Concerning the relatisticly rotating and translating wheel:
Two comments:
1. To the observer translating relative to the
wheel, each part of the wheel is in a different reference frame.
2. Special relativity does not really apply to a
any part of the rim of a rotating wheel because each piece is
accelerating, and NOT an inertial frame.
But it is an interesting problem. On first thought,
the path to the solution would seem to be to consider a non- rotating
wheel next to the rotating one ( or a non-rotating groove containing the
rotating ring with a very small clearance). Thus it would seem
that the shapes of the rotating and non-rotating wheels must vary in the
same way. BUT there is the problem of simultaneity, like the problem of
something longer than the barn in one frame being momentarily enclosed by
the barn in another frame.