Understanding the Impact of Angular Velocity on Mass in Rotating Objects

[Chris Hillman]

Spinning up, spinning down

Hi, pervect,

I thought about these issues some years ago, and wish I could better recall my cogitations off the top of my head. I do remember that my conclusion was that I could not find a single paper or book which fully recognized the depth of the problems, which involved boundary conditions, reversibility, and subtle problems which some might characterize as philosophical, although I would say "operational", regarding sensibly interpreting attempted comparisions of "before" and "after" states.

I would only expect this happy state of affairs to apply if the spin-up and spin-down process is reversible. Examples of how this reversibility assumption can fail would include plastic flow in the CD due to stress (making the state of the CD a function of its history), or if the spin-up or spin-down process heated up the CD (then the state of the CD would include its temperature).

It would be very hard to spin up any object as if it were a rigid body, so if one actually tried to model the process of spinning up or spinning down, dealing with elasticity seems unavoidable. As you know (we discussed this at Wikipedia), a relativistic treatment of elasticity requires some care even in str, since Hooke's law is manifestly inconsistent with str.

As I think about this, I am making another important assumption. This is that it is possible to define angular momentum in GR. I think this is true, but I don't actually know the details offhand.

You've discussed Komar integrals, so you must know that I had in mind an isolated object so that our spacetime model is asymptotically flat. In the case of a stationary asymptotically flat spacetime, as you know, mass and angular momentum can be defined. In the case of dynamic asymptotically flat spacetimes, things get a bit tricky since one wants to be able to track energy sent in "far away" to affect the object of interest, while acknowledging that the total mass-energy should be conserved. In non-asymptotically flat spacetimes, everything becomes much more tricky. Here too it helps to recall that the EFE is nonlinear, so we should expect it to be difficult to unravel everything with the facility of Newton.

For other readers, the textbook by Carroll is a good source of information. A good first way to understand the basic idea is to consider the far field of the Kerr vacuum solution (with the standard parameterization of the metric functions, written in the usual Boyer-Lindquist chart) and to recall how one identifies the two parameters as mass and specific angular momentum by comparing with Newtonian gravitation.

Because CD's are more complicated than black holes, as far as I know nobody has actually written down any specific functions for J(omega) or E(omega) given some sort of specific "model" of the CD. The sci.physics.faq on the rotating rigid disk, for instance, does not give any references and in facts asks anyone who finds one to write to the author.

Are we talking about the essay by Michael Weiss which can be found for example at http://www2.corepower.com:8080/~relfaq/rigid_disk.html or http://www.math.ucr.edu/home/baez/physics/Relativity/SR/rigid_disk.html? That does have references.

I could give an exhaustive list of references, but the most important one to get started with is probably the review paper by Gron; the citation is given both in Michael's essay and in http://en.wikipedia.org/w/index.php?title=Ehrenfest_paradox&oldid=58681705

I know there are some papers with relaxed defintions of rigidity that do allow rotation, but I haven't read them. (I stumbled across them looking for elementary English-language treatments of Born rigidity). I'm not sure how well accepted these defintions are.

There's no royal road: you need to read many papers with a very critical eye. Unfortunately, as I have already remarked at Wikipedia, quite a few papers in this area seem to consist of recommitting old errors which were cleared up decades ago.

Another approach to a CD model would be to write down what the mechanical enigineers call "constituitive relationships" between stress and strain. Given these constitutive relationships, one should be able (in principle) to calculate J(omega) and E(omega). But I'm not aware of anyone who has done this.

Well, now I am confused, since that is one approach which we were discussing in Wikipedia! I think your functions above would pretty much have to come from a relavistically reasonable modification of constitutive relationships.

While we haven't said much about these functions, I think we can safely say that E is going to monotonically increase with omega for any reasonable reversible CD model, which means that the mass (E/c^2) of the CD is going to monotonically increase with omega.

At least at first, yes, that's what I 'd expect, and of course the energy would have to be supplied. I forgot to stress that it would probably help greatly to begin with weak-field theory. I expect that dealing with elastic or plastic deformations of the disk as it is spun up would be quite challenging enough even in that context.

I am also suggesting that we can import some of the language that is used
in textbooks to talk about how the mass of black holes changes with its "spin" (angular momentum) as a guide to how to talk about how the mass of a CD changes with its angular momentum, because there is literature that talks about the former (and not much literature that talks about the later).

One reason why I have tried to partially redress this imbalance.

This is a rather subtle point that may cause some argument, so I'm going to give a reference:

http://arxiv.org/abs/physics/0505004

The stress-energy tensor is always covariant. The energy-momentum of a physical system with a finite volume is covariant only if the system is isolated.

Yes, I hope that everyone here understands the local vs. global distinction, and is aware of the standard remarks in MTW and other textbooks about trying to integrate the stress-energy tensor over some region, which gets very tricky once we leave the domain of weak-field theory.

Chris Hillman

 

[pervect]

Hi pervect,

I thought about these issues some years ago, and wish I could better recall my cogitations off the top of my head. I do remember that my conclusion was that I could not find a single paper or book which fully recognized the depth of the problems, which involved boundary conditions, reversibility, and subtle problems which some might characterize as philosophical, although I would say "operational", regarding sensibly interpreting attempted comparisions of "before" and "after" states.


It would be very hard to spin up any object as if it were a rigid body, so if one actually tried to model the process of spinning up or spinning down, dealing with elasticity seems unavoidable. As you know (we discussed this at Wikipedia), a relativistic treatment of elasticity requires some care even in str, since Hooke's law is manifestly inconsistent with str.

I recall reading something in Wikipedia about that topic, now that you mention it, but I don't think I was involved directly in that discussion. Perhaps you could post a link so I could refresh my memory? Google didn't find it for me.

You've discussed Komar integrals, so you must know that I had in mind an isolated object so that our spacetime model is asymptotically flat.

I probably should have known that, but I didn't. Doing some more reading clears up the issue for me, so that I now see that we can define angular momentum using the Komar approach without any ambiguity, as long as we have a stationary space-time. So that clears that issue up.

Are we talking about the essay by Michael Weiss which can be found for example at http://www2.corepower.com:8080/~relfaq/rigid_disk.html or http://www.math.ucr.edu/home/baez/physics/Relativity/SR/rigid_disk.html? That does have references.


I could give an exhaustive list of references, but the most important one to get started with is probably the review paper by Gron; the citation is given both in Michael's essay and in http://en.wikipedia.org/w/index.php?title=Ehrenfest_paradox&oldid=58681705

Yes, that's the essay I was talking about, the one with the following quote:

To settle the question definitively, it seems one has to perform a full-blown, hairy GR calculation. Perhaps someone has done this; perhaps someone has turned the vague notion of "infinitely rigid" into a formula for a stress-energy tensor, plugged that into the Einstein field equations, and solved. If the Gentle Reader knows of a reference, please let me know.

I missed the Gron reference somehow, I'll put it on my list of things to look at the next time I make it to a library with access.

Well, now I am confused, since that is one approach which we were discussing in Wikipedia! I think your functions above would pretty much have to come from a relavistically reasonable modification of constitutive relationships.

I do recall reading some of your remarks about this, now that you mention it, but as I said, I don't think I was directly involved in that discussion (if I was, I have forgotten it :-(). I should probably refresh my memory before commenting further, do you recall exactly where this was discussed.

Yes, I hope that everyone here understands the local vs. global distinction, and is aware of the standard remarks in MTW and other textbooks about trying to integrate the stress-energy tensor over some region, which gets very tricky once we leave the domain of weak-field theory.

That's probably too much to hope for -- some of our readers are still struggling with outdated notions of relativistic mass and have probably not quite grasped the fact that there is no such thing as an absolute velocity. Of course, many other of our readers are much more advanced. Such a wide audience and difference in backgrounds makes discussion difficult.

 

[Stingray]

Probably the best general framework for discussing these issues is Dixon's formalism. This was developed back in the 1970's to treat extended bodies in GR without approximation. The theory defined a momentum vector with particularly nice properties for any stress-energy tensor with bounded support satisfying the appropriate conservation law (with both gravitational and electromagnetic fields in general).

The norm of this momentum was naturally identified as a mass. Call it M. One may uniquely write this in the form

<br /> M=m+\frac{1}{4} S^{ab} \Omega_{ab} + \Phi<br />

\Phi is a scalar which is naturally interpreted as the body's gravitational potential energy. S^{ab}=S^{[ab]} is the angular momentum tensor defined naturally in the theory, and \Omega_{ab}=\Omega_{[ab]} may be interpreted as the body's "mean angular velocity." It is defined in terms of the angular momentum and an inertia tensor defined from the body's quadrupole moment.

m is probably the most important quantity here, and is identified as the "total internal energy. There is a sense in which it is a "minimum energy" of the body, but it's complicated to explain.

In general, none of these components are constant (nor is their sum). But it may be shown that all objects which are rigid in an appropriate sense will have constant m. Rigidity here is defined to be condition whereby all of the object's multipole moments remain constant in an appropriate corotating frame. This never violates stress-energy conservation, but such objects are likely to contain singularities in general, which would violate the conditions under which all of this formalism was derived.

Edit: All of this may also be related to the ADM mass if the spacetime is stationary. But I'd have to look up the paper for details.

 

[Chris Hillman]

Quadrupole moment?

Hi, Stingray,

I take it that this is in the context of asymptotically flat spacetimes? If you give the citation I will make some effort to find his paper.

Chris Hillman

 

[Stingray]

I take it that this is in the context of asymptotically flat spacetimes? If you give the citation I will make some effort to find his paper.

No. The formalism does not depend on asymptotic flatness. Everything is defined (quasi)locally. The quadrupole moment ends up being defined as an integral over a particular spacelike hypersurface which involves the stress-energy tensor and various geometric quantities.

Unfortunately, the formalism is quite involved and spread over several long papers. The simplest starting point is probably Ehlers and Rudolph, Gen. Rel. Grav. 8, 197 (1977). After that, you might want to look at Dixon, Proc. R. Soc. London A314, 499 (1970). The main work was done in Dixon, Phil. Trans. R. Soc. London A277, 59 (1974). There is also a review article by Dixon in the conference proceedings "Isolated Gravitating Systems in General Relativity;" published in 1979 (and edited by Ehlers). Some further improvements have been made since, but the vast majority of the material is contained in these papers.

 

[Chris Hillman]

Thanks, stingray!

Chris Hillman