Translation:The Principle of Relativity (Laue, Philosophy)

by Max von Laue

§ 4. The kinematics of Einstein.

Now we pass to the conclusions from the Lorentz transformation, by starting with some consequences from the relativity of time.

Let a clock be at rest in the valid reference system $K'$. We want to compare its indications with the indications of clocks of the same kind at rest in system $K$, namely always with such ones, at which it momentarily passes by. Its indication $t'_{1}$ thus corresponds to the reading $t_{1}$ of a clock that is momentarily at rest in $K$ at the same location, and its indication $t'_{2}$ is analogous to the reading $t_{2}$. Its coordinates in the primed system $K'$ are invariable by definition. Consequently we conclude from the last of equations 7)

$t={\frac {t'+{\frac {v}{c^{2}}}x'}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}},$

that the difference is

$t{}_{2}-t{}_{1}={\frac {t'_{2}-t'_{1}}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}$

Let us assume, that in the first moment the two clock to be compared are in agreement, i.e. it would be $t'_{1}=t_{1}=0$; then for the later moment $t_{2}$, the clock that is at rest in the primed system (= the clock in motion relative to $K$), would only indicate

$t'_{2}=t_{2}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}},$

i.e. it would be retarded. A clock moving with velocity $q$ is thus slower in the ratio ${\sqrt {1-{\tfrac {v^{2}}{c^{2}}}}}$ as the same clock, when it is at rest.^[1] This theorem is of course true, independent to which of the valid systems we relate the expressions "rest" or "motion". We are allowed to define the term "clock" as far as at page 108, i.e. all physical, chemical, and biological processes must show in their course the same dilatation. A life form in motion is less aging, and even the thought process of a man must occur slower, when the relativity principle shall be correct. However, he won't experience anything of this, as long as his velocity is not changing, since he is also at rest in a valid reference system. On the contrary, he will believe with some justification, that he thinks faster than the men who are moving with respect to him. The expressions "slower" and "faster" are thus without absolute meaning, and only get a certain sense by the indication of the reference system.

Then we consider two equal clocks that at first are resting at the same location in the valid references system $K$. We bring the second one to the velocity $q$, then for some time we let it travel together with the other one, and then (by the same or a different velocity) we bring it back to the origin, where it again comes to rest. If the clocks were in agreement in the beginning, then the second one is now retarded with respect to the first one, since the clock's rate was slower during the motion according to the previous theorem. Namely these differences in the indications of both clock are "absolute", i.e. valid for all reference systems.

The fact of different pointer positions of two clocks at the same location cannot be transformed away. A life form is returning at a younger age, than its earlier coeval. — It's understandable, that many objections were raised against this consequence of the theory of relativity; some are denying it as a consequence of the theory, the others want to make of it an inner contradiction of the theory of relativity, since in a pure kinematic consideration of the mutual motion it cannot be decided, which of the clocks is at rest, and which is in motion. In a pure kinematic sense the latter claim is correct. However, by our presuppositions, one clock is at rest in one valid reference system during the time of separation, while the second one is at rest in valid reference systems both during the forward- and the backward motion, but necessarily in two different ones. Therefore the two fates differ physically. If we would let remain the second clock in the motion which was given to it at the start, and if we send after it the first clock after some time by a greater velocity, then at the encounter the first one would be retarded with respect to the second one; since now it was the first one that was at rest in two different systems during the separation.^[2] With respect to the velocities that are ordinarily at our disposal, those dilatations are of course too small to be observable, yet in the last decade we have discovered (during the investigation of electrical conduction through rarefied gases) corpuscular rays, with respect to which the atoms are moving with much more considerable velocities. For example, in canal rays the atoms of hydrogen or mercury are moving with velocities of about ¹⁄₁₀₀ of the speed of light. At the same time, oscillations of high regularity occur within them, of which we were informed by the sharp lines of their spectra. Every oscillating structure is a clock in our sense. By spectroscopic observations, whose precision (as we know) can be largely increased, we can measure its oscillation number and the influence upon it by the velocity, and although the difficulties of such an experiment have not been overcome yet, there is justified hope, that later a verification of the theory, which is directly related to the relativity of time measure, will be possible.

(...)

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1. In the Michelson experiment, the Lorentz contraction (according to our presentation in § 2) removes the difference between the two times of return. Their common value ${\tfrac {2lc}{\sqrt {c^{2}-q^{2}}}}$ remains, however, dependent on velocity $q$. This time indication is related to the system, in which the sphere has the velocity $q$. On the other hand, a clock that is carried by the observer shows (according to the above) the time ${\tfrac {2lc}{\sqrt {c^{2}-q^{2}}}}\cdot {\sqrt {1-{\tfrac {q^{2}}{c^{2}}}}}={\tfrac {2l}{c}}$. The observer cannot find out the velocity $q$ by its reading.
2. The objection which is near at hand, that we cannot say anything about the rate of a clock during a velocity change, can be met most simply by the allusion, that we can render the times of uniform motion arbitrarily great with respect to acceleration.