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If I take a rod of some radius $r$ and length $L$, and I spin this rod with angular velocity $\omega$. How would the geometry of the rod appear to an observer as one converges to $c$? What are the consequences of this to, say, electrons in a solenoid?

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Angular velocity $\omega$ has different dimension from light speed $c$, hence can't really approach it. You want $\omega \rightarrow c r$? – qazwsx Jan 12 '13 at 3:17
@user1664196 Yeah, I'd be happy with whatever the limiting speed is for angular velocity (there must be one, right?). – Kevin Jan 12 '13 at 3:24
@Raindrop Right, I noticed the paper you referenced, but it doesn't help with the specific questions I have: How would the geometry of the rod appear to an observer as one converges to c? What are the consequences of this to, say, electrons in a solenoid? – Kevin Jan 12 '13 at 3:50
Check this research out: [Relativistic Hall Effect][2] "We examine manifestations of the relativistic Hall effect in quantum vortices and mechanical flywheels and also discuss various fundamental aspects of this phenomenon." quoted from the abstract. [2]: – raindrop Jan 12 '13 at 6:54
Relativistic contraction and related effects in noninertial frames – raindrop Jan 12 '13 at 7:20

I believe the rod would appear to bend backwards against the direction of rotation in a line where the mass is travelling near the relativistic limit. As the rod is rotated the faster the bend moves inward, as it slows down the bend moves outward until it is straight again. It remains bent at relative speed where the radius and angular velocity at the limit. Radius becomes distorted (shortened) to maintain the velocity limit.

If not viewed top down, but head on, it will will recede on one cycle and approach on the other like Einstein's clock.

I'm not a physicist, but I play one on the internet.

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Based on Relativistic description of a rotating disk O. Gron, Am. J. Phys. 43, 869 (1975), DOI:10.1119/1.9969 [I got the link from Wikipedia references on the Ehrenfest article] I think that for a rod instead of a disk: An observer S ("momentarily at rest relative to the disk") "measures an elliptical shape for the" path of the tip of the rod, "and finds that each point of it describes a cycloid-like path, while its center moves along a straight line with constant velocity. S' ("an accelerated observer ... rotating with the" rod) observes a rod at rest, while the surroundings are rotating. He measures a circular shape for the path of the tip of the rod.

I have no idea how electrons would behave in a solenoid coiled around a rod with angular acceleration, $\omega \rightarrow cr$.

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