Uniform rotation and Lorentz contraction.
By Max Planck
In the discussion on a note published in this journal by P. Ehrenfest, entitled: "Uniform Rotation of Rigid Bodies and the Theory of Relativity", I repeatedly encountered the view that the case considered by Ehrenfest: the uniform rotation of a body around a fixed axis, brings with it a difficulty for the theory of relativity. It is therefore perhaps not entirely superfluous if I allude with a few words to the misunderstanding that is the basis for such a view.
The theorem that the volume of a body moving with velocity q, appears to be smaller for an observer at rest in the ratio than for a co-moving observer with velocity q, must be distinguished from the other theorem, that the volume of a body is diminished in the ratio when it is brought from 0 to speed q. The first theorem is one of the basic requirements of relativity theory, but the latter theorem is incorrect, at least in this generality. Because the volume change of a body element in any change of state is always very much conditioned by the external forces that are exerted on its surface, particularly by the behavior of pressure and heat supply. As long nothing is known about these influences, as long nothing can be said about the contraction of the volume. Most directly, this can be seen at slightly compressible bodies, as in gases, but it applies equally well to solid bodies. A body that has an "independent" volume, i.e. independent of external influences, does not exist in all of nature, and even less in relativity theory. At least I think, that the attempt to make the abstraction of the rigid body (which is so important for ordinary mechanics) also useful for the theory of relativity, does not promise any real success.
The task, to determine the deformation of a somehow accelerated body is therefore essentially a problem of elasticity, in relativity theory and in ordinary mechanics. For the special case of quasi-stationary translation I have given some time ago the solution. Besides other things, it follows that the above mentioned Lorentz contraction always occurs at isobaric-adiabatic acceleration of a body, but otherwise generally not. For a rotating body, the acceleration is in any case not isobaric for all individual body elements; but also in this case the problem can be easily solved as soon as the kinetic potential of the elastic deformation is known. In the exploration of the expression for the kinetic potential it must of course be noted primarily, that this function which is based on unit volume, is invariant for all Lorentz transformations.
(Received March 10, 1910.)
- ↑ This journal, 10, 918, 1909.
- ↑ In a note published in the meantime (this Journal. 11, 233, 1910) Born limits the definition of rigidity to a single electron. Against this we surely cannot object from the outset, but whether it brings us physically further than the general principles of relativity theory, might well escape each test for the time being. (Note added in proof.)
- ↑ Ann. d. Phys. 26, 24, 1908.
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