). According to Clark, in such a stiff solid, we should have
$$\rho = \frac{\mu}{1+n}, \; p = n \; \rho$$
where $n$ is "the dilitation", by which I assume he means the strain, which here is one-dimensional, and $\mu$ is the rest state density of the material. The very first Rindler type Ansatz I tried quickly leads to a solution in which we use n as the master variable, so that everything is expressed in terms of n and n_x.

 

[Chris Hillman]

I have to interrupt my work on this for a few hours, but I just wanted to say that Greg Egan has just put up his analysis of his own model of a relavistic hoop treated as an elastic solid at http://gregegan.customer.netspace.net.au/SCIENCE/Rings/Rings.html
His analysis confirms my intuition: the hoop expands as per Newtonian analysis, but as the angular velocity increases, the counteracting relativistic effect first noticed by Einstein means that the rate of increase is slower than in Newtonian physics.

Note that the unit vectors he writes
$$\vec{e}_t, \; \vec{e}_r, \; \vec{e}_\Phi$$
are the ones I wrote (in the cylindrical chart for Minkowski vacuum)
$$\vec{e}_1 = \partial_t, \; \vec{e}_3 = \partial_r, \; \vec{e}_4 = \frac{1}{r} \, \partial_\phi$$
He is treating an elastic solid in Minkowski vacuum, i.e. ignoring gravitational effects in order to clarify the dynamics of a rotating disk or annnulus or hoop. His models should relate to ones considered by Clark in his earlier Proc. Cambr. Phil. Soc. paper, which is cited in the later paper I cited.

I am still mulling my putative exact solution modeling a linearly accelerated elastic solid body (using Clark's "stiff solid" material model in which the speed of p-waves equals the speed of light, while the speed of s-waves is smaller).

 

[Chris Hillman]

Referring the my post #46: MW coordinates for a Langevin observer are a pain to work with, but the eprint cited gives a nice figure. To obtain the MW chart determined by a proper time (T) parameterized timelike curve in Minkowski vacuum, given two events on the curve with T=T_1 and T=T_2, where T_1 < T_2, draw the forward light cone at T=T_1 and the past light cone at T=T_2, and observe that their intersection is a boosted two-sphere. All the events on this two-sphere have MW coordinates
$$\tau = \frac{T_2+T_1}{2}, \; \sigma = \frac{T_2-T_1}{2}$$
and the other two MW coordinates are determined by putting some chart on the sphere. Similarly for all other events. Exercise: the MW constant time surfaces are always orthogonal to the world line used to define them. Why doesn't this contradict the Frobenius theorem if it belongs to a congruence with nonvanishing vorticity.

 

[pervect]

I'll have to compare Greg Egan's results with mine in more detail. A couple of interesting points stand out to me, though from an initial reading.

Egan predicts that hoops can shrink when they are spun up. There is nothing to constrain the angular velocity for the relativistic case in his model (or in my rigid model), therfeore one can increase omega as much as one wishes. If r did not shrink, a point on the circumference of the hoop would exceed the speed of light at some value of omega. One must either have r shrinking or some natural limit on omega (a limit which does not appear to exist).

I would argue with Greg Egan's wording when he concludes the fact that r eventually shrinks means the ring is under compression, however - the ring always be in tension in its own frame field.

 

[gregegan]

I would argue with Greg Egan's wording when he concludes the fact that r eventually shrinks means the ring is under compression, however - the ring always be in tension in its own frame field.

You're quite right, of course! The ring will always be under tension. I've amended the web page, and added a curve to the final plot showing the tension, which continues to increase but appears to approach an asymptote as omega goes to infinity.

 

[gregegan]

You're quite right, of course! The ring will always be under tension. I've amended the web page, and added a curve to the final plot showing the tension, which continues to increase but appears to approach an asymptote as omega goes to infinity.

The tension does approach a horizontal asymptote, which turns out to be connected to the weak energy condition. So in principle omega can increase indefinitely, but any real material would have a breaking strain that fell short of the absolute ceiling imposed by the weak energy condition, so there would be some finite omega at which it would have to be torn apart.

Still, it's nice to imagine spinning a sufficiently strong ring so rapidly that it managed to fit inside a smaller radius than it started out with!

 

[pervect]

Hi, and welcome to PF, gregegan!

I believe that your equations can be solved exactly for the rotating hoop if one takes n as the primary variable.

1/n is (for lurkers) the amount by which the hoop stretches, i.e. n=.5 represents the material elongating to twice its length, see Greg's original webpage at

http://gregegan.customer.netspace.net.au/SCIENCE/Rings/Rings.html

I'm getting the following. Let $\rho_0$ be the initial density of the hoop, and let k be the Young's modulus as per the above webpage

Then we can write the stress-energy tensor in your u,r,w frame which we've been calling the Langevian frame here using geometric units as $\rho$,P, where $\rho$ is the density and P is the pressure

$$\rho = n \rho_0 + \frac{k}{2} n \left(1-\frac{1}{n}\right)^2$$

$$P = k \left(1-\frac{1}{n}\right)$$

this follows most directly from your other webpage

http://gregegan.customer.netspace.net.au/SCIENCE/Rindler/SimpleElasticity.html

We observe that there is a minimum n factor for which the weak energy condition is satisfied, this is

$$n_{min} = \sqrt{\frac{k}{2 \rho_0 + k}}$$

While n is a minimum, 1/n is the amount by which the hoop expands, so this represents a condition of maximum expansion.

Given the above, can then solve for the tangential velocity of the hoop via the relationship:

$$v^2 = -P/\rho$$

This relationship between presssure(tension), density, and the radial velocity v of the hoop was derived by setting the divergence of the stress-energy tensor to zero.

This simplifies (using Maple) to
$$v^2 = {\frac {2 \,k \left( 1-n \right) }{k \left( 1-n \right) ^{2}+2\,{n}^{2} {\it \rho_0}}}$$

We notice that when n=nmin, v=1, and that for n>nmin, a series expansion puts v^2<1, as it should be. So for any velocity <1, the positivity of the stress energy tensor is (just barely) satisfied.

Now we can find the radius of the hoop explicitly via the relationship

$$r = \frac{r_0}{n} \sqrt{1-v^2}$$

How this was derived takes a little explaining. Basically, the difference in angles between any two points on the hoop is assumed to be constant as the hoop is spun up. So two points 1 degree apart initially in the lab frame will always be 1 degree apart. This is part of the spinup process which was discussed at some length in previous posts.

We can then say that the intial proper length of a small section of the hoop before spinup was $r_0 d\theta$. After spin up, the proper length is $$\frac{r d \theta}{\sqrt{1-v^2}}$$

But we know that the ratio must equal 1/n, hence the above equation.

We can find r explicitly by substituting the value we just calculated for v^2 into the formula above. We can also find $\omega = v/r$ as a function of n the same way.

I get
$$r = \frac {r_0}{n}\,\sqrt {{\frac {2\,n^{2}{\it \rho_0}+k n^{2}-k}{2\,n^{2}{ \it \rho_0}+k n^{2}-2\,kn+k}}$$

$$\omega = \frac{n}{r_0}\sqrt {\frac {2 \, k \left( 1-n \right) }{2\,{n}^{2}{\it \rho_0}-k \left( 1-{n}^{2} \right) }}$$

we note that $\omega$ goes to infinity just as n=nmin when the weak energy condition is violated.

Thus I also find that a hyperelastic material satisfying the weak energy condition can (just barely) be "spun up" to a point where it shrinks.

I think that the total energy E and angular momentum J of the spun-up hoop are also of some interest - in fact, this was what originally motivated me to try and perform these caclulations.

This can be found by converting the stress-energy tensor back into the lab frame, and integrating over the volume. I have not yet done these calculations for the hyperelastic hoop yet. I think that my simpler "Born rigid" model may even provide more insight here, being more tractable (though less physical).

 

[pervect]

While the formulas above should now be stable (it took me a while to get some of the latex to come out correctly), and I have checked them against my worksheet for typos, they still probably need to be checked to make sure they actually satisfy Greg Egan's original differential equations. I haven't done that yet, I'm not quite sure of the best way to proceed.

 

[gregegan]

While the formulas above should now be stable (it took me a while to get some of the latex to come out correctly), and I have checked them against my worksheet for typos, they still probably need to be checked to make sure they actually satisfy Greg Egan's original differential equations. I haven't done that yet, I'm not quite sure of the best way to proceed.

Hi pervect, there's no need to check your formulas against the differential equation; because you're looking at a zero-width "hoop", not a finite-width ring, you can just plug them into the scalar equation I derived for that special case.

I did this, and they do satisfy the equation. It is also possible to solve the equation analytically for r in terms of r0 (a pretty messy cubic for r^2) or for r0 in terms of r (which I give explicitly, it's just a quadratic), but your way of parameterising the solution space has a nice physical meaning.

 

[pervect]

Great! Glad to hear we're getting the same results.

As far as the energy and momentum goes for the rotating hoop go

The energy (if geometric units are used, this is also the mass) of the hoop when it was not spun up was just

$$2 \pi r_0 \rho_0 A$$

where we've introduced A, the cross-sectional area of the hoop.

Using the results I got way back in post #10 (also in #19)

the energy of the spun-up hoop should be

$$2 \pi r \rho A (1+v^2)$$

A shouldn't change as the hoop is spun up, and we can use the expressions for [tex]\rho[/itex], r and v^2 as a function of n already derived.

If there were no tension in the hoop, we would see a factor of $\gamma^2$ term multiplying the density, rather than (1+v^2).

Similarly, the angular momentum should be just be p x r, where p is the linear momentum (the integral of the linear momentum density $$T^{0i}$$ ) or

$$L = 2 \pi r^2 v \rho A = 2 \pi r^3 \omega \rho A$$

Here the tension in the disk introduces terms which exactly cancel the usual factor of $\gamma^2$ so that it does not appear in the final expression.

 

[pmb_phy]

I'm not sure whether you addressed this or not but let me mention it just in case it didn't come up. Any physical loop must expand when it is spun up. You'd have to take a counter action to force the radius to maintain a radius of your desire.

Pete

 

[pervect]

If you spin up a hoop of constant radius, and mark two points on the surface of the hoop, you will find that the proper distance between these points increases at high enough relativistic velocities.

Thus a "Born rigid" hoop will have its radius contract when it is spun up, but of course such a hoop is not very physical.

Adding elasticity to the hoop requires a more detailed model, which Greg Egan has supplied, in the form of a hyperelastic hoop. Hyperelastic hoops are more physical than Born hoops, however the caclulations show they still shrink at high enough velocities, just as the less physical Born rigid hoops did.

Are hyperelastic hoops "physical"? It's hard to say, but in the region above, where the hoop is contracting, they still satisfy the weak energy condition, and they also still have a dynamic "speed of sound" in the material lower than 'c'.

We don't have any materials nearly strong enough to actually exhibit this sort of effect, however - one would need materials strong enough that their tension could approach their density.

I did some back of the envelope calculations for a carbon nanotube, for instance, and determined that elastically they'd stretch about 10% in length at their yield point, which would optimistically be 8-9 km/sec radial velocity. At this velocity there would be aroudn a part per billion relativistic contraction.

 

[pervect]

Here's one of many possible points where the radius just starts to shrink with increasing n or increasing omega, i.e. dr/dn = 0.

rho0=1
k = .5730994
n = .7

This represents a material where the velocity of sound is about 3/4 the speed of light

This gives rho = .737 and P = -.246, so it's within the weak energy condition by a fairly large margin.

I haven't fully explored the solution space (dr/dn=0). I've seen possibilities with a lower speed of sound in the material that are closer to violating the weak energy condition because of a higher stetch factor, and materials which have a lower ratio of P/rho that have a higher sound velocity.

 

[MeJennifer]

Thus a "Born rigid" hoop will have its radius contract when it is spun up, but of course such a hoop is not very physical.

There is no such thing as a Born rigid rotating hoop, not even in theory!

In an accelerating Born rigid rod all surfaces orthogonal to the direction of acceleration enjoy a coherent proper distance but an incoherent proper time. Alternatively an accelerating rod that maintains a coherent proper time between these surfaces would undergo stress and possibly break since the distances between the surfaces cannot be held coherent.

But in the case of a rotating hoop there are no surfaces of coherent distance or coherent time, so Born rigid rotation is not just practically but also theoretically impossible.

 

[pervect]

There is no such thing as a Born rigid rotating hoop, not even in theory!

In an accelerating Born rigid rod all surfaces orthogonal to the direction of acceleration enjoy a coherent proper distance but an incoherent proper time. Alternatively an accelerating rod that maintains a coherent proper time between these surfaces would undergo stress and possibly break since the distances between the surfaces cannot be held coherent.

But in the case of a rotating hoop there are no surfaces of coherent distance or coherent time, so Born rigid motion is not just practically but also theoretically impossible for a rotating hoop.

I think my answer to this (in case you didn't guess, I disagree) is pretty much on record in my earlier posts. Perhaps we can entice Greg Egan into giving his view on this matter. (And, perhaps not, we'll see.).

 

[gregegan]

Using the results I got way back in post #10 (also in #19)

the energy of the spun-up hoop should be

2 pi r rho A (1+v^2)

A shouldn't change as the hoop is spun up, and we can use the expressions for rho, r and v^2 as a function of n already derived.

Using the full elastic stress-energy tensor, I get the same result for the total energy of the hoop, in the centroid frame, as the approach you've outlined here. I've added an explicit formula for this (in terms of your nice parameterisation by n) to my web page, as well as some plots:

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

The plots show something interesting: as omega increases, the centroid-frame energy rises, reaches a maximum, and then falls, eventually dropping below the original total rest mass. So for high enough omega, there is effectively a kind of (negative) "binding energy" in this system which is greater (in magnitude) than the kinetic energy! In principle (if strong enough materials existed), it might be possible to extract energy from a system like this, though there might well be some pitfall -- beyond the requirement for ludicrously high breaking strains -- that I haven't identified.

 

[pervect]

Using the full elastic stress-energy tensor, I get the same result for the total energy of the hoop, in the centroid frame, as the approach you've outlined here. I've added an explicit formula for this (in terms of your nice parameterisation by n) to my web page, as well as some plots:

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

The plots show something interesting: as omega increases, the centroid-frame energy rises, reaches a maximum, and then falls, eventually dropping below the original total rest mass. So for high enough omega, there is effectively a kind of (negative) "binding energy" in this system which is greater (in magnitude) than the kinetic energy! In principle (if strong enough materials existed), it might be possible to extract energy from a system like this, though there might well be some pitfall -- beyond the requirement for ludicrously high breaking strains -- that I haven't identified.

OK, now *that* convinces me that something must be wrong, though I admit I don't know what it is.

I picked some random parameters, and also observed similar behavior to your plots.

A plot of L also shows that angular momentum eventually starts dropping as n decreases (meaning more stretch) along with the energy. In the particular case I looked at, this happened at exactly the same point that E started dropping. I suspect this isn't a coincidence. If the hoop is losing energy, it should be doing work, which means that it should be generating torque rather than requiring torque.

This peak in E and L did not happen at the same point that r started shrinking, however, it happened afterwards.

I suspect that there may be some problem with the hyperelastic model.
There may be some other issues with our assumption of circular symmetry, but I don't see how some sort of oscillatory instability could do anything to fix this issue - all it could do is create even lower energy states, and the problem we have is that we're getting energies lower than the rest energy.

 

[gregegan]

OK, now *that* convinces me that something must be wrong, though I admit I don't know what it is.

Though I was very surprised by this result, I'm not yet convinced that this behaviour is unphysical. It doesn't contradict any basic principle to be able to extract rest-mass energy from a system; it happens all the time in GR and nuclear physics. And if we're getting even halfway to the weak energy condition, what that says is that the energy density is bounded below by zero, not by the rest mass. There's a lot of territory in between the two. It makes sense that extreme tension will bring us in this direction, and I can't see any fundamental principle that stops us before we get at least close to zero.

One thing I've just been playing with is trying to get the same behaviour with rings, i.e. including the reality of a finite range of radii. The material would be under compression radially, so I've switched material models for the radial component to one that's better behaved for high compressions.

So far I haven't been able to show anything conclusive, but if I can come up with a trustworthy numerical solution for a ring exhibiting less total energy when spun-up than its total rest mass, I'll post the result.

I suspect that there may be some problem with the hyperelastic model.

So long as it stays clear of the weak energy condition, and the speed of sound is less than lightspeed, it's difficult for me to see how the hyperelastic model can be unphysical (although of course it's totally implausible for any real material I can conceive of).

 

[Ich]

Though I was very surprised by this result, I'm not yet convinced that this behaviour is unphysical.

OK, just ignore me if I'm talking nonsense - I'm harldy more than a layman.
But wouldn't this hoop refuse to take energy from the outside world? Wouldn't it show negative inertia?
If it were inhomogenous, wouldn't it lose energy to gravitational waves, spin up beyond any limit and finally collapse? Without a horizon to hide all the violations of various conservation laws?

 

[gregegan]

OK, just ignore me if I'm talking nonsense - I'm harldy more than a layman.
But wouldn't this hoop refuse to take energy from the outside world? Wouldn't it show negative inertia?
If it were inhomogenous, wouldn't it lose energy to gravitational waves, spin up beyond any limit and finally collapse? Without a horizon to hide all the violations of various conservation laws?

First, forget about gravitational waves or any other real-world forces. This is a thought experiment in special relativity, so by definition there is no gravitational force. The only force that we admit the existence of is whatever is responsible for giving tension to the material of the hoop. That force, and the material itself, is certainly behaving -- in detail -- like no real-world force I know of, in order to maintain a linear relationship between tension and the stretching of the material, no matter how much it is stretched. So in terms of the details of real-world physics, I don't doubt for a moment that the hypothesis is massively unlikely.

What we're contemplating is what SR predicts would happen if the force we have hypothesised as the source of the material's elasticity did exist.

There are no conservation laws being violated; the whole model is based on the stress-energy tensor having zero divergence, which is the definition of conservation of energy and momentum in SR.

I can't see any reason why the hoop would "refuse to take energy from the outside world", or refuse to give it up; you ought to be able to scatter particles off it in a manner that does either, just as you can donate or receive energy from any other rotating body that has states with either more or less energy still accessible to it.

My initial intuition when I saw this result was ... err, what if the hoop breaks and flies apart? Where does it get back the rest-mass energy it needs to do that? But of course the attractive force that's giving the material its ridiculous elasticity simply won't, and can't, let it "break and fly apart" at this point, any more than a helium nucleus can "break and fly apart". This isn't like the force that holds a bicycle wheel together.

 

[Ich]

First, forget about gravitational waves or any other real-world forces.

This was only meant to add some kind of friction - extract energy to spin up the hoop.

I can't see any reason why the hoop would "refuse to take energy from the outside world", or refuse to give it up; you ought to be able to scatter particles off it in a manner that does either, just as you can donate or receive energy from any other rotating body that has states with either more or less energy still accessible to it.

That's the point: I can't see any reason either, but this hoop's energy has an upper limit.
If you scatter particles that are faster than the hoop's particles, you would add energy. And you would increase the speed of the hoop's particles, therefore increase omega. Increasing omega means reducing the hoop's energy. (Does it increase angular momentum? That would be a contradiction, wouldn't it?)
Same thing if you decelerate the hoop. It behaves as if it had negative mass, which sounds unphysical to me.

What we're contemplating is what SR predicts would happen if the force we have hypothesised as the source of the material's elasticity did exist.

So if it did exist, it would lead to rather weird conclusions. I look forward to following your further discussion. Silently, of course, lest Chris Hillmann wishes to exclude the public.

 

[gregegan]

That's the point: I can't see any reason either, but this hoop's energy has an upper limit.
If you scatter particles that are faster than the hoop's particles, you would add energy. And you would increase the speed of the hoop's particles, therefore increase omega. Increasing omega means reducing the hoop's energy. (Does it increase angular momentum? That would be a contradiction, wouldn't it?)
Same thing if you decelerate the hoop. It behaves as if it had negative mass, which sounds unphysical to me.

The hoop itself wouldn't really have an upper limit to its energy; where the maximum occurs is in a plot of energy vs omega for a restricted set of states: those with perfect axial symmetry. I expect you could excite various modes of radial vibration if you hit the thing in the right way. (Actually, I guess it's conceivable that they might grab all the energy once you're over the hump; I'll have to try to look at that possibility more closely.)

The hoop doesn't really behave as if it has negative mass, any more than, say, a planet orbiting the sun does. Don't take this analogy too precisely, but a planet orbiting closer to the sun has greater angular and linear velocity than one orbiting further out, but less total energy. If you take orbital energy away from the Earth in a manner that keeps its orbit close to circular, it ends up moving faster. This doesn't violate any laws or cause anything terribly weird to happen, but it's certainly counterintuitive when you first come across it (or at least it used to confuse the hell out of me). And just as things get more complicated if you consider elliptical orbits for planets, things would get more complicated if we included radial vibrations of the hoop.

The angular momentum of the hoop goes up and down with the energy; it doesn't increase with omega when the energy is falling.

I can't see that there are any basic laws being violated by this scenario. You can't extract more energy than the rest mass, whatever you do. No quantity becomes infinite, unless you idealise the hoop to have no thickness at all, so it could shrink to a literal point. The weak energy condition is upheld. The only potential flaw I can think of is some form of instability; you could avoid any kind of azimuthal instability by making the hoop a thin cylindrical shell rather than an essentially planar object, but radial vibrations might not be damped the way they would in an ordinary hoop spinning under tension.

 

[pervect]

The hoop itself wouldn't really have an upper limit to its energy; where the maximum occurs is in a plot of energy vs omega for a restricted set of states: those with perfect axial symmetry. I expect you could excite various modes of radial vibration if you hit the thing in the right way. (Actually, I guess it's conceivable that they might grab all the energy once you're over the hump; I'll have to try to look at that possibility more closely.)

I had very similar suspicions about possible (probable) instabilities of circular motion when energy starts going down vs omega, but I wasn't sure how to explain why succinctly. It greatly simplifies the analysis to assume circular symmetry, and the Newtonian case is stable, so we've been assuming circular motion, but that's an added assumption.

My initial intuition when I saw this result was ... err, what if the hoop breaks and flies apart? Where does it get back the rest-mass energy it needs to do that? But of course the attractive force that's giving the material its ridiculous elasticity simply won't, and can't, let it "break and fly apart" at this point, any more than a helium nucleus can "break and fly apart". This isn't like the force that holds a bicycle wheel together.

A question: in its own frame, the hoop is elongating, not shrinking. And there is a limit to the elongation factor. Previously, we had been assuming that the hoop simply broke when the maximum elongation factor was exceeded. But now, you are suggesting that that can't happen. So what does happen when the hoop speeds up enough that n < n_min?

One final point: the Born rigid hoop suffers from some of these same issues. So the hyperelastic model may not be the problem.

 

[Ich]

The hoop itself wouldn't really have an upper limit to its energy; where the maximum occurs is in a plot of energy vs omega for a restricted set of states: those with perfect axial symmetry. I expect you could excite various modes of radial vibration if you hit the thing in the right way. (Actually, I guess it's conceivable that they might grab all the energy once you're over the hump; I'll have to try to look at that possibility more closely.)

Glad you feel uneasy about this limit too - I really don't trust my intuition when it comes to GR.

The hoop doesn't really behave as if it has negative mass, any more than, say, a planet orbiting the sun does. Don't take this analogy too precisely, but a planet orbiting closer to the sun has greater angular and linear velocity than one orbiting further out, but less total energy. If you take orbital energy away from the Earth in a manner that keeps its orbit close to circular, it ends up moving faster. This doesn't violate any laws or cause anything terribly weird to happen, but it's certainly counterintuitive when you first come across it (or at least it used to confuse the hell out of me). And just as things get more complicated if you consider elliptical orbits for planets, things would get more complicated if we included radial vibrations of the hoop.

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.

The angular momentum of the hoop goes up and down with the energy; it doesn't increase with omega when the energy is falling.

So somehow this whole scenario could be really consistent. I'm curious what will follow.

Thanks for the answers.

 

[gregegan]

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.

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.