Ehrenfest Paradox

The Ehrenfest paradox concerns the rotation of a "rigid" disc in the theory of relativity.

In its original formulation as presented by Paul Ehrenfest 1909 in the Physikalische Zeitschrift, 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 R0.

(Note that a cylinder was considered in order to circumvent the possibility of a disc "dishing" out of its plane of rotation and trivially satisfying C<2πR. Subsequently when a rotating disc is substituted it is assumed that this distortion possibility is also excluded).

The paradox has been deepened further by later reasoning 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.

The Ehrenfest paradox may be the most basic phenomenon in relativity which is still not completely resolved, that is, different interpretations are still being published in peer-reviewed journals.

Any rigid object made from real materials, which is rotating with the transverse velocity close to the speed of sound in this material, must exceed the point of rupture due to centrifugal force because centrifugal pressure can not exceed shear modulus of material

where is speed of sound, is density and is shear modulus. Therefore, when considering velocities close to the speed of light, it is only a thought experiment. Neutron-degenerate matter allows velocities close to speed of light, because e.g. the speed of neutron-star oscillations is relativistic; however; these bodies cannot strictly be said to be "rigid".