Understanding the Impact of Angular Velocity on Mass in Rotating Objects

[sazzles]

Hi, I know that a moving object has a greater mass due to relativistic effects. This may be a stupid question, but will a rotating or spinning object's mass increase as its angular velocity increases?
Thank you for any help.

 

[Chen]

When the angular velocity increases so does the "normal" velocity of the object ($$v = \omega r$$). So yes, if you change the angular velocity the mass of the object will change as well.

But bear in mind that in constant circular motion, both $$\omega$$ and $$v$$ are constant. So the mass of the object changes, relative to you, only when it first started to rotate and when it stops.

 

[cookiemonster]

It's actual mass won't increase. It's rest mass is always constant.

The whole concept of "relativistic mass" of "effective mass" and "mass increasing" comes from the relativistic corrected momentum, i.e.

$$p = \gamma m_0 v$$

So an objects mass never increases.

However, an object with angular velocity is related to its linear velocity by

$$\omega = \frac{v}{r}$$

so it will experience relativistic effects for large enough radius.

cookiemonster

 

[sazzles]

Thanks, I think I understand now.
Just one small thing, I'm still confused about, does this mean the rest mass of an electron is its mass when it is not spinning, even though it is always spinning?

 

[cookiemonster]

Although there is a property of electrons called "spin," they are not actually spinning like you would spin a tennis ball. Spin as it applies to subatomic particles is a little bit more abstract.

And yes, an electron's rest mass never changes.

cookiemonster

 

[sazzles]

Oh dear, now I am totally lost. If electrons are not actually spinning, how does the Pauli Exclusion principle work?

 

[Chen]

They have a spin, but it's not the same kind of spin as that of a tennis ball or Earth.

Looks like someone thought this was an important enough subject to warrant its very own website:
http://www.electronspin.org/
(Ahh, sorry, apparently it's a site about a book with the same name.)

 

[sazzles]

Damn my textbook, it says

We can visualise spin as an electron rotating at a fixed rate

.
The link doesn't seem to work, are you sure it is right?

 

[sazzles]

oops, the link's fine, it's just my PC

 

[pmb_phy]

Originally posted by sazzles
Hi, I know that a moving object has a greater mass due to relativistic effects. This may be a stupid question, but will a rotating or spinning object's mass increase as its angular velocity increases?
Thank you for any help.

The mass of which you speak is relativistic mass which, as you indicate, is a function of speed.

If you have a body which has a spatial extent (i.e. it is not a point particle) and as such it is meaningful to speak of it in the classical sense of spining then each part of the body, except those parts on the axis of rotation, is moving while the body as a whole stays at rest. The mass of the body is the sum of the masses so you simply integrate (i.e. add up) the contributions of all parts of the body. The result will be a function of the body's angular velocity.

For an explicit example see
http://www.geocities.com/physics_world/sr/rotating_cylinder.htm

Chen wrote
When the angular velocity increases so does the "normal" velocity of the object..

sazzles was speaking of a spinning object, not an object which has a translational motion. That means, if I understood him correctly, the center of mass of the object remains at rest while other parts of the body move around the axis of rotation.

cookiemonster wrote
It's actual mass won't increase. It's rest mass is always constant... The whole concept of "relativistic mass" of "effective mass" and "mass increasing" comes from the relativistic corrected momentum, ... So an objects mass never increases.

sazzles question indicates that she is speaking of relativistic mass and not rest mass (aka proper mass). Which one you call "actual mass" is a matter of taste and the subject of a long going debate in physics. E.g. Many people prefer to think of relativistic mass as "actual mass" and think of "rest mass" as "intrinsic mass."

However since the translational motion of the object is zero it makes no difference which one sazzles is speaking of since, in this case, even when the body has no translational motion it is still spinning and if the body is spinning even its rest mass increases. This can readily be seen since the "rest mass" of a macroscopic body is given by the "total energy"/c² as measured in the zero momentum frame. Since a spinning body has more energy than the same body which is not spinning then it's "rest mass" increases as well. The mass derived in the above link and given in Eq. (8) is actually the rest mass of the object.

Also the electron's spin is not a classical spin in that different parts of the electron are moving.

 

[sazzles]

Wow, thanks pmb_phy for that detailed reply. It was really useful. One minor point, I'm a 'she' not a 'he', I thought my avatar and user name might have conveyed that.

 

[pmb_phy]

Originally posted by sazzles
Wow, thanks pmb_phy for that detailed reply. It was really useful. One minor point, I'm a 'she' not a 'he', I thought my avatar and user name might have conveyed that.

Oops! Sorry. I corrected it. And you're quite welcome. This topic came up elsewhere in this forum. I think someone was curious about this "mass of a moving body" thing compared to "mass = rest mass" and in investigating the difference they asked a similar question. So I expected this question to arise again so I did out an explicit example and placed in on my web site. I'm glad to see that it was useful.

 

[Chris Hillman]

When we spin up a CD, does its mass increase?

Hi, sazzles, you raised an interesting question way back in 2004:

Hi, I know that a moving object has a greater mass due to relativistic effects. This may be a stupid question, but will a rotating or spinning object's mass increase as its angular velocity increases?

Consider the question I raised in the title of this post: "When we spin up a CD, does its mass increase?" This question is analgous to "if we heat up a saucepan, does its mass increase"?

The best short answer to both questions is: yes! You might well ask: "doesn't this violate conservation of mass?" The answer is "no, because we did work when we spun up the CD and when we heated up the saucepan, and the energy we added slightly increased the effective gravitational mass".

Needless to say, these effects are very small when you use the CD drive in your computer or cook your dinner.

It's actual mass won't increase. It's rest mass is always constant.

Particle by particle, that is true. In these examples, however, if we think of the saucepan and CD as composed of atoms (cautiously trying to avoid quantum mechanics as far as possible, however), then we can see that when we heat the pan, we make the individual atoms move faster, even though the pan as a whole is sitting on our burner. The increased kinetic energy however adds to the gravitational mass we assign to the pan. Similar remarks apply to the CD.

In short, I agree that it is best to think of the "rest mass" (or better, the "mass", unqualified) of a particle as an invariant, and the so-called "relativistic mass" as representing the mass plus the relativistic kinetic energy, but in gtr (and in many competing relativistic classical field theories of gravitation), all forms of mass-energy gravitate, so it should not be terribly surprising that simply making the constituents move in any way we can without making the body as a whole move will increase its effective gravitational mass.

Chris Hillman

 

[tehno]

Gravitational field of the rotating object will increase.
Speaking about its' mass you have to accept or not to accept definition of a relativistic mass.

 

[pervect]

Measured in a frame in which the average momentum of the CD is zero, there is no difference between the invariant mass and the relativistic mass. Both of them increase.

Look at the defintion of invariant mass:

m = sqrt(E^2 - (pc)^2) / c^2

This defintion applies to systems in special relativity as well as point particles, if one just adds up the total energy E, and takes the magniutde of the total linear momentum vector, p.

(There are some unpleasant wrinkles that result from this when the system in question is not isolated, but we don't need to go into that, they don't affect the argument).

When you spin up the CD, you add energy to the system, while the momentum stays constant (zero), thus the invariant mass of the CD has increased.

 

[jtbell]

One minor point, I'm a 'she' not a 'he', I thought my avatar and user name might have conveyed that.

What avatar?

And your user name doesn't have any female connotation to me, which may simply mean that I'm old enough to be totally out of touch with your generation's vocabulary or culture. :uhh:

Speaking about its' mass you have to accept or not to accept definition of a relativistic mass.

You don't need the concept of relativistic mass in order to say that a rotating object has a larger mass than a non-rotating one. An extended object is a bound system of particles. The invariant mass (a.k.a. "rest mass") of each individual particle is related to its energy and momentum by

$$E^2 = (pc)^2 + (m_0 c^2)^2$$

which gives

$$m_0 c^2 = \sqrt { E^2 - (pc)^2 }$$

The invariant mass of the system is related to the total energy and total momentum in the same way:

$$m_0 c^2 = \sqrt { E_{total}^2 - (p_{total} c)^2 }$$

In the reference frame in which the object's motion is purely rotational, ${\vec p}_{total} = 0$ (remember momentum is a vector), so $m_0 c^2 = E_{total}$ in that frame. But you get the same value for $m_0 c^2$ in any inertial reference frame; it's simply easiest to calculate in the frame in which the object has no translational motion.

A rotating object has more non-translational energy (in the form of rotational kinetic energy) than a non-rotating one, so it has a larger invariant mass.

 

[selfAdjoint]

From this reply and Chris's I take it that the "increase in gravitational mass due to the added kinetic energy of the parts" is accepted terminology. But I was taught to make a careful distinction between the gravitating mass and the total mass-energy tensor, which includes other forms of energy, not to mention momentum-stress. In this terminology what would increase would be the energy, and that would indeed contribute to the curvature. Am I out of date? Too inflexible? What?

On this same topic, my copy of Diagrammatica arrived yesterday, and I discovered that Veltman, writing in 1994, uses ict in his Minkowski metric. Yeee Gods! He devotes part of an appendix to the issue, but here's what he says in the text:

Let us emphasize that there is no physics in the choice of metric. Some physicists prefer to work with real space/time but define their dot-product with a metric involving minus signs. It is really of no relevance where you hide your minus signs, at most it is a matter of convenience. Which is usually what you are used to. It is a matter though that you can debate hotly at lunch time (real time).

It occurs to me that this attitude, rigorously enforced, would shorten an awful lot of threads on thei forum.

 

[jtbell]

Chris can address the applicability of this to gravitational mass... I was thinking purely in terms of SR.

 

[RandallB]

The invariant mass (a.k.a. "rest mass") of each individual particle is related to its energy and momentum by

$$E^2 = (pc)^2 + (m_0 c^2)^2$$

which gives

$$m_0 c^2 = \sqrt { E^2 - (pc)^2 }$$

The invariant mass of the system is related to the total energy and total momentum in the same way:

$$m_0 c^2 = \sqrt { E_{total}^2 - (p_{total} c)^2 }$$

In the reference frame in which the object's motion is purely rotational, ${\vec p}_{total} = 0$ (remember momentum is a vector), so $m_0 c^2 = E_{total}$ in that frame. But you get the same value for $m_0 c^2$ in any inertial reference frame; it's simply easiest to calculate in the frame in which the object has no translational motion.

A rotating object has more non-translational energy (in the form of rotational kinetic energy) than a non-rotating one, so it has a larger invariant mass.

I disagree:
If your last formula is correct then the first could be revised to:

$$E^2 = (p_{total} c)^2 + (m_0 c^2)^2$$

BUT this leaves you in the same position as Newton’s error in debating Leibniz on “Vis-viva”. In effect your arguing the Energy disappears as the momentum does. This was resolved by the French (du Chatelet) in the mid 18th Cent.
Basically you cannot sum the vectors of the many momentums and then square them to obtain Energy (vis-viva) you must square each momentum and then sum all the energies (note what happens to the momentum vector sign when squaring).

Then you can see that $m_0 c^2 = E_{total}$ is Not True!

Rest Mass, Invariant Mass, and IMO Gravitational Mass as well; all remain the same no matter how fast the object spins, or how much Kinetic Energy is stored in the rotation.

So I agree with selfAdjoint that mass remains unchanged and therefore does not affect the gravitational mass. But I’m not convinced that accounting for a ‘spin energy’ will produce any additional gravitation or gravitational curvature from some form of an energy tensor. I would need to see some experimental evidence to be convinced of that.

 

[Chris Hillman]

Gravitational energy or mass?

From this reply and Chris's I take it that the "increase in gravitational mass due to the added kinetic energy of the parts" is accepted terminology. But I was taught to make a careful distinction between the gravitating mass and the total mass-energy tensor, which includes other forms of energy, not to mention momentum-stress. In this terminology what would increase would be the energy, and that would indeed contribute to the curvature. Am I out of date? Too inflexible? What?

I've read a lot of papers (especially recent ones) but I'm not a physicist by training, so take my reading for what it is worth. A notorious difficulty with constructing thought experiments in gtr is that all forms of mass-energy gravitate, so to model a CD which is originally nonspinning and which is then spun up to some new spinning equilibrium state, one would presumably have to model something like a laser pulse coming in from infinity, striking the rim, and transfering momentum, which would be rather awful. Even in this case, since we are discussing changing the geometry of spacetime (i.e. changing the gravitational field), it can be tricky to compare the initial and final states "location by location". Nonetheless, I suspect that most researchers would speak of "augmented active gravitational mass" of the CD rather than "augmented gravitation due to the augmented kinetic energy of the bits of matter in the CD".

Hmm... here's a new conundrum: is the inertial mass of an operating CD player greater, due to the effect just mentioned? (I am thinking of a battery-operated model.)

On this same topic, my copy of Diagrammatica arrived yesterday, and I discovered that Veltman, writing in 1994, uses ict in his Minkowski metric. Yeee Gods! He devotes part of an appendix to the issue, but here's what he says in the text:

It occurs to me that this attitude, rigorously enforced, would shorten an awful lot of threads on thei forum.

Ye gods indeed! This particular choice of notation is now so nonstandard that it is likely to annoy many of his readers. History shows that otherwise valuable books which insist on employing highly nonstandard notation have less impact than they would have otherwise, so I feel his choice may prove to be self-defeating. This is actually quite a serious issue for any student who might plan to adopt his notation and who might later try to read some gtr literature. I have in mind important techniques like NP tetrads, where the notation is notoriously delicate, and I can think of many other places where huge changes in notation would make it much harder to avoid sign errors in "translating" from a paper by Chandrasekhar, say, to the notation preferred by Veltmann. It's hard enough to avoid errors of this kind even if one does not make changes of the kind he advocates!

I happen to feel that there are many good reasons for prefering real variables here, quite apart from the issue of consistency with the huge literature on gtr, almost all of which uses the terminology and notation of semi-Riemannian geometry. I admit that the standard terminology does lead to some common student misconceptions, e.g. the "metric" structure we are imposing to form a Lorentzian manifold (by bundling an indefinite but nondegenerate quadratic form) certainly does not play well with "metric topology" as in elementary courses on general topology, unlike Riemannian manifolds (obtained by bundling a positive definite quadratic form), but these misconceptions are easily cleared up.

Chris Hillman

 

[pervect]

Some insight might be able to be gained from considering the case of black holes, rather than CD's.

It's rather well known (see for example MTW, pg 913) that

$$M^2 = \left( M_{ir}+ \frac{Q^2}{4 M_{ir}} \right)^2 + \frac{S^2}{4 M_{ir}^2}$$

M is the mass of the black hole
Q is the charge of the black hole
S is the angular momentum of the black hole
$$M_{ir}$$ is the irreducible mass of the black hole

(A purist might additionally specify which particular concept of mass was meant by M, i.e. Komar mass, ADM mass, etc., but MTW doesn't.)

There are some labels on the equations, with parts labelled 'irreducible contribution to mass', 'electromagnetic contribution to mass', and 'rotational contribution to mass'.

So it's pretty clear that rotation contributes to mass in some general sense, as per the part of the equation labelled 'rotational contribution to mass'. Note also that charge also contributes to mass.

As far as interaction with the environment goes, it does require interaction to 'spin up' a black hole. The form of the equation suggests that one consider such spin-ups that are performed so as to keep $M_{ir}$ constant. As MTW says "A black hole transformation that holds fixed the irreducible mass is reversible.", so we can call this sort of spin-up a "reversible" spin-up process.

While this is nice as far as it goes, and does give some insight, it doesn't totally answer all the original questions.

 

[Chris Hillman]

Hi, pervect, MTW is sketchy here because the concepts of "black hole mechanics" were very new concepts when the book was published in 1973. The irreducible mass is DEFINED by $$M_{\rm{irred}}^2 = 1/2 \, \left( M^2 + \sqrt{M^4-J^2} \right)$$; see Wald, General Relativity, section 12.4. The popular book by Wald also discusses this topic. And by a happy coincidence, there is also this brand new review paper: http://www.arxiv.org/abs/gr-qc/0611129 I haven't had a chance to read that yet, but it should be useful.

Chris Hillman

 

[Los Bobos]

I don't want to be annoying, but there is a reason why energy-momentum tensor is called energy-momentum tensor and not mass scalar.

 

[Chris Hillman]

Hi, Los Bobos,

I can't tell whether or not you were commenting on what I wrote, but if so, can you describe your objection more specifically?

Thanks,

Chris Hillman

 

[Los Bobos]

Hi Chris, this was related to the difference between the invariant mass and the Newtonian momentum. Equivalent effects do not mean equivalent physics in every case. And again let's use only invariant mass. And 4-momentum :).

 

[aeroboyo]

I have two questions... I'm new to this so sorry if these sound silly...

If a disk is rotating then there's only a relativistic change in mass for an observer when the object is accelerating and deccelerating right... but not when its spinning at a constant speed? Also, when the objects 'relativistic mass' changes, does that change the curvature of spacetime around the object? Because mass curves spacetime. Or is there just a kind of relativistic change in spacetime geometry due to this relativistic change in mass... Or no change at all?

 

[pervect]

I have two questions... I'm new to this so sorry if these sound silly...

If a disk is rotating then there's only a relativistic change in mass for an observer when the object is accelerating and deccelerating right... but not when its spinning at a constant speed? Also, when the objects 'relativistic mass' changes, does that change the curvature of spacetime around the object? Because mass curves spacetime. Or is there just a kind of relativistic change in spacetime geometry due to this relativistic change in mass... Or no change at all?

A disk spinning at a constant speed has a constant mass, if that's what you're asking - at least if the disk is an isolated system.

Have you read any of the FAQ's about mass? Such as http://math.ucr.edu/home/baez/physics/Relativity/SR/mass.html
?

It seems to me that we've covered some of your questions in this thread already, but perhaps we are talking past each other. If you read some of the FAQ's, they might answer your questions.

As for the gravitational field of a moving object, you might try reading the thread https://www.physicsforums.com/showthread.php?t=144817. For technical reasons, it's much easier to talk about the tidal gravity of a moving object, which is the approach taken in this thread.

 

[pervect]

Hi, pervect, MTW is sketchy here because the concepts of "black hole mechanics" were very new concepts when the book was published in 1973. The irreducible mass is DEFINED by $$M_{\rm{irred}}^2 = 1/2 \, \left( M^2 + \sqrt{M^4-J^2} \right)$$; see Wald, General Relativity, section 12.4. The popular book by Wald also discusses this topic. And by a happy coincidence, there is also this brand new review paper: http://www.arxiv.org/abs/gr-qc/0611129 I haven't had a chance to read that yet, but it should be useful.

Chris Hillman

I took a quick look at the writup in Wald, but it didn't particularly inspire me :-(.

I have some general comments, though that hopefully will clarify some of my remarks.

The first is that on physical grounds, if we have a CD that can be "spun up" and "spun down" in a reversible fashion, I expect that there should be two specific functions that characterize the CD. One function should give its total angular momentum J as a function of omega (the rotational velocity of the CD), and another function that gives its total energy E as a function of omega.

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).

So I thought that the "reversibility" comment in MTW was quite illuminating as an important quality that the CD's must have to make the analysis simple. This basically says that some abstract model of the CD exists and that the only important parameter in this model is the angular velocity omega, given this single parameter it is possible to find the total energy and total angular momentum of the CD.

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. I recall reading a bit about this issue in Held "General Relativity and Gravitation", apparently there are some tricky aspects, but I don't recall the details. Of course we can always take the SR limit if there are problems. Maybe you can comment on this issue?

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.

Modelling the CD is going to be tricky - we can't use a Born rigid CD, for instance, because that sort of rigidity doesn't allow rotation. (This is also mentioned in the sci.physics.faq).

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.

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.

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.

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 final issue is that even in special relativity (SR), the mass of a system is well behaved (defined in a manner that's indepenent of the frame of reference) only when the system is isolated. During spin-up, the CD isn't going to be isolated, so I don't think there will be a well-defined reference-independent notion of the mass of the CD alone (though there could be a defined notion of the mass of a larger system which includes the CD as a part).

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 energy-momentum of an object with finite volume is not a covariant physical entity because of the relativity of simultaneity.

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.

 

[Chris Hillman]

Talking past one another?

Hi, Los Bobos,

this was related to the difference between the invariant mass and the Newtonian momentum. Equivalent effects do not mean equivalent physics in every case. And again let's use only invariant mass. And 4-momentum

Sorry, I still can't understand what you are trying to say, but it seems possible that we do not actually disagree about anything. I and others have argued that in the context of str it is best to follow the mainstream and use "mass" (or as some would say, "invariant mass") and "kinetic energy" rather than "relativistic mass". But in gtr, as one might expect, everything acquires several more layers of subtlety.

I have the impression that some of the confusion over what I was saying (?) might arise from failure to recognize that there is nothing inconsistent in saying in effect that the most appropriate notion of "mass" (and "angular momentum") of an isolated object in gtr might not be additive. This shouldn't be surprising if you recall that the field equation of gtr is nonlinear.

Chris Hillman

 

[Chris Hillman]

Original 2004 thread on str; in 2006 I added a gtr twist

Hi, jt,

Chris can address the applicability of this to gravitational mass... I was thinking purely in terms of SR.

I now wish I had taken pains to emphasize that I was resurrecting a thread from 2004, which concerned str, because I wanted to add an interesting gtr twist. It seems that this omission has caused considerable unneccessary confusion, for which I apologize.

Chris Hillman

 

[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

$$M=m+\frac{1}{4} S^{ab} \Omega_{ab} + \Phi$$

$\Phi$ 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.