electromagnetism - An electromagnetic twist on Ehrenfest's paradox

Suppose you have two rings of equal radius exactly overlaid on each other. One of them has uniform charge density $+\lambda$ and other uniform charge density $-\lambda$. Clearly the charges will simply cancel and there will be no electric or magnetic fields anywhere.

Now suppose you start the positive ring rotating in place at relativistic speed. There will now be current flow and thus a magnetic field, but for simplicity I'm going to ignore that and just consider the Lorentz force on a charged particle $q$ at rest with respect to the non-spinning ring. Naively, I might expect the positive ring to get Lorentz-contracted and therefore appear to increase its linear charge density, thus creating a net outward electric field at $q$ and repelling it. But this can't be right, because the total charge across any fixed constant-time slice is both conserved and Lorentz-invariant, so it must stay zero. Why doesn't the positive ring Lorentz-contract and appear to gain charge and repel the charge $q$?

Answer

Let $\theta$ parameterize locations on the fixed ring, and let $a(\theta,t)$ be the acceleration vector of the point on the moving ring at location $\theta$ and time $t$. If the acceleration "looks the same" at every point in the lab frame --- that is, if, for every $t$ and every $\theta$, $a(\theta,t)$ is just $a(0,t)$ rotated by $\theta$ --- then clearly the ring cannont contract in the lab frame. (It suffices to think about the limiting case where every point on the ring jumps from speed $0$ to speed $v$ all at once.)

So what happened to the Lorentz contraction? Answer: Imagine a traveler on the ring, holding a short circular-arc-shaped meter stick that lies on the ring. According to that traveler, the front of his stick will have started accelerating before the back of his stick started accelerating. Therefore the stick has stretched. The observer in the lab frame sees no stretching, hence sees a shorter stick than the observer on the ring.

Bottom line: In the lab frame, the ring is Lorentz contracted, in the sense that any given small arc looks smaller than it looks to a traveler on the ring, but at the same time, that arc (and the entire ring) looks no smaller than it did before it started moving. Hence no change in the charge density.