Essence of the paradox
Imagine a disk of radius R rotating with constant angular velocity ω. Let us fix the reference frame to the stationary center of the disk. Then the magnitude of the relative velocity of any point in the disk circumference is ωR. So the circumference will undergo Lorenz contraction by a factor of (1- (ω R)^2/C^2)^0,5. However the radius, being perpendicular to the direction of motion, will not undergo any contraction. So we have circumference/diameter = (2πR (1- (ω R)^2/C^2)^0,5) / 2R = π (1- (ω R)^2/C^2)^0,5. This is paradoxical, since Euclidean geometry tells us it should be exactly equal to π. Ehrenfest considered an ideally rigid cylinder that is made to rotate.
Assuming that the cylinder does not expand or contract, its radius stays the same. But measuring rods laid out along the circumference 2πR should be Lorentz-contracted to a smaller value than at rest, by the usual factor γ. This leads to the paradox that the rigid measuring rods would have to separate from one another due to Lorentz contraction; the discrepancy noted by Ehrenfest seems to suggest that a rotated rigid disk should shatter.
Even to this day, there are conflicting explanations for this 'paradox'. The simplest way to look at it is from a perspective of simultaneity. There is no way to define simultaneity for the spinning disk as a whole. In simpler words, if we synchronize a clock sitting at the center of the disk with a clock at the perimeter of the stationary disk and then spin the disk, the two clocks will go out of synchronization, just like the clocks and calendars of the twins in the twin paradox did and when two observers cannot agree on the time, they will not agree on the measured lengths of moving objects.
Interpretation of paradox solution in MT.
The matter exposed further corroborates considerations of MT about effective physical consistence of space time distortions, both in SR and in GR. However these are spatial contractions and temporal dilatations measured from different (inertial or not inertial) reference systems. In the local reference system is not appreciated any de synchronization and then there isn't any lenght contraction.
Sequentially doesn't exist any curvature gradient in the disk frame and therefore it doesn't shatter.
This involves that assignement of the freefall motion and, sequentially, of the bodies weight to the space time curvature remains an arbitrary postulate.