Ehrenfest paradox

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

In its original 1909 formulation as presented by Paul Ehrenfest in relation to the concept of Born rigidity within special relativity,[1] it discusses an ideally rigid cylinder that is made to rotate about its axis of symmetry.

[2] 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 indicates that geometry is non-Euclidean for rotating observers, and was important for Einstein's development of general relativity.

Imagine a disk of radius R rotating with constant angular velocity

So This is paradoxical, since in accordance with Euclidean geometry, it should be exactly equal to π. Ehrenfest considered an ideal Born-rigid cylinder that is made to rotate.

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 Born rigid disk should shatter.

Thus Ehrenfest argued by reductio ad absurdum that Born rigidity is not generally compatible with special relativity.

According to special relativity an object cannot be spun up from a non-rotating state while maintaining Born rigidity, but once it has achieved a constant nonzero angular velocity it does maintain Born rigidity without violating special relativity, and then (as Einstein later showed) a disk-riding observer will measure a circumference:[3]

The rotating disc and its connection with rigidity was also an important thought experiment for Albert Einstein in developing general relativity.

[4] He referred to it in several publications in 1912, 1916, 1917, 1922 and drew the insight from it, that the geometry of the disc becomes non-Euclidean for a co-rotating observer.

We suppose these laid in series along the periphery and the diameter of the circle, at rest relatively to K'.

We therefore arrive at the result: the gravitational field influences and even determines the metrical laws of the space-time continuum.

If the laws of configuration of ideal rigid bodies are to be expressed geometrically, then in the presence of a gravitational field the geometry is not Euclidean.Citations to the papers mentioned below (and many which are not) can be found in a paper by Øyvind Grøn which is available on-line.

[3] Grøn states that the resolution of the paradox stems from the impossibility of synchronizing clocks in a rotating reference frame.

The modern resolution can be briefly summarized as follows: Some other "paradoxes" in special relativity