What are the consequences of relativistic angular velocities? 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?
 A: 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, $\omega \rightarrow cr$.
A: 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.
