Chris Hillman said:
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.