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?
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, $\omega \rightarrow cr$.