The evolution of scientific thought from Newton to Einstein

by A. D'Abro

Again let us put the paradox in the following alternative form: Our twin brother covers a distance of thirty-two light years (when he goes to the star and then returns), and the duration of his trip is forty years; hence his speed along the distance is four-fifths that of light. How, then, could he ever be misled into believing that he had required only twenty years to cover this total distance? Would not this belief suggest to him that his speed had been greater than that of light, and would not this fact conflict not only with relativity but also with the assertion which he would voluntarily offer, that his speed with respect to us was only four-fifths that of light?

But here, again, the criticism is occasioned by another erroneous obsession, that of absolute distance. The critic does not grasp the fact that the distance from earth to star, just like any other distance, is indeterminate and has no meaning in itself. While it is true that, so far as we are concerned, this distance is sixteen light years, yet in the opinion of the traveller it is only eight light years; and his opinion expresses exactly the same measure of reality as does our own. We can only repeat once more that absolute duration and distance are incompatible with Einstein’s premises; and if we insist upon preserving our classical obsessions, a study of Einstein’s theory will be little better than a nightmare.

Let us now pass in review a totally different order of criticism. It is contended, for example, that the principle of relativity permits us to interpret any problem of relative motion between two bodies in two different ways: by considering either one body or the other as being at rest. Then, according to the critic, since in the problem of the two twins it is the twin’s motion that is the cause of the slowing down of his time, all we need do is to consider him at rest and his brother in motion, in order to reach the ridiculous conclusion that either twin is younger than his brother.

This argument is radically incorrect for two separate reasons. In the first place, the special principle of relativity does not state that, in the general case, when two bodies are in relative motion we can consider either one of them as being at rest. The special principle emphasises the fact that a reversibility of this kind is possible only when the motions of both bodies are Galilean.

In the present case the state of the twin remaining on earth is truly Galilean (if we neglect the earth’s rotation and the curvature of its orbit); but this Galilean condition no longer applies to the twin who reverses his course when he reaches the star. In this latter case there is acceleration, and the well-known symptoms of accelerated motion, namely, forces of inertia, would make their appearance and would bring about well-defined physical disturbances.

As a matter of fact, we can amend the problem of the two twins so as to eliminate acceleration by assuming that the so-called travelling twin passes both earth and star without ever stopping or reversing his course. In this case the motions of both twins are Galilean; and, true enough, the special principle of relativity allows us to consider either one of the twins as at rest and the other as in motion. But, again, we should see that exactly the same relative agings followed, whether we considered one twin or the other as being at rest; hence the critic’s statement that motion produces a slowing down of time is a loose one unless explained more fully.

We may make these points clearer by examining the slightly modified example we have suggested. First, suppose that the earth twin, the earth and the star are all at rest; and that the so-called travelling twin travels without a stop past earth and star. We will further suppose that a rod extends from earth to star. To return to the numerical values selected at the beginning of the chapter, calculation would show that the duration of the passage of the travelling twin from earth to star would be twenty years in the opinion of the earth observer and only ten in the opinion of the wanderer. Why this difference?

Because, as we have already mentioned, since a relative motion exists between the so-called wanderer and the rod, the distance from earth to star, or the length of the rod, is contracted to half its length when computed by him. And here we must note that this contraction is not due to the so-called wanderer’s motion (for motion, unless accelerated, has no absolute significance). It is due to the relative motion between the wanderer and the rod, or the earth-star distance. If, therefore, the wanderer were to consider himself at rest and the earth-twin, together with the earth and star, travelling past him, exactly the same conclusions would endure; since once again the rod, or earth-star distance, would be animated with exactly the same relative velocity in respect to the so-called wanderer, and it is relative velocity, and not absolute velocity, that counts. We see, therefore, that no inconsistencies can arise whether we consider one twin or the other as at rest.

Let us now revert to the complete problem where the wanderer returns to earth. In this case, the wanderer has led an entirely different life history from his brother, since he, and he alone, has suffered acceleration.

It would not be legitimate to attempt to apply the special principle of relativity, and to argue as though the wanderer had always been at rest; for by so doing we should be assuming that he had never suffered acceleration, and we should therefore have to credit his brother with having experienced this acceleration. The problem would be completely modified; and, indeed, the relative aging would then be reversed. But no argument against the theory could be invoked, for the simple reason that we should be considering a totally different problem, one in which the circumstances of acceleration were no longer the same.