The evolution of scientific thought from Newton to Einstein

by A. D'Abro

Let us abandon these generalities, and consider the clock experiment in a more vivid form. We refer to what is known as the problem of the twins. In this we consider twin brothers one of whom remains on earth, while the other steps on to a magic carpet and visits distant Arcturus. On his return he finds that his brother has grown old and decrepit, while he himself has preserved his youth.

This particular consequence of the theory has been one of the stumbling blocks of practically every lay writer who has devoted his time to criticising the theory of relativity. Some have stated that Einstein never upheld any such absurdity, and that the whole trouble must be ascribed to his too enthusiastic followers. Others have stated that the conclusions violated the rules of logic in that they led to inconsistencies; and that if Einstein’s theory necessitated such results, it should be regarded as a gigantic hoax from beginning to end.

Now, in order to avoid any misconception, we may mention that in our example both twins hold in their hands clocks which, when placed side by side, beat out their seconds and minutes in perfect unison. Einstein’s theory indicates that the clock carried by the travelling twin will be behind time when brought back to earth and compared again with the clock that has been left there, in the keeping of the stationary twin. Some critics, such as Bergson, while accepting the postulate of the invariant velocity of light, deny that this slowing down in the beatings of the moving clock is required by the theory in any real sense. It is a mere appearance, Bergson maintains; a mathematical illusion. Just as the gradual shrinking of a man’s apparent size as he moves away from us would not connote any real decrease in his stature, so now, when the travelling twin returns with his clock to his brother on earth, the illusion would vanish, and both twins would recognise that their respective clocks were still marking the same time in perfect agreement, one with the other.

Now, it cannot be stressed too strongly that this interpretation of the problem is worse than incorrect. Not only is it incorrect from the standpoint of Einstein’s theory, but it denotes on the part of the critic a colossal ignorance of the significance of a theory of mathematical physics. Here it must be recalled that the comparative retardation of the travelling twin’s clock is an inevitable consequence of the Lorentz-Einstein transformations. To deny, therefore, that these transformations correspond to what would actually occur, to what would be detected and measured by the most accurate instruments, would be to deny that the transformations possess any physical basis. If this be the stand we wish to take, well and good; but the logical outcome of this attitude will be to deny the legitimacy of the entire theory; it would be quite impossible to retain the theory while contesting the real significance of the transformations. Once again a theory of mathematical physics is not one of pure mathematics. Its aim and its raison d’être are not solely to construct the rational scheme of some possible world, but to construct that particular rational scheme of the particular real world in which we live and breathe. It is for this reason that a theory of mathematical physics, in contradistinction to one of pure mathematics, is constantly subjected to the control of experiment. If, therefore, the Lorentz-Einstein transformations failed to yield the results that would be actually measured, Einstein’s theory as a means of physical discovery, as a co-ordination of physical facts, would be not only useless, but entirely incorrect, and we should have to abandon it.

Furthermore, it should also be noted that every precise physical experiment that it has been possible to accomplish has verified the anticipations of the Lorentz-Einstein transformations; it is owing to this fact that the theory was not abandoned years ago. This in itself should warn us against asserting that the transformations possess no physical significance.

Thus far we have been concerned with generalities and have merely attempted to show why Bergson’s solution of the problem must be discarded on first principles. Leaving aside this aspect of the question, it is instructive to consider the formal proof which he presents in support of his contentions.

He appears to realise that the entire matter hinges on the measure of physical reality which should be credited to the Lorentz-Einstein transformations. Accordingly, he attempts to prove that these transformations do not represent reality, but possess a mere fictitious mathematical significance having nothing in common with the physical world. He calls them “phantasmatical,” contending that mathematicians, not being trained in philosophy, are severely handicapped when they attempt to interpret their equations, and are incapable of distinguishing mathematical fictions from physical reality. So Bergson proceeds to demonstrate the fictitious significance of the transformations.

He first serves us with a series of arguments based on the principle of the relativity of motion itself. In the very first paragraph, however, he confuses the relativity of Galilean motion with the complete relativity of all motion, Newtonian and special relativity with visual relativity, velocity with acceleration. All these errors relate to classical science itself; and it is obviously quite impossible to discuss Einstein’s theory intelligently until we have acquired at least an elementary knowledge of classical mechanics. Following this philosophical demonstration, he goes on to give us a mathematical one; but all he succeeds in proving is that he has not the slightest conception of the most elementary properties of mathematical transformations.

Now, without going into mathematical details, let us make an effort to see why it is that the clock of the travelling twin must necessarily have suffered retardation. To understand the reason, we must revert to Einstein’s premises, and in particular to the postulate of the invariant velocity of light. This postulate, as we have explained on several occasions, states that all Galilean observers situated in their respective Galilean frames will discover, as a result of measurement with rigid rods, that waves of light invariably pass through their frames with the same constant speed of 300,000 kilometres per second. If this postulate be accepted, we must infer that in every Galilean frame, time can be measured by the distance in space covered by a wave of light (the distance being measured with rigid rods at rest in the frame). As a result the postulate allows us to construct a sort of optical clock. We may, for example, consider two parallel mirrors situated at a fixed distance apart of 1.5 kilometres in our Galilean frame, and we may suppose that a wave of light is reflected back and forth from one mirror to the other. These oscillations of the ray of light, covering always the same distance, require equal durations; and in the present case 100,000 such double oscillations would be seen to define a duration of one second for that Galilean frame in which the optical apparatus was situated. We must henceforth consider every Galilean frame as possessing an optical clock of this kind defining the passage of time for its own frame.

Let us now analyse the problem of the two twins; and let us suppose that we as observers are stationed on the earth, say, at the North Pole. To a high degree of approximation we may consider ourselves, therefore, as at rest in a non-rotating or Galilean frame. The moving observer is also situated in a Galilean frame, moving away from us with a speed which we shall assume to be about four-fifths that of light. He visits a distant star which we should recognise as being sixteen light years distant[73] from us, then reverses his velocity and returns to earth.