Now, in order to answer the criticism which charges Einstein’s selection of optical clocks with being artificial, we must proceed to investigate whether the definitions of congruent durations marked out by the optical clocks are justifiable. And, first, what do we mean by a justifiable definition of the equality of two successive durations? Would it have been justifiable to define equal durations by the durations a body falling vertically towards the earth would require to cover equal Euclidean distances in space? Obviously not.
We saw, when discussing congruence generally, that mathematical analysis by itself afforded us no means of exalting above all others one particular definition of congruent stretches either for space or for time; but that when we combined physics with mathematics, certain precise methods of defining congruence imposed themselves immediately in any given Galilean frame of reference. In the case of time-congruence we found it necessary to assume that any periodic phenomenon at rest in our frame, in a complete state of isolation from all external influences, would have to be regarded as passing at regular congruent intervals of time through the same initial state. This assumption was demanded by our belief in causality, and by a justifiable definition of time-congruence we mean to imply one which would satisfy these consequences of causality.
When, therefore, we stop to consider whether Einstein’s optical clocks yield us a justifiable definition of equal successive durations, the whole question simmers down to deciding whether or not the oscillations of a ray of light in an optical clock constitute an isolated phenomenon, or whether, according to the motion of the Galilean frame in which the clock is situated, certain perturbating influences may not interfere with the state of isolation we credit to the working of the clock.
Until such time as the effect of the ether on an object moving through it had been settled, there was every reason to suppose that in the case of a Galilean frame moving through the ether, the effect of the ether wind would be to disturb the free oscillations of the light rays between the mirrors of the clock. And so the clock could not be considered isolated, and its readings would be valueless, or at least in need of correction. But Einstein’s special principle of relativity has precisely for its object to inform us that the ether behaves as though it were non-existent, so that any perturbative effect which classical science had suspected was ipso facto eliminated. In other words, if we accept the special principle of relativity, the optical clocks may be regarded as perfectly isolated; and their readings must therefore afford us with perfectly reliable measures of congruent time-stretches, as required by the principle of causality.
It follows that if in a Galilean frame we place side by side, at rest, an optical clock, a vibrating atom, a piece of radium gradually losing its mass, and a top revolving without friction, the time determinations defined by all these phenomena will be identical, since all these are perfectly isolated from external influences. To be more precise, if a sodium atom vibrates so many times per second, as measured by the optical clock of its Galilean system, it should still vibrate exactly the same number of times per second (as measured by the clock of the frame) even if the constant velocity of the Galilean frame (with respect to some standard Galilean frame) has been changed to some other constant velocity. In short, all physical phenomena at rest in a frame would evolve at exactly the same rate in terms of the optical clock of the frame, regardless of the magnitude of the Galilean motion of the frame; since, according to the special principle of relativity, this absolute Galilean motion is entirely meaningless.
If observation, by any chance, should prove that the relative rates of vibration of the atom and the clock were to vary according to the Galilean frame in which we set them, we should have a physical means of differentiating one Galilean frame from another, hence of determining the velocity of our frame through the ether; and the special principle of relativity would have to be abandoned. Assuming, therefore, Einstein’s premises to be correct, we must agree that the vibrations of the optical clock constitute as perfect a means of determining congruent intervals of time as it is possible for physicists to obtain. Finally, we see that in any Galilean frame exactly the same time-determinations would be obtained whether we made use of an optical clock, a top spinning on its axis without friction, a vibrating atom, or a mechanical clock, provided all these bodies were at rest as a whole in our Galilean frame.
Consider, then, two Galilean frames, each containing an optical clock and a top spinning without friction on its axis. If, when the two frames are brought to relative rest, side by side, the two tops are spinning at exactly the same rate, both executing
rotations per second when measured in terms of their respective clocks, then exactly the same correlation must exist when the Galilean frames are moving apart. It is therefore a matter of indifference whether the observers in the frames measure time by means of their optical clocks or by means of the rotations of their tops.
Now a planet revolving on its axis in space is nothing but a huge top; and if, for instance, we stand at the non-rotating point of a planet, say the North Pole (so as to be in a Galilean frame), we should be perfectly justified in measuring our unit durations by the successive rotations of our planet with respect to the fixed stars. Consider, then, two planets rotating at exactly the same speed when held side by side, and let us call one of these planets the earth. Regardless of whether these two planets be set flying apart or held together, the observer on either planet will be justified in measuring his unit time-stretches in terms of his own planetary rotations. Classical science assumed that since time was one and universal, both planets would always beat out exactly the same unit time through their rotations, whether they were in relative motion or at rest, so it mattered little which particular planet was selected. Under the circumstances, from motives of convenience, the earth was chosen as standard.