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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.

mlg556
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    Possible duplicate of Euclidean geometry in non-inertial frame or Distance in relativistic circular motion in invariant spacetime – AccidentalFourierTransform Jan 14 '16 at 22:08
  • "So the circumference changes, and the radius doesn't." I think this is the main problem. The particles forming the radius move all in the same direction, but with different velocities and different accelerations. The Lorentz contraction certainly does not apply to them as there is no inertial frame in which the particles are at rest, but that by itself does not mean the radius has the same length as it had when the disk was at rest. The disk will deform and expand even in non-relativistic regime and it may deform even differently in a relativistic regime. – Ján Lalinský Jan 15 '16 at 03:26
  • I think I didn't explain myself properly, my main problem is that while proving "the geometry is non-euclidean", we used an assumption which only works in Euclidean geometries, and we shouldn't be able to do that. – mlg556 Jan 15 '16 at 18:01

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