Part 2 — CHAPTER XXI THE REALITY OF THE CONTRACTION OF LENGTHS AND OF THE LENGTHENING OF DURATIONS

LET us now pass to the question of lengths and durations. Consider a rigid rod placed on a table, and suppose that both table and rod are drawn away from us with constant speed along a straight line. The contraction of lengths implies that both rod and table will appear to us to be contracted, owing to their relative motion with respect to us.

Here we must emphasise the fact that this contraction has nothing to do with the motion of the rod through space or the ether. As we know, motion, or at least Galilean motion, through space has no meaning. It is purely a relative; so that exactly the same effects of contraction would ensue were we to conceive of the table and rod as at rest and of the observer as moving away from them. It follows that an observer attached to the rod would perceive no contraction, although his friend in motion with respect to him would assert that a contraction had taken place.

The nature of this contraction is obviously rather mysterious, and the question is to decide as to its reality. Here we are met with the difficulty of defining what we wish to convey by the word “reality”; in some respects the contraction of the rod is a physical reality, in others it is a fiction, but in no case is it an illusion.

Let us consider the problem of illusions. For instance, we experience an aggregate of visual impressions which we unhesitatingly ascribe to the presence of a table situated “there in the room.” Yet all we have any right to be positive about is that certain visual impressions have been experienced; our belief in the existence of the table situated “there” results from an inference, a judgment formulated in an attempt to ascribe a cause to our visual sensations. By an association of ideas and by recalling past experience, we feel justified in claiming the existence of the table “there” as we see it. In the majority of cases our judgment will be correct, but in a certain number of instances it will be erroneous. Such would be the case were we to be viewing the reflection of a table in the mirror. An illusion is therefore an error of judgment. And what prompts us to state that our belief in the object behind the mirror is an illusion? Obviously, the bare fact that on testing the matter out in a variety of different ways it appears impossible to credit the table with any existence at the spot where we claim to see it in the common objective world of science. In much the same way we see a mirage and interpret our sensations as connoting the presence of water. Yet were a meteor to fall into the water it would not appear to be extinguished by the waters of the lake. Kindred observations of an indirect order would finally compel us to recognise that our initial judgment was at fault, and that no water lay before us.

And now let us consider the FitzGerald contraction. Is it an illusion? To this question our reply must unhesitatingly be in the negative. Not only should we perceive the contraction were our eyes capable of yielding a snap-shot picture of a rapidly moving object, but in addition every conceivable physical experiment performed in our frame would indicate that the contraction was physically present. And yet we have said that the contraction differed essentially from the one contemplated by Lorentz, generated as was the latter by a real compression caused by the motion of the object through the stagnant ether. The explanation of the dilemma is as follows:

In Einstein’s theory, the rod is contracted, but we must add that this contraction holds only in the space of the frame with respect to which the rod is in motion. In the space of the frame accompanying the rod in its motion, no contraction would occur. We see, then, that as there exists no privileged space, only “my space,” “your space,” “his space,” there is no particular length to the rod, only a length “for me,” “for you” and “for him.” The contraction is thus embodied in a two-term relationship extending between the observer and the observed; it is a “relative.” But, once again, for a given observer it is as physically real as a chair or a table. Only when we consider the contraction from an impersonal standpoint, from that of space-time, without mentioning the conditions of observation, is the contraction not an illusion, but downright meaningless. It is, as we see, the concept of spatial magnitude, of primary qualities, which is at stake, in a form entirely different from anything science or philosophy had ever dreamed.

Now although length, size and duration are relatives, the theory of relativity has discovered absolutes which transcend the observer, remaining the same for all. They are to be found in the absolute world of space-time; and it is this thought that Minkowski desired to convey when he said: “Henceforth space and time by themselves are mere shadows.” In the absence of some specified observer, space and time become indeterminate, and with them length and duration; that which endures and transcends the observer is a something in space-time, and this is neither a length nor a duration. Or, again, lengths and durations are the projections of something in absolute space-time on to the space and time directions of the observer. These latter are indeterminate (until the frame of the observer is specified), since no one space and no one time exists above all others. Space and time being indeterminate, the same will apply to those projections which we call length and duration.

Although we may be repeating ourselves unnecessarily, we cannot help but feel that the easiest way for the layman to grasp these difficult concepts will be to return to the well-known illustration of the relativity of a visual angle. When we perceive an object, say a telegraph pole, we see it under a certain angle, which we may call the visual angle. Is this visual angle real? Of course it is; we can measure it and define it physically as the angle subtended by two Euclidean straight lines extending between our eye and the two extremities of the pole. It is real but it expresses a relationship—a relationship between the positions of the observer and pole. It is certainly not inherent in the pole, since as the observer changes his position the visual angle changes in magnitude, though nothing happens to the pole in the meantime.

Visual angle in classical science is analogous to distance in Einstein’s theory. Our rod has no true length any more than the telegraph pole has any true visual angle. The whole significance of relativity is precisely to deprive length and duration of the absolute characteristics which classical science attributed to them. In short, when we are asked whether the rod has really been contracted and lost part of its length, we might counter by asking whether the telegraph pole has really been modified and lost part of its visual angle. Just as, if we omit to stipulate the position of the observer, the visual angle of the pole has no meaning and can teach us nothing about the pole, so now, in relativity, the length of the rod has no meaning in itself and can teach us nothing about the rod.

All these arguments developed on the score of length apply in similar fashion to duration. Both length and duration are relatives having no absolute significance in the universe.

The critic will generally balk at these statements and be inclined to say: “What nonsense! The length of the rod is immanent in the rod and can in no wise be affected by the observer’s motion or by the frame’s existence. The observer might die, the frame might disappear; what has all this to do with the length of the concrete entity we call the rod? The length of the rod may appear longer or shorter according to its relative motion; but we must not confuse appearance and reality. Thus, a man standing 100 yards away appears to us shorter than when he is close up, but it would never enter our minds to say that he was really reduced in stature. Just as the height of the man is independent of our position and of our existence as observers, so is the length of the rod independent of our relative velocity.”

Now we might readily concede to the critic the right to maintain that the existence and absolute characteristics of the rod are in no wise dependent on the motion or the existence of an observer. But the critic is begging the question when he assumes without further ado that these absolute characteristics are expressed by the length of the rod. This length enters into existence only when a definite space is specified; till then it is meaningless. And in the absence of an observer there is no such thing as a space, but merely four-dimensional space-time. The absolute characteristics of the rod thus refer to space-time, and this eliminates all primary qualities such as length. The situation is of course not easy to grasp at first blush and is still less easy to explain, but in view of the extreme importance of the matter we shall do our best to clarify it by slightly varying the presentation.

Let us first examine the problem from the standpoint of classical science. Consider an observer who walks away from a point on the surface of the earth. He finally comes to a stop after having walked a distance from the point . He turns round, looks back and sees a black line sticking up from the earth at the point , whence he started. The height of the line appears to him to have a certain magnitude; and in order to register this magnitude accurately, he measures with an instrument the precise visual angle under which he sees the black line. He then continues his journey, stops, turns round again and finds that the black line is now seen under a smaller visual angle, hence appears to him to be reduced in height. His first impulse would be to assume that the black line was due to some squirming mirage and, while a reality in so far as it affected his eyesight, yet could be endowed with no objective existence in any impersonal way. Yet, without even returning to the point and investigating the matter more closely, our observer could soon find out that the black line must be produced by some existent entity. He would make this discovery in the following way: By combining the visual angle under which the black line appeared to him with his distance from the point , he would find that wherever he stood, a certain mathematical expression (i.e., ) representative of a length maintained the same invariant value. Something truly objective would have been discovered since, in contrast to visual angle and distance which varied with his position, here was a length which was an absolute since its value transcended his position. He would immediately attribute this absolute to the objective existence of a real rigid pole of length standing at the point . In short, the test of objectivity is invariance for all observers, and this illustration has enabled us to separate the absolute length of the pole from the relative visual angle and distance . Classical science assumed that this test of objectivity would hold in all cases, regardless of whether our observers were stationary or in motion. The theory of relativity tells us that this is not so. The observer in relative motion would always perceive the pole under a smaller visual angle, so that the mathematical expression purporting to give the length of the pole would no longer be an invariant.[71] Under the circumstances we should again be thrown back on the perplexing problem of whether there could exist any kind of universal objectivity holding for all observers in this universe.

One thing is certain. We could no longer attribute an invariant length to the pole. This length would no longer be immanent in the pole, since it is impossible to discover an invariant expression of spatial length holding for all observers, hence independent of their existence. We cannot, therefore, escape the conclusion that spatial length by itself can have no more universal existence than the apparent height or visual angle of classical science. Unless we abandon all hope of discovering a something universally objective, we much search for some substitute for that absolute spatial distance in which classical science believed.[72]

Minkowski found that such an absolute could be constructed; but it was made up of the relative length and duration under which every observer perceived any two instantaneous events. In exactly the same way as classical science had constructed the absolute spatial magnitude, so now Minkowski constructed the absolute space-time magnitude, i.e., the Einsteinian interval. Objectivity is at last rediscovered, though this objectivity refers neither to space magnitudes nor to durations, but to a combination of both. In other words, the analogue of the absolute objective length of the pole in classical science is now given by the absolute objective space-time interval. The analogue of the relative visual angle, or apparent height of the pole, of classical science is given by its relativistic length as measured by the observer; and this length is no more an illusion than was the visual angle. By omitting to analyse the meaning of his words, the critic has argued as though the visual angle, or apparent height, in classical science were an illusion. But the visual angle is not an illusion. The illusion arises only from faulty inferences, when, having identified visual angle with apparent height, we then proceed to confuse the apparent height with the real height of the pole, thereby crediting it with too short a length. It is this last confusion that is unjustified, for it amounts to confusing visual angle, or apparent height (which, though real for the individual, has no universal significance, hence which is relative) with real length (which is absolute in classical science).

In the same way the FitzGerald contraction is not an illusion. It can be measured, and manifests itself physically in every conceivable manner. But again it is a relative for the very concept of spatial magnitude is essentially a relative, since, according to our motion, it will have one value or another. That which is absolute, that which endures unchanged, is the Einsteinian interval; and unless we wish to continue to talk about such shadows as space and time, it is in the space-time world and there alone that we shall obtain universal objectivity.

Let us now state the precise nature of those modifications in lengths and durations which will arise as a result of relative motion. The Einstein-Lorentz equations show that a rigid material object, say a yardstick, which, when at rest in our frame, would measure out as one yard in length, would appear to contract more and more in value in the direction of its relative motion (FitzGerald contraction) as this relative velocity increased. Could the relative velocity reach that of light, the rod would become completely flattened down to nothing.

Likewise, an observer fixed to a clock would note that the durations between the successive second-beats of the clock were truly seconds. But with an increase in the relative motion existing between clock and observer, these beats would measure out as lengthened; and for a relative speed equal to that of light an infinite duration would separate two successive beats.

It must be noted that this lengthening in the durations defined by the tickings of the clock must not be attributed to the fact that the observer is moving farther and farther from the clock and therefore that longer and longer periods of time are necessary for him to take cognisance of these tickings. The results mentioned are those which would be found to hold after all the necessary corrections had been performed. It is solely the relative motion of the observer and the clock which is involved, and not the variations in their mutual distance. Our results would be exactly the same whether we were approaching or receding from the clock, so long as the same magnitude of our relative motion was maintained.

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.