Chapter 7
Section 7 of Einstein and the Universe by Charles Nordman, translated by Joseph Martin McCabe. This LibriVox recording is in the public domain. Chapter 3, Einstein's Solution, continued. Formerly, before the Einsteinian Higura, before the relativist era opened, everybody was convinced that the portion of space occupied by an object was sufficiently and explicitly defined by its dimensions, length, breadth, and height. These are what are called the three dimensions of an object,
just as we speak to use a different expression of the longitude, latitude, and altitude of each of its points, or as we speak in astronomy of its right ascension, declination, and distance. It was quite understood that we had, in addition, to indicate the epoch, the moment to which these data correspond. If I define the position of an aeroplane by its longitude, latitude, and altitude, these indications are only correct for a certain moment, because the aeroplane is moving relatively
to the observer, and the moment also must be indicated. In this sense, it has long been known that space depends upon time. But the relativist theory shows that it depends upon time in a much more intimate and deeper manner, and that time and space are as closely connected as those twin monsters which the surgeon cannot separate without killing both. The dimensions of an object, its shape, the apparent space occupied by it, depend upon its velocity, that is to say, upon the time
which the observer takes to traverse a certain distance relatively to the object. Here we have space already depending upon time. In addition, the observer measures the time with a chronometer, the seconds of which are more or less accelerated according to his velocity. Hence, it is impossible to define space without time. That is why we now say that time is the fourth dimension of space, or that the space in which we live has four dimensions. It is remarkable that there were
able men in the past who had a more or less clear intuition of this. Thus, we find Diderot, in 1777, writing in the Encyclopédie, in the article Dimension, quote, I have already said that it is impossible to conceive more than three dimensions. A learned man of my acquaintance, however, believes that one might regard duration as a fourth dimension, and that the product of time by solidity would be, in a sense, a product of four dimensions. The idea may not be admitted,
but it seems to be not without merit if it be only the merit of originality, unquote. It was algebra, undoubtedly, that gave rise to the idea of a space with more than three dimensions, since in point of fact, lines or spaces of one dimension are represented by algebraic expressions of the first degree, surfaces or spaces of two dimensions by formulae of the second degree, and volumes or spaces of three dimensions by expressions of the third degree.
It was natural to ask oneself if formulae of the fourth and higher degrees are not also the algebraic representation of some form of space with four or more dimensions. The four-dimensional space of the relativists is, however, not quite what Diderot imagined. It is not the product of time by extension, for a diminution of time is not compensated in it by an increase of space.
Quite the contrary. take to events such as the successive passage of our Pullman car through two stations. For a passenger in the car, the distance between the two stations, measured by the length of the track covered, is, as we saw, shorter than for a person who is standing stationary beside the line.
The time between passing through the two stations is likewise less for the first observer. The number of seconds and fractions of seconds marked by his chronometer is smaller for him, as we saw. In a word, distance in time and distance in space diminish simultaneously when the velocity of the observer increases, and both increase when the velocity of the observer lessens. Thus velocity, velocity relative to the things observed, we must always remember,
acts in a sense as a double break, lessening durations, and shortening lengths. If a different illustration be preferred, velocity enables us to see both spaces and times more obliquely, at an increasingly sharp angle. Space and time are therefore only changing effects of perspective. Can we conceive space of four dimensions? That is to say, can we imagine or visualize it? Even if we cannot, it proves nothing as regards the reality of such space.
During ages, no one conceived such a thing as the Hertzian waves, and even today we have no direct sense impression of them. They exist nonetheless. As a matter of fact, we find it difficult to conceive space of three dimensions. If it were not for our muscular changes, we should know nothing about it.
A paralyzed and one-eyed man, that is to say, a man without the sensation of relief which we get from binocular vision, and even this is, in the first place, a muscular sensation, would, with his single eye, see all objects on the same plane, as on the drop scene of a theater. He could have no perception of three-dimensional space. I believe there are people who can form an idea of four-dimensional space.
The successive appearances of a flower in its various phases of growth, from the day when it is but a frail green bud until the time when its exhausted petals fall sadly to the ground, and the successive changes of its corolla under the influence of the wind, give us a globular image of the flower in four-dimensional space.
Are there any who can see all this together? I believe that there are, especially amongst good chess players. When a skillful player plays well, it is because he can take in with a single glance of his mental eye the whole chronological and spatial series of moves that may follow the first move, with all their effects on the board.
He sees the whole series simultaneously. The words I have italicized look contradictory. That is because we are in a province where it is all but impossible to express the fine shades of things in words. One might just as well attempt to define verbally all that there is in a symphony of Beethoven.
Quote, the translator is a traitor, unquote. If there is any truth in the proverb, it is because words are the organ of translation. We have reached a point in our gradual progress into relativist physics where we have before our eyes merely a battlefield strewn with corpses and ruins. We have regarded time and space as hooks solidly fastened to the wall behind which lurks reality, and on these we hang our floating ideas of the material world, just as we hang our coats
on the rack. Now they lie, torn down and crumpled, amongst the rubbish of ancient theories, victims of the hammer blows of the new physics. We knew quite well, of course, that the souls of men were inscrutable to us, but we did think that we saw their faces. Now, as we approach them, we find that it is only masks we saw. The material world, as Einstein shows it to us, is a sort of masked ball, and by a deceptive irony, it is we ourselves who have made the black velvet masks and the gay
costumes. Instead of revealing reality to us, space and time are, according to Einstein, only moving veils, woven by ourselves, which hide it from us. Yet, strange and melancholy reflection, we can no more conceive the world without space and time than we can observe certain microbes under the microscope without first injecting coloring matter into them. Are time and space, then, merely hallucinations? And if so, what is real? No, once the relativist has thrown down
the tottering ruins, he begins to reconstruct. Behind the veils, now torn down and trodden underfoot, a new and more subtle reality is about to appear. If we describe the universe in the usual way, in separate categories of space and time, we see that its aspect depends upon the observer. Happily, it is not the same when we describe it in the unique category of the four dimensional continuum in which Einstein locates phenomena and in which space and time are
inseparably united. If I may venture to use this illustration, time and space are like two mirrors, one convex, the other concave, the curvature of which is accentuated in proportion to the velocity of the observer. Each of these mirrors gives us, separately, a distorted picture of the succession of things. But this is fortunately compensated for by the fact that, when we combine the two mirrors so that one reflects the rays received by the other, the picture of the succession of things
is restored in its unaltered reality. The distance in time and the distance in space of two given events which are close to each other both increase or decrease when the velocity of the observer decreases or increases. We have shown that. But an easy calculation, easy on account of the formula given previously to express the Lorentz-Fitzgerald contraction, shows that there is a constant relation between these concomitant variations of time and space. To be precise, the distance in
time and the distance in space between two contiguous events are numerically to each other as the hypotenuse and another side of a rectangular triangle are to the third side, which remains invariable. Footnote. In the geometrical calculus or representation that may be substituted for this, the hypotenuse of the triangle is the distance in time, each second being represented by 300,000 kilometers. End footnote. Taking this third side for base, the other two will describe, above it,
a triangle more or less elevated according as the velocity of the observer is more or less reduced. This fixed base of the triangle, of which the other two sides, the spatial distance and the chronological distance, vary simultaneously with the velocity of the observer, is, therefore, a quantity independent of the velocity.
It is this quantity which Einstein has called the interval of events. This interval of things in four-dimensional space-time is a sort of conglomerate of space and time, an amalgam of the two. Its components may vary, but it remains itself invariable. It is the constant resultant of two changing vectors. The interval of events thus defined gives us for the first time, according to relativist physics, an impersonal representation
of the universe. In the striking words of Minkowski, quote, space and time are mere phantoms. All that exists in reality is a sort of intimate union of these entities, end quote. The sole reality accessible to man in the external world, the one really objective and impersonal thing which is comprehensible, is the Einsteinian interval as we have defined it. The interval of events is to relativists the sole perceptible part of the real. Apart from that, there is
something, perhaps, but nothing that we can know. Strange destiny of human thought. The principle of relativity has, in virtue of the discoveries of modern physics, spread its wings much farther than it did before, and has reached summits which were thought beyond the range of its soaring flight. Yet it is to this we owe, perhaps, our first real perception of our weakness in regard to the world of sense, in regard to reality. Einstein's system, of which we have now to see
the constructive part, will disappear someday, like the others, for in science there are merely theories with provisional titles, never theories with definitive titles. Possibly that is the reason of its many victories. The idea of the interval of things will, no doubt, survive all these changes. The science of the future must be built upon it. The bold structure of the science of our time rises upon it daily. It must, in fine, be clearly understood that the Einsteinian
interval tells us nothing about the absolute, about things in themselves. It, like all others, shows us only relations between things, but the relations which it discloses seem to be real and unvarying. They share the degree of objective truth which classic science attributed, with perhaps unfounded assurance, to the chronological and spatial relations of phenomena. In the view of the new physics, these were but false scales. The Einsteinian interval alone shows us what can
be known of reality. Einstein's system, therefore, takes pride in having lifted for all future time a corner of the veil which conceals from us the sacred nudity of nature. End of section 7. End of chapter 3.
