On Ehrenfest's paradox.
By V. Varićak.
The occurrence of Ehrenfest's paradox is understandable, when one clings to the standpoint taken by Lorentz in the formulation of his contraction hypothesis, i.e., when one sees the contraction of moving rigid bodies in the direction of motion as a change which takes place in an objective way. Every element of the periphery will be changed independently of the observer according to Lorentz, while the elements of the radius remain non-contracted.
However, if one employs Einstein's standpoint, according to whom the mentioned contraction is only an apparent, subjective phenomenon, caused by the manner of our clock-regulation and length-measurement, then this contradiction doesn't appear to be justified.
That Ehrenfest took the Lorentzian standpoint in his argumentation is concluded by me from the questions directed by him to v. Ignatowsky, and mainly therefrom that he thinks to find this contradiction at the tracing images and as well. It seems to me that those tracing images must be identical; they will have the same radius and the same periphery.
To justify this, it shall be allowed for me to take recourse to the uniform translation of a rigid body, at which that contraction is ordinarily demonstrated as a concomitant of that translation. A mirror shall be fixed at the front end , and a light source at the back end . The doubled length of the rod is measured by the time required by a light signal, to come from to and back to . In order of don't becoming to spacious, I allude e.g. to the work of Lewis and Tolman, who especially emphasized the radical difference in the views of Lorentz and Einstein. There one can also see, by which considerations the stationary observer is forced to assume the contraction of the moving rod. But he remains conscious, that this contraction is so to speak only a psychological, not a physical fact, i.e., that the body experienced no change in reality.
Now, the stationary observer shall execute with this rod the same experiment, that according to Ehrenfest shall be executed by him with the rotating disc. There are marks at both ends of the rod. While the rod is at rest, the stationary observer holds a tracing paper above him, and traces the marks upon the resting paper.
While the rod is uniformly moving forwards in a straight line, the resting observer holds a tracing paper above him and traces in the moment, when his clock indicates , at one stroke both marks upon the resting paper.
Eventually, the resting observer measures the distance of those marks at the resting tracing images and .
I believe, that they find the same distance in both cases, because the rod hasn't become shorter in reality.
The mentioned procedure of the resting observer is surely identical with the mechanical adjustment of the measuring rod to the object to be measured; yet this is not the same operation as measuring the length by the aid of optical signals.
I still want to mention in short, that it's known that the clocks in points and of the moving rod, although they have the same rate, indicate different times when the clock of the resting observer indicates .
Only a historical remark shall still be given. After Lorentz stated his hypotheses, that all bodies suffer a contraction of their dimensions in the direction of Earth's motion, the question was near at hand, whether this deformation or compression shall not be accompanied by double refraction. The corresponding experiments of Rayleigh and Brace gave a negative result.
According to Einstein's relativity principle, one wouldn't come to this question at all.
In the mentioned report of Laub, one can also read about other similar experiments, to which one is quite consequently led by Lorentz's standpoint, yet which can have no meaning in Einstein's theory.
Eventually I would like to remark, that it is not allowed without further ado, to transfer that contraction arising at a translation, to the case of rotation. The occurrence of that contraction is in connection with the change of clock rate due to motion. In rotation, however, the time parameter changes in another way, and changes not at all when the observer is in the center, as it is easy to show.
Agram, February 5, 1911.
(Received February 8, 1911.)
- This journal, 11, 1129, 1910.
- G. N. Lewis and R. C. Tolman, The Principle of Relativity, and Non-Newtonian Mechanics, Proceedings of the American Academy of Arts and Sciences 44, 711, 1909. Recently, J. Laub emphatically alluded to the fact, that in the theory of Lorentz the interpretation of this phenomenon is essentially different than that of Einstein. See Jahrb. der Radioaktivität und Elektronik 7, 430, 1910.
- I leave open the question, whether this experiment is to be considered as possible from the relativistic standpoint.