What does a rotating hoop, with each point moving at a velocity close to the speed of light, appear like with respect from a stationary observers perspective. For example how does the shape of the hoop change? (Note, I'm not intending to ask about optical effects, but rather what physically happens, analogous to length contraction of a moving train.)
By symmetry, it'll look circular. Just look at it from above, along the axis. You can synchronize clocks along the rim by a signal from the center. Then have those rim clocks all emit signals simultaneously. In the rest frame, their signals will arrive at the same time as those of stationary clocks positioned around the "orbit".
Or imagine encasing the whole thing in a hollow toroidal container at rest. Like the proton bunches going around the LHC. No ambiguity about what the encasing torus looks like.