Question:

(asked by Lung Ko, Oct 95)

Take a rigid disk of radius R and spin it up to angular velocity . As seen by an observer S that is at rest in the center of the disk, the radius is still R, but the circumference is , with . How is this possible?

More physically, if I place a fixed ring just outside the spinning disk and place equally spaced markers on the rim of the disk and on the fixed ring, I know by symmetry that, when one marker on the disk is aligned with a marker on the ring, all pairs of markers must be aligned. This contradicts the fact that, for observer S, the distance between successive markers on the disk is reduced by .

Answer:

After asking many people, I received a message from David Djajaputra on 19 Nov 95 that reads in part:

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It seems that Ko's rotating disk paradox (it turned out to be Ehrenfest's
paradox) has been extensively analyzed by many people (including Einstein
himself, who developed general relativity to answer this problem, as one
author speculates...). This I found from a nice paper :

       O. Gron, "Relativistic description of a rotating disk"
       Am. J. Phys. V43, 869 (1975),

and all the references therein.

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The key sentence in Grøn's paper is at the end of Section IV:

"By definition a Born rigid motion of a body leaves lenghts unchanged, when measured in the body's proper frame . (...) A Born rigid motion is not a material property of the body, but the result of a specific program of forces designed to set the body in motion without introducing stresses. (...) A transition of the disk from rest to rotational motion, while it satisfies Born's definition of rigidity, is a kinematic impossibility"

The simplest "program of forces" achieves the motion described by

With this kinematics the radius is R and the circumference is as measured by observer S (lab frame), but an observer riding on the disk will measure a distance R to the center and a distance around the circumference (he can do this measurement by slowly walking around the spinning disk with a meter tape). This is consistent with the usual Lorentz contraction . The point is that this is NOT a Born rigid motion.

There is much more in Grøn's paper.