What would happen if something *rotated* with a tangential velocity near speed of light? There's length contraction and time dilation when traveling in a straight line, but what if a spherical object like a white dwarf through some interaction rotated at a speed near the speed of light? What would be the effects? Would it appear smaller from all directions or just along its equator? Would time slow down around it or just its surface? Or would nothing happen at all because angles aren't dimensional units? 
 A: Let us consider a stationary observer hovering on the equatorial plane of the rotating spherical object. SR (special relativity) is applicable locally, i.e. comparing the observer to an inertial reference frame instantaneously at rest with a limited region of the rotating body.  
In the equatorial plane, for the limited regions rotating along a direction normal to the radial distance to the body (in front of us) we measure a length contraction. However for the limited regions rotating along a direction approximately parallel to the radial distance there is no contraction as measured transversally. That means that the transversal outer dimension of the body as measured by the observer is unchanged. As for the outer dimension measured along the rotation axis, in that direction there is no velocity component, so no contraction either.
So, from an equatorial plane the outer dimensions of the rotating sphere do not change.  
If the observer hovers along the rotating axis, for similar considerations,  no change in the outer dimensions either.  
As for time dilation it occurs according to the local Lorentz factor, that is time runs slower on the surface of the star due to the speed. No time dilation on the axis of rotation as the speed is nil there.  
Note: In what above I assumed the dimension of the rotating body negligible compared to the distance observer-body. The curvature of spacetime negligible as well.
