What happens to a ball spinning with peripheral speed near to the speed of light? I can't imagine such phenomenon. Would it becomes an ellipsoid, or maybe a straight line?
 A: What you are referring to is a special case of the Ehrenfest Paradox

In its original formulation as presented by Paul Ehrenfest 1909 in
  relation to the concept of Born rigidity within special relativity,1
  it discusses an ideally rigid cylinder that is made to rotate about
  its axis of symmetry. The radius R as seen in the laboratory frame is
  always perpendicular to its motion and should therefore be equal to
  its value R0 when stationary. However, the circumference (2πR) should
  appear Lorentz-contracted to a smaller value than at rest, by the
  usual factor γ. This leads to the contradiction that R=R0 and R
  
  The paradox has been deepened further by Albert Einstein, who showed
  that since measuring rods aligned along the periphery and moving with
  it should appear contracted, more would fit around the circumference,
  which would thus measure greater than 2πR. This indicates that
  geometry is non-Euclidean for rotating observers, and was important
  for Einstein's development of general relativity.

