This question made me curious. It refers to a ball which is rotating so that its equator is approaching the speed of light, then talks about what it would look like if the observer moved. What I'm curious about is if the ball itself could move much at all. If the edge is already moving very near the speed of light, would its rotation have to slow down for the whole ball to accelerate significantly in any direction?
For clarification, the way I'm thinking of this is that you'd have a ball (made of unobtanium - this questions isn't about whether or not it's possible) spinning so that the top speed of any point on it is very close to $c$ and otherwise not moving. Then you give it a strong shove (parallel to a line from its center to the "equator" of spin, for simplicity) while it continues to spin. Then, instead of all points along its equator moving at an equal speed, the side spinning towards the direction of travel would be changing position faster than the side spinning away from that direction. Assuming your shove was strong enough, that could push the forward-spinning edge across the speed of light.
Of course that can't happen, so the question is what would happen? Would the spin of the ball need to slow down to compensate, so that no part of it travels faster than $c$? Would it resist being pushed at all, the same way a gyro resists when you try to move it? Something else entirely? Am I even envisioning the physics of it correctly? If not, please give an explanation of where my reasoning is wrong.