BETADespite the number of articles and web pages on the Ehrenfest paradox, hardly any provide a direct explanation of how the paradox can be resolved. The following is my interpretation of the physics, I’m open to comments if I have erred anywhere or if my language is excessively lax.
The argument that leads to the Ehrenfest paradox is as follows:
- Assume that a rotating disc is a certain size in its own rotating reference frame.
- Divide the circumference into small segments of angle dφ and length dσ = Rdφ.
- Consider each segment in its own frame, which is instantaneously travelling tangentially at v = ωR relative to a stationary observer.
- Assume that the length of the segment is still dσ in the segment’s frame.
- Transform from the segment’s frame to the stationary frame, obtaining a contracted length dσ/γ (where γ is the Lorentz factor).
- The segments do not cover the circumference of the disc in the stationary frame.
In my opinion the problem with this argument, surprisingly, is point 4. Whereas the segment was previously considered from the point of view of a rotating observer at the center of the disc (with angular velocity but no linear velocity), it is now being considered from the point of view of an observer moving with the segment (with linear velocity).
In fact it can be shown that an observer in the segment’s frame1 measures longer lengths locally, according to the Langevin-Landau-Lifschitz metric:
![\[ d\sigma'^2 = dz^2 + dr^2 + \frac{r^2 d\phi^2}{1-\frac{\omega^2r^2}{c^2}} \]](http://www.zmatt.net/wp-content/ql-cache/quicklatex.com-f52ced98e27274152f3971acb121bc5c_l3.png "Rendered by QuickLaTeX.com")
Measuring the segment (of angle dφ) along the rim we obtain:
![\[ d\sigma' = \frac{R d\phi}{\sqrt{1-\frac{\omega^2R^2}{c^2}}} = \frac{d\sigma}{\sqrt{1-\frac{v^2}{c^2}}} = \gamma d\sigma \]](http://www.zmatt.net/wp-content/ql-cache/quicklatex.com-39dec57483db9b6ff26bd2086d62b87b_l3.png "Rendered by QuickLaTeX.com")
Thus, the corrected argument would be:
- An observer in the segment’s frame would measure the length of the segment as γdσ.
- Transform from the segment’s frame to the stationary frame, obtaining a contracted length dσ.
- There is no paradox in the stationary frame.
Note that while each segment-riding observer locally measures an arc length greater than what might be expected, the same observer would see length contraction if observing the far side of the disc. Thus it is not meaningful to add up each observer’s local measurements into a whole greater than 2πR, and there is no paradox there either.
1 The observer in the segment’s frame is usually known as a Langevin observer after Paul Langevin who originally considered the geometry of this reference frame. Nathan Rosen showed that the same metric also applies to the inertial frame that is instantaneously co-moving alongside the Langevin observer, which we can be more confident applying Lorentz transformations to.
