As a result of the Ehrenfest Paradox, the geometry of a rotating disc is non-Euclidean.
However, while reaching this conclusion, we assumed that "the radius doesn't undergo Lorentz contraction", because "the radius is always perpendicular to the velocity vector", which is equivalent to : "in a circle, the radius is always perpendicular to the tangent at that point."
But this is an Euclidean assumption (We have to use the parallel postulate to prove it.) Therefore it doesn't work in Non-Euclidean geometries, and we shouldn't be able to use it.
