OK I thought about this some more, and I think I may know how it works. (Chronos will probably pipe in and tell me I’m wrong.)

The issue at hand is that when you are up near c, differences that would be significant when an object is not moving (say one side of the disk moving forward at 1m/sec and the other side moving backward at 1m/sec) are no longer significant because everything is moving at near c, not c + 1m/sec or c - 1m/sec.

For rotational ease let V the forward velocity of the disk (very close to c) and R be the rotational velocity of the edge of the disk V>>R.

Scenario 1, The axis of rotation is perpendicular to the direction of motion, like a drill

If we look at the atoms on the rim of the disk. They are all moving at the same speed with v in the direction of travel and r perpendicular to the direction of travel. If Newton’s laws applied then the total velocity vector would be sqrt(V^2+R^2) approximately equal to V+(r^2)/V. However since we are in a relativistic realm, the result of adding the additional r velocity will be much slower. So while we would observed the forward velocity v, the r velocity would appear very slow. Thus the disk would appear to be rotating much slower, almost as if time slowed down for the disk. Hmmn, didn’t I remember reading about time relativistic time distortion? That time appears to slow down as they move faster? It all works out to the same thing.

Scenario 2: The axis of rotation is perpendicular to the velocity, like a Frisbee.

For an atom of the front or back of the disk, the direction of rotation will be perpendicular to the direction of motion so you’ll get the same result, a general slowing that can be represented by a time distortion. However on the left and right edges, the nominal speed would be V+R much bigger jump than when it was perpendicular. Thus, to keep everything below c, the edges apparent velocity has to slow down even more, so that the side of the disk is just barely creeping forward relative to the center of the disk. Also for similar reasons the other side of the disk is also creeping backwards. Now in order to have the disk uniformly rotate s, while having the left and right sides move much slower than the front and back , without having a massive pile up somwehere in between, the atoms on the side must have a shorter distance to travel to rotate than the ones on the front and back, so rather than a circle, the disk must turn into an oval with short sides and a broad middle. Hey, didn’t I remember reading about relativistic distance distortion where things get crunched in the direction of travel? Again it all works out the same.

For angles in between the two the shape of the disk will be more smooshed the closer it is to a frisbee, and less smooshed the closer it is to a drill, while the rate of rotation will be uniformly slower regardless of its orientation.