Paradox of Ehrenfest

The paradox of Ehrenfest is a paradox noted in the study of the revolving reference marks and more especially here in the study of the revolving discs. When one takes into account the restricted Relativité it is noted that the geometry seems different in the inertial reference mark and the turning reference mark whereas it is about the same physical space.

It is shown (see the Effet Sagnac) that the circumference of a turning disc is different sight in the inertial reference mark R and the reference mark attached R' attached to the revolving disc because of the contraction of Lorentz.

But as the ray of the disc is perpendicular to the rotation movement, it does not undergo contraction of Lorentz.

Consequently, the L/R relationship between the perimeter and the ray are different from 2 \ pi in one of the two reference marks.

R being an inertial reference mark, space is Euclidean, as shows it the restricted relativity (contrary to the space time which is not Euclidean but of Minkowski). And the disc, in rotation or not, is a circle obeying the geometry of Euclide with a report/ratio L/R=2 \ pi. It is the relation between the circumference of the disc and its ray, which we all saw at the school.

And thus:

L'/R' \ 2 \ pi

It is the paradox of Ehrenfest: in a reference mark in rotation, the relationship between the circumference and the diameter are different from \ pi. If the geometry were Euclidean in R', the disc would be " voilé" or " déchiré".

But it is not it, on the one hand because we consider a rotation " in bloc" but especially because this disc is " virtuel" : what imports us it is the reference mark in rotation, the frame of reference in rotation, and not especially a physical disc which one would put in rotation at high speed. Moreover a O' observer placed at the edge of the disc can turn very well in circle without needing this disc!

The geometry is thus not Euclidean in R'!

With the simple reasoning seen in the Effect Sagnac, one notes that the circumference is larger in R'. The geometry is thus hyperbolic there.

The geometry being nonEuclidean, it is difficult to guarantee that to divide the length of the circle (considered from an Euclidean point of view) by run time around the circle leads well to a speed identical at a speed measured locally. That justifies the term apparent speed which was used in the article on the Effet Sagnac.

In addition, the word paradox quite selected because one of the postulates is used in this simple reasoning, but as in restricted relativity, is as space is Euclidean. However, we point out that restricted relativity is built for inertial reference marks (or accelerated in infinitesimal fields where it is always possible to find a space flat tangent, just like there exists always a tangent line in a point of the circle). It may be that for accelerated reference marks this postulate is not bearable.

To the whole beginning we said that the paradox was due to the fact that physical space considered (the space occupied by the disc) was the same one in the two reference marks. But actually, we know that in restricted relativity we cannot arbitrarily separate space from time. Thus, which is identical for the two reference marks is the physical space time made up of the continuum of events (for example the passage of O' in a point given to a given moment) and described by the geometry of Minkowski. We do not have any guarantee that, by considering only the space part of the space time, space remains Euclidean in R'.

Thorough analysis of what master key can be found in the Calcul of the Sagnac effect in restricted relativity and in the Géométrie of the space time in the revolving reference marks.

See too

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