Deriving Møller's Relativistic Minimum Radius for Rotating Bodies

[michael879]

Can someone either derive or point me to a derivation of Møller's formula for the relativistic minimum radius of a rotating body? I've been searching for about an hour and it's driving me crazy!

The only "minimum radius" equation I've seen imposes the speed limit c on a classical rotating body (v = L/mR < c). This is only semi-relativistic since it takes L=r x p and p=mv. The two results are identical, but from the references I can find Møller's formula is supposedly fully relativistic...

 

[michael879]

anyone? I wouldve expected this to be a quick answer.. To bad I posted this seconds before they shut the site down for migration >.<

 

[pervect]

Try:
C. Moller, Commun. Dublin Inst. Adv. Stud., A5, 1 (1949).
C. Moller, ”The Theory of Relativity”, 2nd ed. (Oxford University Press, 1972), p. 176.

The second (textbook) reference will probably be easier to find. The secondary source for these references was http://www.phys.lsu.edu/faculty/oconnell/PDFfiles/311. Rotation and Spin in Physics.pdf in the section quoted below.

This is also connected to the fact that, as Moller has shown [12, 13], a spinning body has a minimum radius equal to ...

 

[michael879]

Thanks pervect, I did come across Moller's stuff but I don't have access through my school (they're too old). I'll check out that second source tho

 

[michael879]

>.< yea your source is just like all the ones I found! They reference Moller's work but only show the conclusion...

 

[George Jones]

See page 338 of the wonderful, advanced book "Special Relativity in General Frames" by Gourgoulhon

https://www.amazon.com/dp/3642372759/?tag=pfamazon01-20

https://www.amazon.com/dp/3642372759/?tag=pfamazon01-20

 

[michael879]

See page 338 of the wonderful, advanced book "Special Relativity in General Frames" by Gourgoulhon

https://www.amazon.com/dp/3642372759/?tag=pfamazon01-20

https://www.amazon.com/dp/3642372759/?tag=pfamazon01-20

Thanks, but I'd really rather not buy an entire textbook for the answer to this one question!

 

[michael879]

Thanks, but I'd really rather not buy an entire textbook for the answer to this one question!

Sorry, let me clarify that a little:

I already have plenty of textbooks on relativity, so buying a new one just for this question seems ridiculous. If there really is no accessible derivation can somebody who has the textbook just sketch the argument out for me?

 

[vanhees71]

Does this already help?

https://en.wikipedia.org/wiki/Center_of_mass_(relativistic)#M.C3.B8ller_world-tube_of_non-covariance

 

[michael879]

Does this already help?

https://en.wikipedia.org/wiki/Center_of_mass_(relativistic)#M.C3.B8ller_world-tube_of_non-covariance

Unless I'm missing something, that just uses the result I am looking for a derivation of. I can't think of any reason why a spinning object would have a minimum radius in a fully relativistic non-quantum theory