It is customary to designate by the names proper length and proper duration the length and duration obtained by an observer at rest with respect to the rod or clock. Beginners appear to have considerable difficulty in convincing themselves that all these durations, whether proper or elongated, are equally real. We have already explained in what measure they are so, but it may be helpful to give the following additional illustration: Suppose we are suffering from a headache and that this headache lasts an hour when timed by our watch. This will be the proper duration of the headache. Now we might be inclined to claim that inasmuch as we alone are suffering from the headache, we alone are in a position to determine its duration; and when some rapidly moving observer tells us that we have held our watch in our hand for one week and that our headache has therefore lasted a week, we might be tempted to tell him to mind his own business. But it must be realised that although the headache is ours and ours alone, and nobody else can feel it for us, yet the duration occupied by the headache is common property, and any other observer is just as much entitled to live it and to measure it as we ourselves are.
CHAPTER XXII
AND THE TRIP TO THE STAR
THE inevitable rational consequences which follow with mathematical certainty when once the existence of space-time as a model of the universe is conceded, are often puzzling to beginners, who feel that these consequences are too paradoxical to be acceptable. Of course, we must distinguish between a paradox of feeling, and a paradox of logic or an inconsistency. Any inconsistencies would be fatal to a rational construction and would denote either errors of reasoning or the presence of some fundamental contradiction in our basic premises. If, therefore, when analysing any given example in Einstein’s theory, we proceed to follow our deductions by adopting successively two different alternatives in our point of view, both of which would be consistent with the premises of the theory, and if we finally compare the results thus obtained and find them to be contradictory, hence inconsistent, the theory of relativity could never survive. However, it is scarcely probable that any such inconsistencies will ever be discovered by the layman, inasmuch as the theory of relativity is the work of mathematicians, for whom, more than for any one else, accurate and consistent reasoning is the indispensable condition of research.
On the other hand, though no inconsistencies or paradoxes of logic can exist in Einstein’s theory, this does not mean that the conclusions of relativity may not appear strange and lead to paradoxes of feeling. But once more it is important to differentiate between strangeness or paradoxes of feeling, and inconsistency or paradoxes of reason.
The important point which must be settled, therefore, is whether our habitual notions of time, space and simultaneity are rational necessities, or whether they are merely feasible notions among a number of equally feasible ones. The question we have to decide is analogous to that which Kant proposed to solve when he stated that Euclidean geometry and three-dimensional space were the a priori forms of pure perception. Kant was wrong, as the discovery of non-Euclidean geometries proved without doubt. To-day Einstein has done for space, time and motion what Riemann and Lobatchewski did for the geometry of space alone. He has furnished us with a model just as rational as the model of classical science; and this model has the additional advantage of being in accord with empirical observation to a high order of precision, instead of being a mere approximate construction as was the classical structure. Indirectly, then, Einstein has proved that our innate classical conceptions of space, time, simultaneity and motion were by no means rational necessities.
Many people, however, are of the mistaken opinion that the theory of relativity leads to paradoxes of logic or inconsistencies. But when we analyse their arguments we are always confronted with the same erroneous method of reasoning. In many cases the critics seem to have a very hazy idea of the premises on which the theory is built. Others, while stating that for the sake of argument they are willing to recognise the validity of these premises, appear to forget their good intentions before reaching the end of their discussion. Inadvertently, in the very middle of their demonstrations, they relapse into the premises of classical science and, of course, are led to insufferable contradictions; whereupon they conclude that the theory of relativity leads to inconsistent results.
Let us consider, for example, the relativity of duration. A number of clocks or atoms which beat at the same rate and mark the same time when placed side by side are displaced along different routes to the same terminal point. The relativity theory then demands that when set side by side once again they shall be marking different times. The critic immediately exclaims: “This is inconsistent!” But where is the inconsistency? There would be inconsistency, indeed, had Einstein started out by positing as an axiom that time was absolute. This, in fact, is precisely what the critic has done. But then, when, as a result of this uncalled-for presupposition, he finds that the theory leads him into inconsistencies, he should not blame Einstein; for the theory is inconsistent not with itself as a logical structure, but solely with the critic’s own presuppositions.
Of course, the source of all the trouble lies in the fact that a belief in the absoluteness of duration has become second nature to us; and it requires a certain effort of introspection on our part for us to realise that this belief presents no rational justification. Eventually, however, our ideas become clarified, and we concede that we have no means of figuring out a priori whether time will be absolute or relative. We realise that it is impossible to lay down the law at the start one way or the other, unless we wish to force nature into a preconceived mould—which may not fit.
Our sole means of settling the problem is, then, to appeal to experiment; to ultra-precise experiment in the present case, since the crude variety would not enable us to reach a decision. The experiment of the clocks, could it be performed with sufficient refinement, would constitute one of these. Obviously it would be unwise to assume that its result could be known with certainty in advance; for were men to labour under any such belief, there would be no need for them to experiment. We may, of course, question the likelihood of a certain result occurring, basing our opinion on previous experiments of a similar nature; but in the final analysis our understanding of nature will issue from an accurate observation of facts and not from a-prioristic metaphysics.
Unfortunately, as we mentioned previously, this experiment with clocks or atoms cannot be performed with sufficient accuracy, so that indirect methods must be considered. But it is precisely because all the indirect experiments thus far attempted have yielded results incompatible with the absoluteness of duration and distance that Einstein was compelled to reject these secular beliefs of mankind. Accordingly, we have every reason to anticipate that the clock experiment would bear out Einstein’s views; in fact, were this not the case, the entire theory would collapse.