EINSTEIN AND THE UNIVERSE

A Popular Exposition of the Famous Theory

By CHARLES NORDMANN

Astronomer to the Paris Observatory.

Translated by JOSEPH McCABE

With a Preface by the Rt. Hon.
THE VISCOUNT HALDANE, O.M.

T. FISHER UNWIN LTD.

LONDON: ADELPHI TERRACE

First published in EnglishApril 1922
Second ImpressionJune 1922

All rights reserved

PREFACE

A distinguished German authority on mathematical physics, writing recently on the theory of Relativity, declared that if his publishers had been willing to allow him sufficient paper and print he could have explained what he wished to convey without using a single mathematical formula. Such success is conceivable. Mathematical methods present, however, two advantages. Their terminology is precise and concentrated, in a fashion which ordinary language cannot afford to adopt. Further, the symbols which result from their employment have implications which, when brought to light, yield new knowledge. This is deductively reached, but it is none the less new knowledge. With greater precision than is usual, ordinary language may be made to do some, if not a great deal, of this work for which mathematical methods are alone quite appropriate. If ordinary language can do part of it an advantage may be gained. The difficulty that attends mathematical symbolism is the accompanying tendency to take the symbol as exhaustively descriptive of reality. Now it is not so descriptive. It always embodies an abstraction. It accordingly leads to the use of metaphors which are inadequate and generally untrue. It is only qualification by descriptive language of a wider range that can keep this tendency in check. A new school of mathematical physicists, still, however, small in number, is beginning to appreciate this.

But for English and German writers the new task is very difficult. Neither Anglo-Saxon nor Saxon genius lends itself readily in this direction. Nor has the task as yet been taken in hand completely, so far as I am aware, in France. Still, in France there is a spirit and a gift of expression which makes the approach to it easier than either for us or for the Germans. Lucidity in expression is an endowment which the best French writers possess in a higher degree than we do. Some of us have accordingly awaited with deep interest French renderings of the difficult doctrine of Einstein.

M. Nordmann, in addition to being a highly qualified astronomer and mathematical-physicist, possesses the gift of his race. The Latin capacity for eliminating abstractness from the description of facts is everywhere apparent in his writing. Individual facts take the places of general conceptions, of Begriffe. The language is that of the Vorstellung, in a way that would hardly be practicable in German. Nor is our own language equal to that of France in delicacy of distinctive description. This book could hardly have been written by an Englishman. But the difficulty in his way would have been one as much of spirit as of letter. It is the lucidity of the French author, in combination with his own gift of expression, that has made it possible for the translator to succeed so well in overcoming the obstacles to giving the exposition in our own tongue this book contains. The rendering seems to me, after reading the book both in French and in English, admirable.

M. Nordmann has presented Einstein’s principle in words which lift the average reader over many of the difficulties he must encounter in trying to take it in. Remembering Goethe’s maxim that he who would accomplish anything must limit himself, he has not aimed at covering the full field to which Einstein’s teaching is directed. But he succeeds in making many abstruse things intelligible to the layman. Perhaps the most brilliant of his efforts in this direction are Chapters [V] and [VI], in which he explains with extraordinary lucidity the new theory of gravitation and of its relation to inertia. I think that M. Nordmann is perhaps less successful in the courageous attack he makes in his [third chapter] on the obscurity which attends the notion of the “Interval.” But that is because the four-dimensional world, which is the basis of experience of space and time for Einstein and Minkowski, is in itself an obscure conception. Mathematicians talk about it gaily and throw its qualities into equations, despite the essential exclusion from it of the measurement and shape which actual experience always in some form involves. They lapse on that account into unconscious metaphysics of a dubious character. This does not destroy the practical value of their equations, but it does make them very unreliable as guides to the character of reality in the meaning which the plain man attaches to it. Here, accordingly, we find the author of this little treatise to be a good man struggling with adversity. If he could make the topic clear he would. But then no one has made it clear excepting as an abstraction which works, but which, despite suggestions made to the contrary, cannot be clothed for us in images.

This, however, is the fault, not of M. Nordmann himself, but of a phase of the subject. With the subject in its other aspects he deals with the incomparable lucidity of a Frenchman. I know no book better adapted than the one now translated to give the average English reader some understanding of a principle, still in its infancy, but destined, as I believe, to transform opinion in more regions of knowledge than those merely of mathematical physics.

Haldane

CONTENTS

INTRODUCTION

This book is not a romance. Nevertheless.... If love is, as Plato says, a soaring toward the infinite, where shall we find more love than in the impassioned curiosity which impels us, with bowed heads and beating hearts, against the wall of mystery that environs our material world? Behind that wall, we feel, there is something sublime. What is it? Science is the outcome of the search for that mysterious something.

A giant blow has recently been struck, by a man of consummate ability, Albert Einstein, upon this wall which conceals reality from us. A little of the light from beyond now comes to us through the breach he has made, and our eyes are enchanted, almost dazzled, by the rays. I propose here to give, as simply and clearly as is possible, some faint reflex of the impression it has made upon us.

Einstein’s theories have brought about a profound revolution in science. In their light the world seems simpler, more co-ordinated, more in unison. We shall henceforward realise better how grandiose and coherent it is, how it is ruled by an inflexible harmony. A little of the ineffable will become clearer to us.

Men, as they pass through the universe, are like those specks of dust which dance for a moment in the golden rays of the sun, then sink into the darkness. Is there a finer or nobler way of spending this life than to fill one’s eyes, one’s mind, one’s heart with the immortal, yet so elusive, rays? What higher pleasure can there be than to contemplate, to seek, to understand, the magnificent and astounding spectacle of the universe?

There is in reality more of the marvellous and the romantic than there is in all our poor dreams. In the thirst for knowledge, in the mystic impulse which urges us toward the deep heart of the Unknown, there is more passion and more sweetness than in all the trivialities which sustain so many literatures. I may be wrong, after all, in saying that this book is not a romance.

I will endeavour in these pages to make the reader understand, accurately, yet without the aid of the esoteric apparatus of the technical writer, the revolution brought about by Einstein. I will try also to fix its limits; to state precisely what, at the most, we can really know to-day about the external world when we regard it through the translucent screen of science.

Every revolution is followed by a reaction, in virtue of the rhythm which seems to be an inherent and eternal law of the mind of man. Einstein is at once the Sieyès, the Mirabeau, and the Danton of the new revolution. But the revolution has already produced its fanatical Marats, who would say to science: “Thus far and no farther.”

Hence we find some resistance to the pretensions of over-zealous apostles of the new scientific gospel. In the Academy of Sciences M. Paul Painlevé takes his place, with all the strength of a vigorous mathematical genius, between Newton, who was supposed to be overthrown, and Einstein. In my final pages I will examine the penetrating criticisms of the great French geometrician. They will help me to fix the precise position, in the evolution of our ideas, of Einstein’s magnificent synthesis. But I would first expound the synthesis itself with all the affection which one must bestow upon things that one would understand.

Science has not completed its task with the work of Einstein. There remains many a depth that is for us unfathomable, waiting for some genius of to-morrow to throw light into it. It is the very essence of the august and lofty grandeur of science that it is perpetually advancing. It is like a torch in the sombre forest of mystery. Man enlarges every day the circle of light which spreads round him, but at the same time, and in virtue of his very advance, he finds himself confronting, at an increasing number of points, the darkness of the Unknown. Few men have borne the shaft of light so deeply into the forest as has Einstein. In spite of the sordid cares which harass us to-day, amid so many grave contingencies, his system reveals to us an element of grandeur.

Our age is like the noisy and unsubstantial froth that crowns, and hides for a moment, the gold of some generous wine. When all the transitory murmur that now fills our ears is over, Einstein’s theory will rise before us as the great lighthouse on the brink of this sad and petty twentieth century of ours.

Charles Nordmann.

EINSTEIN AND THE UNIVERSE

CHAPTER I

THE METAMORPHOSES OF
SPACE AND TIME

Removing the mathematical difficulties—The pillars of knowledge—Absolute time and space, from Aristotle to Newton—Relative time and space, from Epicurus to Poincaré and Einstein—Classical Relativity—Antinomy of stellar aberration and the Michelson experiment.

“Have you read Baruch?” La Fontaine used to cry, enthusiastically. To-day he would have troubled his friends with the question “Have you read Einstein?”

But, whereas one needs only a little Latin to gain access to Spinoza, frightful monsters keep guard before Einstein, and their horrible grimaces seem to forbid us to approach him. They stand behind strange moving bars, sometimes rectangular and sometimes curvilinear, which are known as “co-ordinates.” They bear names as frightful as themselves—“contravariant and covariant vectors, tensors, scalars, determinants, orthogonal vectors, generalised symbols of three signs,” and so on.

These strange beings, brought from the wildest depths of the mathematical jungle, join together or part from each other with a remarkable promiscuity, by means of some astonishing surgery which is called integration and differentiation.

In a word, Einstein may be a treasure, but there is a fearsome troop of mathematical reptiles keeping inquisitive folk away from it; though there can be no doubt that they have, like our Gothic gargoyles, a hidden beauty of their own. Let us, however, drive them off with the whip of simple terminology, and approach the splendour of Einstein’s theory.

Who is this physicist Einstein? That is a question of no importance here. It is enough to know that he refused to sign the infamous manifesto of the professors, and thus brought upon himself persecution from the Pan-Germanists.[1] Mathematical truths and scientific discoveries have an intrinsic value, and this must be judged and appreciated impartially, whoever their author may chance to be. Had Pythagoras been the lowest of criminals, the fact would not in the least detract from the validity of the square of the hypotenuse. A theory is either true or false, whether the nose of its author has the aquiline contour of the nose of the children of Sem, or the flattened shape of that of the children of Cham, or the straightness of that of the children of Japhet. Do we feel that humanity is perfect when we hear it said occasionally: “Tell me what church you frequent, and I will tell you if your geometry is sound.” Truth has no need of a civil status. Let us get on.

All our ideas, all science, and even the whole of our practical life, are based upon the way in which we picture to ourselves the successive aspects of things. Our mind, with the aid of our senses, chiefly ranges these under the headings of time and space, which thus become the two frames in which we dispose all that is apparent to us of the material world. When we write a letter, we put at the head of it the name of the place and the date. When we open a newspaper, we find the same indications at the beginning of each piece of telegraphic news. It is the same in everything and for everything. Time and space, the situation and the period of things, are thus seen to be the twin pillars of all knowledge, the two columns which sustain the edifice of men’s understanding.

So felt Leconte de Lisle when, addressing himself to “divine death,” he wrote, in his profound, philosophic way:

Free us from time, number, and space:

Grant us the rest that life hath spoiled.

He inserts the word “number” only in order to define time and space quantitatively. What he has finely expressed in these famous and superb lines is the fact that all that there is for us in this vast universe, all that we know and see, all the ineffable and agitated flow of phenomena, presents to us no definite aspect, no precise form, until it has passed through those two filters which are interposed by the mind, time and space.

The work of Einstein derives its importance from the fact that he has shown, as we shall see, that we have entirely to revise our ideas of time and space. If that is so, the whole of science, including psychology, will have to be reconstructed. That is the first part of Einstein’s work, but it goes further. If that were the whole of his work it would be merely negative.

Once he had removed from the structure of human knowledge what had been regarded as an indispensable wall of it, though it was really only a frail scaffolding that hid the harmony of its proportions, he began to reconstruct. He made in the structure large windows which allow us now to see the treasures it contains. In a word, Einstein showed, on the one hand, with astonishing acuteness and depth, that the foundation of our knowledge seems to be different from what we had thought, and that it needs repairing with a new kind of cement. On the other hand, he has reconstructed the edifice on this new basis, and he has given it a bold and remarkably beautiful and harmonious form.

I have now to show in detail, concretely, and as accurately as possible, the meaning of these generalities. But I must first insist on a point which is of considerable importance: if Einstein had confined himself to the first part of his work, as I have described it, the part which shatters the classical ideas of time and space, he would never have attained the fame which now makes his name great in the world of thought.

The point is important because most of those—apart from experts—who have written on Einstein have chiefly, often exclusively, emphasised this more or less “destructive” side of his work. But, as we shall see, from this point of view Einstein was not the first, and he is not alone. All that he has done is to sharpen, and press a little deeper between the badly joined stones of classical science, a chisel which others, especially the great Henri Poincaré, had used long before him. My next point is to explain, if I can, the real, the immortal, title of Einstein to the gratitude of men: to show how he has by his own powers rebuilt the structure in a new and magnificent form after his critical work. In this he shares his glory with none.

The whole of science, from the days of Aristotle until our own, has been based upon the hypothesis—properly speaking, the hypotheses—that there is an absolute time and an absolute space. In other words, our ideas rested upon the supposition that an interval of time and an interval of space between two given phenomena are always the same, for every observer whatsoever, and whatever the conditions of observation may be. For instance, it would never have occurred to anybody as long as classical science was predominant, that the interval of time, the number of seconds, which lies between two successive eclipses of the sun, may not be the fixed and identically same number of seconds for an observer on the earth as for an observer in Sirius (assuming that the second is defined for both by the same chronometer). Similarly, no one would have imagined that the distance in metres between two objects, for instance the distance of the earth from the sun at a given moment, measured by trigonometry, may not be the same for an observer on the earth as for an observer in Sirius (the metre being defined for both by the same rule).

“There is,” says Aristotle, “one single and invariable time, which flows in two movements in an identical and simultaneous manner; and if these two sorts of time were not simultaneous, they would nevertheless be of the same nature.... Thus, in regard to movements which take place simultaneously, there is one and the same time, whether or no the movements are equal in rapidity; and this is true even if one of them is a local movement and the other an alteration.... It follows that even if the movements differ from each other, and arise independently, the time is absolutely the same for both.”[2] This Aristotelic definition of physical time is more than two thousand years old, yet it clearly represents the idea of time which has been used in classic science, especially in the mechanics of Galileo and Newton, until quite recent years.

It seems, however, that in spite of Aristotle, Epicurus outlined the position which Einstein would later adopt in antagonism to Newton. To translate liberally the words in which Lucretius expounds the teaching of Epicurus:

“Time has no existence of itself, but only in material objects, from which we get the idea of past, present, and future. It is impossible to conceive time in itself independently of the movement or rest of things.”[3]

Both space and time have been regarded by science ever since Aristotle as invariable, fixed, rigid, absolute data. Newton thought that he was saying something obvious, a platitude, when he wrote in his celebrated Scholion: “Absolute, true, and mathematical time, taken in itself and without relation to any material object, flows uniformly of its own nature.... Absolute space, on the other hand, independent by its own nature of any relation to external objects, remains always unchangeable and immovable.”

The whole of science, the whole of physics and mechanics, as they are still taught in our colleges and in most of our universities, are based entirely upon these propositions, these ideas of an absolute time and space, taken by themselves and without any reference to an external object, independent by their very nature.

In a word—if I may venture to use this figure—time in classical science was like a river bearing phenomena as a stream bears boats, flowing on just the same whether there were phenomena or not. Space, similarly, was rather like the bank of the river, indifferent to the ships that passed.

From the time of Newton, however, if not from the time of Aristotle, any thoughtful metaphysician might have noticed that there was something wrong in these definitions. Absolute time and absolute space are “things in themselves,” and these the human mind has always regarded as not directly accessible to it. The specifications of space and time, those numbered labels which we attach to objects of the material world, as we put labels on parcels at the station so that they may not be lost (a precaution that does not always suffice), are given us by our senses, whether aided by instruments or not, only when we receive concrete impressions. Should we have any idea of them if there were no bodies attached to them, or rather to which we attach the labels? To answer this in the affirmative, as Aristotle, Newton, and classical science do, is to make a very bold assumption, and one that is not obviously justified.

The only time of which we have any idea apart from all objects is the psychological time so luminously studied by M. Bergson: a time which has nothing except the name in common with the time of physicists, of science.

It is really to Henri Poincaré, the great Frenchman whose death has left a void that will never be filled, that we must accord the merit of having first proved, with the greatest lucidity and the most prudent audacity, that time and space, as we know them, can only be relative. A few quotations from his works will not be out of place. They will show that the credit for most of the things which are currently attributed to Einstein is, in reality, due to Poincaré. To prove this is not in any way to detract from the merit of Einstein, for that is, as we shall see, in other fields.

This is how Poincaré, whose ideas still dominate the minds of thoughtful men, though his mortal frame perished years ago, expressed himself, the triumphant sweep of his wings reaching further every day:

“One cannot form any idea of empty space.... From that follows the undeniable relativity of space. Any man who talks of absolute space uses words which have no meaning. I am at a particular spot in Paris—the Place du Panthéon, let us suppose—and I say: ‘I will come back here to-morrow.’ If anyone asks me whether I mean that I will return to the same point in space, I am tempted to reply, ‘Yes.’ I should, however, be wrong, because between this and to-morrow the earth will have travelled, taking the Place du Panthéon with it, so that to-morrow the square will be more than 2,000,000 kilometres away from where it is now. And it would be no use my attempting to use precise language, because these 2,000,000 kilometres are part of our earth’s journey round the sun, but the sun itself has moved in relation to the Milky Way, and the Milky Way in turn is doubtless moving at a speed which we cannot learn. Thus we are entirely ignorant, and always will be ignorant, how far the Place du Panthéon shifts its position in space in a single day. What I really meant to say was: ‘To-morrow I shall again see the dome and façade of the Panthéon.’ If there were no Panthéon, there would be no meaning in my words, and space would disappear.”

Poincaré works out his idea in this way:

“Suppose all the dimensions of the universe were increased a thousandfold in a night. The world would remain the same, giving the word ‘same’ the meaning it has in the third book of geometry. Nevertheless, an object that had measured a metre in length will henceforward be a kilometre in length; a thing that had measured a millimetre will now measure a metre. The bed on which I lie and the body which lies on it will increase in size to exactly the same extent. What sort of feelings will I have when I awake in the morning, in face of such an amazing transformation? Well, I shall know nothing about it. The most precise measurements would tell me nothing about the revolution, because the tape I use for measuring will have changed to the same extent as the objects I wish to measure. As a matter of fact, there would be no revolution except in the mind of those who reason as if space were absolute. If I have argued for a moment as they do, it was only in order to show more clearly that their position is contradictory.”

It would be easy to develop Poincaré’s argument. If all the objects in the universe were to become, for instance, a thousand times taller, a thousand times broader, we should be quite unable to detect it, because we ourselves—our retina and our measuring rod—would be transformed to the same extent at the same time. Indeed, if all the things in the universe were to experience an absolutely irregular spatial deformation—if some invisible and all-powerful spirit were to distort the universe in any fashion, drawing it out as if it were rubber—we should have no means of knowing the fact. There could be no better proof that space is relative, and that we cannot conceive space apart from the things which we use to measure it. When there is no measuring rod, there is no space.

Poincaré pushed his reasoning on this subject so far that he came to say that even the revolution of the earth round the sun is merely a more convenient hypothesis than the contrary supposition, but not a truer hypothesis, unless we imply the existence of absolute space.

It may be remembered that certain unwary controversialists have tried to infer from Poincaré’s argument that the condemnation of Galileo was justified. Nothing could be more amusing than the way in which the distinguished mathematician-philosopher defended himself against this interpretation, though one must admit that his defence was not wholly convincing. He did not take sufficiently into account the agnostic element.

Poincaré, in any case, is the leader of those who regard space as a mere property which we ascribe to objects. In this view our idea of it is only, so to say, the hereditary outcome of those efforts of our senses by means of which we strive to embrace the material world at a given moment.

It is the same with time. Here again the objections of philosophic Relativists were raised long ago, but it was Poincaré who gave them their definitive shape. His luminous demonstrations are, however, well known, and we need not reproduce them here. It is enough to observe that, in regard to time as well as space, it is possible to imagine either a contraction or an enlargement of the scale which would be completely imperceptible to us; and this seems to show that man cannot conceive an absolute time. If some malicious spirit were to amuse itself some night by making all the phenomena of the universe a thousand times slower, we should not, when we awake, have any means of detecting the change. The world would seem to us unchanged. Yet every hour recorded by our watches would be a thousand times longer than hours had previously been. Men would live a thousand times as long, yet they would be unaware of the fact, as their sensations would be slower in the same proportion.

When Lamartine appealed to time to “suspend its flight,” he said a very charming, but perhaps meaningless, thing. If time had obeyed his passionate appeal, neither Lamartine nor Elvire would have known and rejoiced over the fact. The boatman who conducted the lovers on the Lac du Bourget would not have asked payment for a single additional hour; yet he would have dipped his oars into the pleasant waters for a far longer time.

I venture to sum up all this in a sentence which will at first sight seem a paradox: in the opinion of the Relativists it is the measuring rods which create space, the clocks which create time. All this was maintained by Poincaré and others long before the time of Einstein, and one does injustice to truth in ascribing the discovery to him. I am quite aware that one lends only to the rich, but one does an injustice to the wealthy themselves in attributing to them what does not belong to them, and what they need not in order to be rich.

There is, moreover, one point at which Galileo and Newton, for all their belief in the existence of absolute space and time, admitted a certain relativity. They recognised that it is impossible to distinguish between uniform movements of translation. They thus admitted the equivalence of all such movements, and therefore the impossibility of proving an absolute movement of translation.

That is what is called the Principle of Classic Relativity.

An unexpected fact served to bring these questions upon a new plane, and led Einstein to give a remarkable extension to the Principle of Relativity of classic mechanics. This was the issue of a famous experiment by Michelson, of which we must give a brief description.

It is well known that rays of light travel across empty space from star to star, otherwise we should be unable to see the stars. From this physicists long ago concluded that the rays travelled in a medium that is devoid of mass and inertia, is infinitely elastic, and offers no resistance to the movement of material bodies, into which it penetrates. This medium has been named ether. Light travels through it as waves spread over the surface of water at a speed of something like 186,000 miles a second: a velocity which we will express by the letter V.

The earth revolves round the sun in a veritable ocean of ether, at a speed of about 18 miles a second. In this respect the rotation of the earth on its axis need not be noticed, as it pushes the surface of the globe through the ether at a speed of less than two miles a second. Now the question had often been asked: Does the earth, in its orbital movement round the sun, take with it the ether which is in contact with it, as a sponge thrown out of a window takes with it the water which it has absorbed? Experiment—or rather, experiments, for many have been tried with the same result—has shown that the question must be answered in the negative.

This was first established by astronomical observation. There is in astronomy a well-known phenomenon discovered by Bradley which is called aberration. It consists in this: when we observe a star with a telescope, the image of the star is not precisely in the direct line of vision. The reason is that, while the luminous rays of the star which have entered the telescope are passing down the length of the tube, the instrument has been slightly displaced, as it shares the movement of the earth. On the other hand, the luminous ray in the tube does not share the earth’s motion, and this gives rise to the very slight deviation which we call aberration. This proves that the medium in which light travels, the ether which fills the instrument and surrounds the earth, does not share the earth’s motion.

Many other experiments have settled beyond question that the ether, which is the vehicle of the waves of light, is not borne along by the earth as it travels. Now, since the earth moves through the ether as a ship moves over a stationary lake (not like one floating on a moving stream), it ought to be possible to detect some evidence of this speed of the earth in relation to the ether.

One of the devices that may be imagined for the purpose is the following. We know that the earth turns on itself from west to east, and travels round the sun in the same way. It follows that in the middle of the night the revolution of the earth round the sun means that Paris will be displaced, in the direction from Auteuil toward Charenton, at a speed of about thirty kilometres a second. During the day, of course, it is precisely the opposite. Paris changes its place round the sun in the direction from Charenton toward Auteuil. Well, let us suppose that at midnight a physicist at Auteuil sends a luminous signal. A physicist receiving this ray of light at Charenton, and measuring its velocity, ought to find that the latter is V + 30 kilometres. We know that, as a result of the earth’s motion, Charenton recedes before the ray of light. Consequently, since light travels in a medium, the ether, which does not share the earth’s motion, the observer at Charenton ought to find that the ray reaches him at a less speed than it would if the earth were stationary. It is much the same as if an observer were travelling on a bicycle in front of an express train. If the express travels at thirty metres a second and the cyclist at three metres a second, the speed of the train in relation to the cyclist will be 30-3 = 27 metres a second. It would be nil if the train and the cyclist were travelling at the same rate.

On the other hand, if the cyclist were going toward the train, the speed of the train in relation to him would be 30 + 3 = 33 metres a second. Similarly, when the physicist at Charenton sends out a luminous message at midnight, and the physicist of Auteuil receives it, the latter ought to find that the ray of light has a velocity of V + 30 kilometres.

All this may be put in a different way. Suppose the distance between the observer at Auteuil and the man at Charenton were exactly twelve kilometres. While the ray of light emitted at Auteuil speeds toward Charenton, that town is receding before it to a small extent. It follows that the ray will have to travel a little more than twelve kilometres before it reaches the man of science at Charenton. It will travel a little less than that distance if we imagine it proceeding in the opposite direction.

Now the American physicist Michelson, borrowing an ingenious idea from the French physicist Fizeau, succeeded, with a high degree of accuracy, in measuring distances by means of the interference-bands of light. Every variation in the distance measured betrays itself by the displacement of a certain number of these bands, and this may easily be detected by a microscope.

Let us next suppose that our two physicists work in a laboratory instead of between Charenton and Auteuil. Let us suppose that they are, by means of the interference-bands, measuring the space traversed by a ray of light produced in the laboratory, according as it travels in the same direction as the earth or in the opposite direction. That is Michelson’s famous experiment, reduced to its essential elements and simplified for the purpose of this essay. In those circumstances Michelson’s delicate apparatus ought to reveal a distinctly measurable difference according as the light travels with the earth or in the opposite direction.

But no such difference was found. Contrary to all expectation, and to the profound astonishment of physicists, it was found that light travels at precisely the same speed whether the man who receives it is receding before it with the velocity of the earth or is approaching it at the same velocity. It is an undeniable consequence of this that the ether shares the motion of the earth. We have, however, seen that other experiments, not less precise, had settled that the ether does not share the motion of the earth.

Out of this contradiction, this conflict of two irreconcilable yet indubitable facts, Einstein’s splendid synthesis, like a spark of light issuing from the clash of flint and steel, came into being.

CHAPTER II

SCIENCE IN A NO-THOROUGHFARE

Scientific truth and mathematics—The precise function of Einstein—Michelson’s experiment, the Gordian knot of science—The hesitations of Poincaré—The strange, but necessary, Fitzgerald-Lorentz hypothesis—The contraction of moving bodies—Philosophical and physical difficulties.

It would be foolish to pretend that we can penetrate the most obscure corners of Einstein’s theories without the aid of mathematics. I believe, however, that we can give in ordinary language—that is to say, by means of illustrations and analogies—a fairly satisfactory idea of these things, the intricacy of which is usually due to the infinitely subtle and supple play of mathematical formulæ and equations.

After all, mathematics is not, never was, and never will be, anything more than a particular kind of language, a sort of shorthand of thought and reasoning. The purpose of it is to cut across the complicated meanderings of long trains of reasoning with a bold rapidity that is unknown to the mediæval slowness of the syllogisms expressed in our words.

However paradoxical this may seem to people who regard mathematics as of itself a means of discovery, the truth is that we can never get from it anything that was not implicitly inherent in the data which were thrust between the jaws of its equations. If I may use a somewhat trivial illustration, mathematical reasoning is very like certain machines which are seen in Chicago—so bold explorers in the United States tell us—into which one puts living animals that emerge at the other end in the shape of appetising prepared meats. No spectator could have, or would wish to have, eaten the animal alive, but in the form in which it issues from the machine it can at once be digested and assimilated. Yet the meat is merely the animal conveniently prepared. That is what mathematics does. By means of a marvellous machinery the mathematician extracts the valuable marrow from the given facts. It is a machinery that is particularly useful in cases where the wheels of verbal argument, the chain of syllogisms, would soon be brought to a halt.

Does it follow that, properly speaking, mathematics is not a science? Does it follow at least that it is only a science in so far as it is based upon reality, and fed with experimental data, since “experience is the sole source of truth.” I refrain from answering the question, as I am one of those who believe that everything is material for science. Still, it was worth while to raise the question because many are too much disposed to regard a purely mathematical education as a scientific education. Nothing could be further from the truth. Pure mathematics is, in itself, merely an abbreviated form of language and of logical thought. It cannot, of its own nature, teach us anything about the external world; it can do so only in proportion as it enters into contact with the world. It is of mathematics in particular that we may say: Naturæ non imperatur nisi parendo.

Are not Einstein’s theories, as some imperfectly informed writers have suggested, only a play of mathematical formulæ (taking the word in the meaning given to it by both mathematicians and philosophers)? If they were only a towering mathematical structure in which the x’s shoot out their volutes in bewildering arabesques, with swan-neck integrals describing Louis XV patterns, they would have no interest whatever for the physicist, for the man who has to examine the nature of things before he talks about it. They would, like all coherent schemes of metaphysics, be merely a more or less agreeable system of thought, the truth or falseness of which could never be demonstrated.

Einstein’s theory is very different from that, and very much more than that. It is based upon facts. It also leads to facts—new facts. No philosophical doctrine or purely formal mathematical construction ever enabled us to discover new phenomena. It is precisely because it has led to such discovery that Einstein’s theory is neither the one nor the other. That is the difference between a scientific theory and a pure speculation, and it is that which, I venture to say, makes the former so superior.

Like some suspension bridge boldly thrown across an abyss, Einstein’s theory rests, on the one side, on experimental phenomena, and it leads, at the other side, to other, and hitherto unsuspected, phenomena, which it has enabled us to discover. Between these two solid experimental columns the mathematical reasoning is like the marvellous network of thousands of steel bars which represent the elegant and translucent structure of the bridge. It is that, and nothing but that. But the arrangement of the beams and bars might have been different, and the bridge—though less light and graceful, perhaps—still have been able to join together the two sets of facts on which it rests.

In a word, mathematical reasoning is only a kind of reasoning in a special language, from experimental premises to conclusions which are verifiable by experience. Now there is no language which cannot in some degree be translated into another language. Even the hieroglyphics of Egypt had to give way before Champollion. I am therefore convinced that the mathematical difficulties of Einstein’s theories will some day be replaced by simpler and more accessible formulæ. I believe, indeed, that it is even now possible to give by means of ordinary speech an idea, rather superficial perhaps, but accurate and substantially complete, of this wonderful Einsteinian structure which ranges all the conquests of science, as in some well-ordered museum, in a new and superb unity. Let us try.

We may resume in the few following words the story of the origin, the starting-point, of Einstein’s system.

1. Observation of the stars proves that interplanetary space is not empty, but is filled with a special medium, ether, in which the waves of light travel.

2. The fact of aberration and other phenomena seems to prove that the ether is not displaced by the earth during its course round the sun.

3. Michelson’s experiment seems to prove, on the contrary, that the earth bears the ether with it in its movement.

This contradiction between facts of equal authority was for years the despair and the wonder of physicists. It was the Gordian knot of science. Long and fruitless efforts were made to untie it until at last Einstein cut it with a single blow of his remarkably acute intelligence.

In order to understand how that was done—which is the vital point of the whole system—we must retrace our steps a little and examine the precise conditions of Michelson’s famous experiment.

I pointed out in the [preceding chapter] that Michelson proposed to study the speed of a ray of light produced in the laboratory and directed either from east to west or west to east: that is to say, in the direction in which the earth itself moves, at a speed of about eighteen miles a second, as it travels round the sun, or in the opposite direction. As a matter of fact, Michelson’s experiment was rather more complicated than that, and we must return to it.

Four mirrors are placed at an equal distance from each other in the laboratory, in pairs which face each other. Two of the opposing mirrors are arranged in the direction east-west, the direction in which the earth moves in consequence of its revolution round the sun. The other two are arranged in a plane perpendicular to the preceding, the direction north-south. Two rays of light are then started in the respective directions of the two pairs of mirrors. The ray coming from the mirror to the east goes to the mirror in the west, is reflected therefrom, and returns to the first mirror. This ray is so arranged that it crosses the path of the light which goes from north to south and back. It interferes with the latter light, causing “fringes of interference” which, as I said, enable us to learn the exact distance traversed by the rays of light reflected between the pairs of mirrors. If anything brought about a difference between the length of the two distances, we should at once see the displacement of a certain number of interference-fringes, and this would give us the magnitude of the difference.

An analogy will help us to understand the matter. Suppose a violent steady east wind blew across London, and an aviator proposed to cross the city about twelve miles from extreme west to east and back: that is to say, going with the wind on his outward journey and against it on the return journey. Suppose another aviator, of equal speed, proposed at the same time to fly from the same starting-point to a point twelve miles to the north and back, the second aviator will fly both ways at right angles to the direction of the wind. If the two start at the same time, and are imagined as turning round instantaneously, will they both reach the starting-point together? And, if not, which of them will have completed his double journey first?

It is clear that if there were no wind, they would get back together, as we suppose that they both do twenty-four miles at the same speed, which we may roughly state to be 200 yards a second.

But it will be different if, as I postulated, there is a wind blowing from east to west. It is easy to see that in such circumstances the man who flies east to west will take longer to complete the journey. In order to get it quite clearly, let us suppose that the wind is travelling at the same speed as the aviator (200 yards a second). The man who flies at right angles to the wind will be blown twelve miles to the west while he is doing his twelve miles from south to north. He will therefore have traversed in the wind a real distance equal to the diagonal of a square measuring twelve miles on each side. Instead of flying twenty-four miles, he will really have flown thirty-four in the wind, the medium in relation to which he has any velocity.

On the other hand, the aviator who flies eastward will never reach his destination, because in each second of time he is driven westward to precisely the same extent as he is travelling eastward. He will remain stationary. To accomplish his journey he would need to cover in the wind an infinite distance.

If, instead of imagining a wind equal in velocity to the aviator (an extreme supposition in order to make the demonstration clearer), I had thought of it as less rapid, we should again find, by a very simple calculation, that the man who flies north and south has less distance to cover in the wind than the man who flies east and west.

Now take rays of light instead of aviators, the ether instead of the wind, and we have very nearly the conditions of the Michelson experiment. A current or wind of ether—since the ether has been already shown to be stationary in relation to the earth’s movement—proceeds from one to the other of our east-west mirrors. Therefore the ray of light which travels between these two mirrors, forth and back, must cover a longer distance in ether than the ray which goes from the south mirror to the north and back. But how are we to detect this difference? It is certainly very minute, because the speed of the earth is ten thousand times less than the velocity of light.

There is a very simple means of doing this: one of those ingenious devices which physicists love, a differential device so elegant and precise that we have entire confidence in the result.

Let us suppose that our four mirrors are fixed rigidly in a sort of square frame, something like those “wheels of fortune” with numbers on them that one sees in country fairs. Let us suppose that we can turn this frame round as we wish, without jerking or displacing it, which is not difficult if it floats in a bath of mercury. I then take a lens and observe the permanent interference-fringes which define the difference between the paths traversed by my two rays of light, north-south and east-west. Then, without losing sight of the bands or fringes, I turn the frame round a quarter of a circle. Owing to this rotation the mirrors which were east-west now become north-south, and vice versa. The double journey made by the north-south ray of light has now taken the direction east-west, and has therefore suddenly been lengthened; the double journey of the east-west ray has become north-south, and has been suddenly shortened. The interference-fringes, which indicate the difference in length between the two paths, which has suddenly changed, must necessarily be displaced, and that, as we can calculate, to no slight extent.

Well, we find no change whatever! The fringes remain unaltered. They are as stationary as stumps of trees. It is bewildering, one would almost say revolting, because the delicacy of the apparatus is such that, even if the earth moved through the ether at a rate of only three kilometres a second (or ten times less than its actual velocity), the displacement of the fringes would be sufficient to indicate the speed.

When the negative result of this experiment was announced, there was something like consternation amongst the physicists of the world. Since the ether was not borne along by the earth, as observation had established, how could it possibly behave as if it did share the earth’s motion? It was a Chinese puzzle. More than one venerable grey head was in despair over it.

It was absolutely necessary to find a way out of this inexplicable contradiction, to end this paradoxical mockery which the facts seemed to oppose to the most rigorous results of calculation. This the men of science succeeded in doing. How? By the method which is generally used in such circumstances—by means of supplementary hypotheses. Hypotheses in science are a kind of soft cement which hardens rapidly in the open air, thus enabling us to join together the separate blocks of the structure, and to fill up the breaches made in the wall by projectiles, with artificial stuff which the superficial observer presently mistakes for stone. It is because hypotheses are something like that in science that the best scientific theories are those which include least hypotheses.

But I am wrong in using the plural in this connection. In the end it was found that one single hypothesis conveniently explained the negative result of the Michelson experiment. That is, by the way, a rare and remarkable experience. Hypotheses usually spring up like mushrooms in every dark corner of science. You get a score of them to explain the slightest obscurity.

This single hypothesis, which seemed to be capable of extricating physicists from the dilemma into which Michelson had put them, was first advanced by the distinguished Irish mathematician Fitzgerald, then taken up and developed by the celebrated Dutch physicist Lorentz, the Poincaré of Holland, one of the most brilliant thinkers of our time. Einstein would no more have attained fame without him than Kepler would without Copernicus and Tycho Brahe.

Let us now see what this Fitzgerald-Lorentz hypothesis, as strange as it is simple, really is.

But we must first glance at a preliminary matter of some importance. A number of able men have declared—after the issue, let it be said—that the result of the Michelson experiment could only be negative a priori. In point of fact, they argue (more or less), the Classic Principle of Relativity, the principle known to Galileo and Newton, implies that it is impossible for an observer who shares the motion of a vehicle to detect the motion of that vehicle by any facts he observes while he is in it. Thus, when two ships or two trains pass each other,[4] it is impossible for the passengers to say which of the two is moving, or moving the more rapidly. All that they can perceive is the relative speed of the trains or ships.

The men of science to whom I have referred say that, if Michelson’s experiment had had a positive result, it would have given us the absolute velocity of the earth in space. This result would have been contrary to the Principle of Relativity of classical philosophy and mechanics, which is a self-evident truth. Therefore the result could only be negative.

This is, as we shall see, ambiguous. There is, if I may say so, a flaw in the argument which has escaped the notice even of distinguished men of science like Professor Eddington, the most erudite of the English Einsteinians. It was he who organised the observations of the solar eclipse of May 29, 1919, which have, as we shall see, furnished the most striking verification of Einstein’s deductions.

In the first place, if Michelson’s experiment had had a positive result, what it would have indicated is the velocity of the earth in relation to the ether. But, for this to be an absolute velocity, the ether would have to be identical with space. This is so far from being necessary that we can easily conceive a space—to put it better, a discontinuity—between two stars that contains no ether and across which neither light nor any other known form of energy would travel.

When Eddington says that “it is legitimate and reasonable,” that it is “inherent in the fundamental laws of nature,” that we cannot detect any movement of bodies in relation to ether, and that this is certain “even if the experimental evidence is inadequate,” he affirms something which would be evident only if space and ether were evidently identical. But this is far from being the case. If Michelson’s experiment had had a positive result, if we had detected a velocity on the part of the earth, should we have discovered a velocity in relation to an absolute standard? Certainly not. It is quite possible that the stellar universe which is known to us, with its hundreds of thousands of galaxies which it takes light millions of years to cross, may be contained in a sphere of ether that rolls in an abyss which is devoid of ether, and is sown here and there with other universes, other giant drops of ether, from which no ray of light or anything else may ever reach us. It is, at all events, not inconceivable. And in that case, assuming that the ether has the properties attributed to it by classic physics, even if we had detected the movement of the earth in relation to it, we should not have discovered an absolute movement, but at the most a movement in relation to the centre of gravity of our particular universe, a standard which we could not refer to some other which would be absolutely stationary. The Classical Principle of Relativity would not be violated.

Hence, whatever may have been said to the contrary, the issue of Michelson’s experiment might, in these hypotheses, be either positive or negative without any detriment to Classical Relativism. As a matter of fact, it was negative, so nothing further need be said. Experiment has pronounced, and it alone had the right to pronounce.

These distinctions were not unknown to Poincaré, and he wrote: “By the real velocity of the earth I understand, not its absolute velocity, which is meaningless, but its velocity in relation to the ether.” Therefore the possibility of the existence of a velocity discoverable in relation to the ether was not regarded as an absurdity by Poincaré. He said: “Any man who speaks of absolute space uses a word that has no meaning.”

It is worth while noticing that in all this the development of Poincaré’s ideas betrays a certain hesitation. Speaking of experiments analogous to those of Michelson, he said:

“I know that it will be said that we are not measuring its absolute velocity, but its velocity in relation to the ether. That is scarcely satisfactory. Is it not clear that, if we conceive the principle in this fashion, we can make no deductions whatever from it?”

From this it is evident that Poincaré, in spite of himself and all his efforts to avoid it, was disposed to find the distinction between space and ether “scarcely satisfactory.”

I must admit that Poincaré’s own argument seems to me not wholly satisfactory, or at least not convincing. “Nature,” says Fresnel, “cares nothing about analytical difficulties.” I imagine that it cares just as little about philosophical or purely physical difficulties. It is hardly an incontestable criterion to suppose that a conception of phenomena is so much nearer to reality the more “satisfactory” it is to us, or the better it is found adapted to the weakness of the human mind. Otherwise we should have to hold, whether we liked or no, that the universe is necessarily adapted to the categories of the mind; that it is constituted with a view to giving us the least possible intellectual trouble. That would be a strange return to anthropocentric finalism and conceit! The fact that vehicles do not pass there, and that pedestrians have to turn back, does not prove that there are no such things as no-thoroughfares in our towns. It is possible, even probable, that the universe also, considered as an object of science, has its no-thoroughfare.

Clearly one may reply to me that it is not the universe that is adapted to our mind, but the mind that has become adapted to the universe in the evolutionary course of their relations to each other. The mind needs in its evolution to adapt itself to the universe, in conformity with the principle of minimum action formulated by Fermat: perhaps the most profound principle of the physical, biological, and moral world. In that respect the simplest and most economical ideas are the nearest to reality.

Yes, but what proof is there that our mental evolution is complete and perfect, especially when we are dealing with phenomena of which our organism is insensible?

Experiment alone has proved, and had the right to prove, that it is impossible to measure the velocity of an object relatively to the ether. At all events, this is now settled. After all, since it is evidently in the very nature of things that we cannot detect an absolute movement, is it not because the velocity of the earth in relation to the ether is an absolute velocity that we have been unable to detect it? Possibly; but it cannot be proved. If it is so—which is not at all certain—it is in the last resort experience, the one source of truth, which thus tends to prove, indirectly, that the ether is really identical with space. In that case, however, a space devoid of ether, or one containing spheres of ether, would no longer be conceivable, and there can be nothing but a single mass of ether with stars floating in it. In a word, the negative result of Michelson’s experiment could not be deduced a priori from the problematical identity of absolute space and the ether; but this negative result does not justify us in denying the identity a posteriori.

Let us return to our proper subject, the Fitzgerald-Lorentz hypothesis which explains the issue of the Michelson experiment, and which was in a sense the spring-board for Einstein’s leap. The hypothesis is as follows.

The result of the experiment is that, whereas when the path of a ray of light between two mirrors is transverse to the earth’s motion through ether, and it is then made parallel to the earth’s motion, the path ought to be longer, we actually find no such lengthening. According to Fitzgerald and Lorentz, this is because the two mirrors approached each other in the second part of the experiment. To put it differently, the frame in which the mirrors were fixed contracted in the direction of the earth’s motion, and the contraction was such in magnitude as to compensate exactly for the lengthening of the path of the ray of light which we ought to have detected.

When we repeat the experiment with all kinds of different apparatus, we find that the result is always the same (no displacement of the fringes). It follows that the character of the material of which the instrument is made—metal, glass, stone, wood, etc.—has nothing to do with the result. Therefore all bodies undergo an equal and similar contraction in the direction of their velocity relatively to the ether. This contraction is such that it exactly compensates for the lengthening of the path of the rays of light between two points of the apparatus. In other words, the contraction is greater in proportion as the velocity of bodies relatively to the ether becomes greater.

That is the explanation proposed by Fitzgerald. At first it seemed to be very strange and arbitrary, yet there was, apparently, no other way of explaining the result of Michelson’s experiment.

Moreover, when you reflect on it this contraction is found to be less extraordinary, less startling, than one’s common sense at first pronounces it. If we throw some non-rigid object, such as one of those little balls with which children play, quickly against an obstacle, we see that it is slightly pushed in at the surface by the obstacle, precisely in the same sense as the Fitzgerald-Lorentz contraction. The ball is no longer round. It is a little flattened, so that its diameter is shortened in the direction of the obstacle. We have much the same phenomenon, though in a more violent form, when a bullet is flattened against a target. Therefore, if solid bodies are thus capable of deformation—as they are, for cold is sufficient of itself to concentrate their molecules more closely—there is nothing absurd or impossible in supposing that a violent wind of ether may press them out of shape.

But it is far less easy to admit that this alteration may be exactly the same, in the given conditions, for all bodies, whatever be the material of which they are composed. The little ball we referred to would by no means be flattened so much if it were made of steel instead of rubber.

Moreover, there is in this explanation something quite improbable, something that shocks both our good sense and that caricature of it which we call common sense. Is it possible to admit that the contraction of bodies always exactly compensates for the optic effect which we seek, whatever be the conditions of the experiment (and they have been greatly varied)? Is it possible to admit that nature acts as if it were playing hide-and-seek with us? By what mysterious chance can there be a special circumstance, providentially and exactly compensating for every phenomenon?

Clearly there must be some affinity, some hidden connection, between this mysterious material contraction of Fitzgerald and the lengthening of the light path for which it compensates. We shall see presently how Einstein has illumined the mystery, revealed the mechanism which connects the two phenomena, and thrown a broad and brilliant light upon the whole subject. But we must not anticipate.

The contraction of the apparatus in Michelson’s experiment is extremely slight. It is so slight that if the length of the instrument were equal to the diameter of the earth—that is to say, 8,000 miles—it would be shortened in the direction of the earth’s motion by only six and a half centimetres! In other words, the contraction would be far too small to be in any way measurable in the laboratory.

There is a further reason for this. Even if Michelson’s apparatus were shortened by several inches—that is to say, if the earth travelled thousands of times as rapidly as it does round the sun—we could not detect and measure it. The measuring rods which we would use for the purpose would contract in the same proportion. The deformation of any object by a Fitzgerald-Lorentz contraction could not be established by any observer on the earth. It could be discovered only by an observer who did not share the movement of the earth: an observer on the sun, for instance, or on a slow-moving planet like Jupiter or Saturn.

Micromegas would, before he left his planet to visit us, have been able to discover, by optical means, that our globe is shortened by several inches in the direction of its orbital movement; supposing that Voltaire’s genial hero were provided with trigonometrical apparatus infinitely more delicate than that used by our surveyors and astronomers. But when he reached the earth, Micromegas, with all his precise apparatus, would have found it impossible to detect the contraction. He would have been greatly surprised—until he met Einstein and heard, as we shall hear, the explanation of the mystery.

I have, unfortunately, neither the time nor the space—it is here, especially, that space is relative, and is constantly shortened by the flow of the pen—to give the dialogue which would have taken place between Micromegas and Einstein. Perhaps, indeed, if we are to be faithful to the Voltairean original, the dialogue would have been very superficial, for—to speak confidentially—I believe that Voltaire never quite understood Newton, though he wrote much about him, and Newton was less difficult to understand than Einstein is. Neither did Mme. du Châtelet, for all the praise that has been lavished upon her translation of the immortal Principia. It swarms with meaningless passages which show that, whether she knew Latin or no, she did not understand Newton. But all this is another story, as Kipling would say.

The movement of the apparatus in the ether varies in speed according to the hour and the month in which the Michelson and similar experiments are made. As the compensation is always precise, we may try to calculate the exact law which governs the contraction as a function of velocities, and makes it, as we find, a precise compensation for the latter. Lorentz has done this. Taking V as the velocity of light and v as the velocity of the body moving in ether, Lorentz found that, in order to have compensation in all cases, the length of the moving body must be shortened, in the plane of its progress, in the proportion of

If we take by way of illustration the case of the orbital movement of the earth, where v is equal to thirty kilometres, we find that the earth contracts in the plane of its orbit in the proportion

The difference between these two numbers is ¹/₂₀₀,₀₀₀,₀₀₀, and the two hundred millionth part of the earth’s diameter is equal to 6½ centimetres. It is the figure we had already found.

This formula, which gives the value of the contraction in all cases, is elementary. Even the inexpert can easily see the meaning of it. It enables us to calculate the extent of contraction for every rate of velocity. We can easily deduce from it that if the earth’s orbital motion were, not 30 kilometres, but 260,000 kilometres a second, it would be shortened by one-half its diameter in the plane of its motion (without any change in its dimensions in the perpendicular). At that speed a sphere becomes a flattened ellipsoid, of which the small axis is only half the length of the larger axis; a square becomes a rectangle, of which the side parallel to the motion is twice as small as the other.

These deformations would be visible to a stationary spectator, but they would be imperceptible to an observer who shares the movement, for the reason already given. The measuring rods and instruments, and even the eye of the observer, would be equally and simultaneously altered.

Think of the distorting mirrors which one sees at times in places of amusement. Some show you a greatly elongated picture of yourself, without altering your breadth. Others show you of your normal height, but grotesquely enlarged in width. Try, now, to measure your height and breadth with a rule, as they are given in these deformed reflections in the mirror. If your real height is 5 feet 6 inches, and your real width 2 feet, the rule will, when you apply it to the strange reflection of yourself in the glass, merely tell you that this figure is 5 feet 6 inches in height and 2 feet in breadth. The rule as seen in the mirror undergoes the same distortion as yourself.

Hence it is that, even if the globe of the earth had the fantastic speed which we suggested above, its inhabitants would have no means of discovering that they and it were shortened by one-half in the plane east to west. A man 5 feet 6 inches in height, lying in a large square bed in the direction north-south, then changing his position to east-west, would, quite unknown to himself, have his length reduced to 2 feet 9 inches. At the same time he would become twice as stout as before, because previously his breadth was orientated from east to west. But the earth travels at the rate of only thirty kilometres a second, and its entire contraction is only a matter of a few centimetres.

In contrast with the earth’s velocity, the speed of our most rapid means of transport is only a small fraction of a kilometre a second. An aeroplane going at 360 kilometres an hour has a speed of only 100 metres a second. Hence the maximum Fitzgerald-Lorentz contraction of our speediest machines can only be such an infinitesimal fraction of an inch that it is entirely imperceptible to us. That is why—that is the only reason why—the solid objects with which we are familiar seem to keep a constant shape, at whatever speed they pass before our eyes. It would be quite otherwise if their speed were hundreds of thousands of times greater.

All this is very strange, very surprising, very fantastic, very difficult to admit. Yet it is a fact, if there really is this Fitzgerald-Lorentz contraction, which has so far proved the only possible explanation of the Michelson experiment. But we have already seen some of the difficulties that we find in entertaining the existence of this contraction.

There are others. If all that we have just said is true, only objects which are stationary in the ether would retain their true shapes, for the shape is altered as soon as there is movement through the ether. Hence, amongst the objects which we think spherical in the material world (planets, stars, projectiles, drops of water, and so on), there would be some that really are spheres, whilst others would, on account of the speed or slowness of their movements, be merely elongated or flattened ellipsoids, altered in shape by their velocity. Amongst the various square objects, some would be really square, while others, travelling at different speeds relatively to the ether, would be rather rectangles, shortened on their longer sides owing to their velocity. And it is supposed that we would have no means of knowing which of these objects moving at different speeds are really shaped as we think and which are shaped otherwise, because, as the Michelson experiment proves, we cannot detect a velocity relatively to the ether.

This we utterly decline to believe, say the Relativists. There are too many difficulties about the matter. Why speak persistently, as Lorentz does, of velocities in relation to the ether, when no experiment can detect such a velocity, yet experiment is the sole source of scientific truth? Why, on the other hand, admit that some of the objects we perceive have the privilege of appearing to us in their real shape, without alteration, while others do not? Why admit such a thing when it is, of its very nature, repugnant to the spirit of science, which is always opposed to exceptions in nature—science deals only with general laws—especially when the exceptions are imperceptible?

That was the state of affairs—very advanced from the point of view of the mathematical expression of phenomena, but very confused, deceptive, contradictory, and troublesome from the physical point of view—when “at length Malherbe arrived” ... I mean Einstein.

CHAPTER III

EINSTEIN’S SOLUTION

Provisional rejection of ether—Relativist interpretation of Michelson’s experiment—New aspect of the speed of light—Explanation of the contraction of moving bodies—Time and the four dimensions of space—Einstein’s “Interval” the only material reality.

Einstein’s first act of intelligent audacity was that, without relegating the ether to the category of those obsolete fluids, such as phlogiston and animal spirits, which obstructed the avenues of science until Lavoisier appeared—without denying all reality to ether, for there must be some sort of support for the rays which reach us from the sun—he observed that, in all that we have as yet seen, there is always question of velocities relatively to the ether.

We have no means whatever of establishing such velocities, and perhaps it would be simpler to leave out of our arguments this entity, real or otherwise, which is inaccessible and merely plays the futile and troublesome part of fifth wheel to the electro-magnetic chariot in the progress of physicists along the ruts of their difficulties.

The first point is then: Einstein begins, provisionally, by omitting the ether from his line of reasoning. He neither denies nor affirms its existence. He begins by ignoring it.

We will now follow his example. We shall no longer, in the course of our demonstration, speak about the medium in which light travels. We shall consider light only in relation to the beings or material objects which emit or receive it. We shall find that our progress becomes at once much easier. For the moment we will relegate the ether of the physicists to the store of useless accessories, along with the suave, formless, vague—but so precious artistically—ether of the poets.

Shortly, what does Michelson’s experiment prove? Only that a ray of light travels at the surface of the earth from west to east at exactly the same speed as from east to west. Let us imagine two similar guns in the middle of a plain, both firing at the same moment, in calm weather, and discharging their shells with the same initial velocity, but one toward the west and the other toward the east. It is clear that the two shells will take the same time to traverse an equal amount of space, one going toward the west and the other toward the east. The rays of light which we produce on the earth behave in this respect, as regards their progress, exactly as the shells do. There would therefore be nothing surprising in the result of the Michelson experiment, if we knew only what experience tells us about the luminous rays.

But let us push the comparison further. Let us consider the shell fired by one of the guns, and imagine that it hits a target at a certain spot, and that, when it reaches the target, the residual velocity of the shell is, let us say, fifty metres a second. I imagine the target mounted on a motor tractor. If the latter is stationary the velocity of the shell in relation to the target will be, as we said, fifty metres a second at the point of impact. But let us suppose that the tractor and the target are moving at a speed of, for instance, ten metres a second toward the gun, so that the target passes to its preceding position exactly at the moment when the shell strikes it. It is clear that the velocity of the shell relatively to the target at the moment of impact will not now be fifty metres, but 50 + 10 = 60 metres a second. It is equally evident that the speed will fall to 50-10 = 40 metres a second if (other things being equal) the target is travelling away from the gun, instead of toward it. If, in the latter case, the velocity of the target were equal to that of the shell, it is clear that the relative velocity of the shell would now be nil.

So much is clear enough. That is how jugglers in the music-halls can catch eggs falling from a height on plates without breaking them. It is enough to give the plate, at the moment of contact, a slight downward velocity, which lessens by so much the velocity of the shock. That is also how skilled boxers make a movement backward before a blow, and thus lessen its effective force, whereas the blow is all the harder if they advance to meet it.

If the luminous rays behaved in all respects like the shells, as they do in the Michelson experiment, what would be the result? When one advances very rapidly to meet a ray of light, one ought to find its velocity increased relatively to the observer, and lessened if the observer recedes before it. If this were the case, all would be simple; the laws of optics would be the same as those of mechanics; there would be no contradiction to sow discord in the peaceful army of our physicists, and Einstein would have had to spend the resources of his genius on other matters.

Unfortunately—perhaps we ought to say fortunately, because, after all, it is the unforeseen and the mysterious that lend some charm to the way of the world—this is not the case. Both physical and astronomical observation show that, under all conditions, when an observer advances rapidly toward luminous waves or recedes rapidly from them, they still show always the same velocity relatively to him. To take a particular case, there are in the heavens stars which recede from us and stars which approach us; that is to say, stars from which we recede, or which we approach, at a speed of tens, and in some cases hundreds, of miles a second. But an astronomer, de Sitter, has proved that the velocity of the light which reaches us is, for us, always exactly the same.

Thus, up to the present it has proved quite impossible for us, by any device or movement, to add to or lessen in the least the velocity with which a ray of light reaches us. The observer finds that the rate of speed of the light is always exactly the same relatively to himself, whether the light comes from a source which rapidly approaches or recedes from him, whether he is advancing toward it or retreating before it. The observer can always increase or lessen, relatively to himself, the speed of a shell, a wave of sound, or any moving object, by pushing toward or moving away from the object. When the moving object is a ray of light, he can do nothing of the kind. The speed of a vehicle cannot in any case be added to that of the light it receives or emits, or be subtracted from it.

This fixed speed of about 186,000 miles a second, which we find always in the case of light, is in many respects analogous to the temperature of 273° below zero which is known as “absolute zero.” This also is, in nature, an impassable limit.

All this proves that the laws which govern optical phenomena are not the same as the classic laws of mechanical phenomena. It was for the purpose of reconciling these apparently contradictory laws that Lorentz, following Fitzgerald, gave us the strange hypothesis of contraction.

But we shall now find Einstein showing us, in luminous fashion, that this contraction is seen to be perfectly natural when we abandon certain conceptions—perhaps erroneous, though classical—which ruled our habitual and traditional way of estimating lengths of space and periods of time.

Take any object—a measuring rod, for instance. What is it that settles for us the apparent length of the rod? It is the image made upon our retina by the two rays that come from the two ends of the rod, and which reach our eye simultaneously.

I italicise the word, because it is the key of the whole matter. If the rod is stationary before us, the case is simple. But if it is moved while we are looking at it, the case is less simple. It is so much less simple that before the work of Einstein most of our learned men and the whole of classic science thought that the instantaneous image of an object that was not subject to change of shape was necessarily and always identical, and independent of the velocities of the object and the observer. The whole of classical science argued as if the spread of light was itself instantaneous—as if it had an infinite velocity—which is not the case.

I stand on the bank by the side of a railway. On the line is a handsome Pullman car, in which it is so pleasant to think that space is relative, in the Galileian sense of the word. Close to the line I have two pegs fixed, one blue, the other red, and they exactly mark the ends of the coach and indicate its length. Then, without leaving my observation-post on the bank, my face turned towards the middle of the coach, I give orders for the coach to be drawn back and coupled to a locomotive of unheard-of power, which is to carry the coach past me at a fantastic speed, millions of times faster than the speed any mere engineer could provide. Such is the potential superiority of the imagination over sober reality! I assume further that my retina is perfect, and is so constituted that the visual impressions will remain on it only as long as the light which causes them. These somewhat arbitrary suppositions count for nothing in the essence of the demonstration. They are only for the sake of convenience.

Now for the question. Will the coach (which I assume to be of some rigid metal), as it passes before me at full speed, seem to me to be exactly the same length as it did when it was at rest? To put it differently, at the moment when I see its front end coincide with the blue peg I had planted, shall I see its back end coincide at the same time with the red peg? To this question Galileo, Newton, and all the supporters of classic science would reply yes. Yet according to Einstein the answer is no.

Here is the simple proof, as we deduce it from Einstein’s general idea.

I am, recollect, on the edge of the track, at an equal distance from both pegs. When the front end of the coach coincides with the blue peg, it sends toward my eye a certain ray of light (which, for convenience, we will call the front ray), and this coincides with the luminous ray coming to me from the blue peg. This front ray reaches my eye at the same time as a certain ray that comes from the back end of the coach (which we will call the back ray). Does the back ray coincide with the ray which comes to me from the red peg? Clearly not. The front ray leaves the front end of the coach at the same speed as the back ray leaves the back end; as any observer in the coach would find who cared to try the Michelson experiment on them. But the front end of the coach is receding from me while the back end is approaching me. Hence the front ray travels toward my eye more slowly than the back ray, though I cannot perceive this, as, when they reach me, I find that they both have the same velocity. Hence the back ray, which reaches my eye at the same time as the front ray, must have left the back end of the coach later than the front ray left the front end of the coach. Therefore, when I see the front end of the coach coincide with the blue peg, I at the same time see the back end of the carriage after it has passed the red peg. Therefore the length of a coach travelling at full speed, and such as it appears to me, is shorter than the distance between the two pegs, which indicated the length of the coach at rest. Q.E.D.

Very little attention is needed for any person to understand this argument, though its elementary simplicity has not been attained without difficulty. It is part of Einstein’s mathematical argument and of his conception of simultaneity.

It follows that the coach, or, in general, any object, seems to be contracted in virtue of its velocity, and in the direction of that velocity, relatively to the spectator. The same thing happens, obviously, if the observer moves in relation to the object, because we can know only relative velocities, in virtue of the Classical Principle of Relativity of Newton and Galileo.

In this new light the Lorentz-Fitzgerald contraction becomes intelligible, or at least admissible. The contraction, thus considered, is not the cause of the negative result of the Michelson experiment: it is an effect of it. It is now quite clear, and we see that there was something wrong with the classical way of estimating the instantaneous dimension of objects.

Certainly the fact that luminous rays, starting out from their sources at different speeds, should have the same speed when they reach our eye, is strange. It upsets our habitual way of looking at things. If I may venture to use a comparison simply for the purpose of provoking reflection, not at all in the way of explanation, we have here something analogous to what happens with the bombs of aviators. Bombs of a given type, whether released at a height of 5,000 or of 10,000 metres, which therefore have very different downward velocities at 5,000 metres from the ground, have always the same residual velocity when they reach the ground. This is due to the moderating and equalising influence of the atmospheric resistance, which prevents the speed from increasing indefinitely, and makes it constant when it has attained a certain value.

Must we suppose that there is round our eye and round objects a sort of field of resistance which sets a similar limit to the light? Who knows? But perhaps such questions have no meaning for the physicist. He can know nothing about the behaviour of light except when it leaves its source or when it reaches the eye, whether armed with instruments or no. He cannot learn how it behaves during its passage across the intermediate space, in which there is no matter.

Indeed, the more deeply we study the new physics, the more we see that it derives almost all its strength from its systematic disdain of all that is beyond phenomena, all that cannot fall under experimental observation. It is because it is solely based upon facts (however contradictory they may be) that our proof of the necessary contraction of objects owing to their velocity relatively to the observer is so strong.

We must understand the profound significance of the Fitzgerald-Lorentz contraction. This apparent contraction is by no means due to the movement of objects relatively to the ether. It is essentially the effect of the movements of objects and observers relatively to each other, or relative movements in the sense of the older mechanics.

The greatest relative velocities to which we are accustomed in our daily life are less than a few kilometres a second. The initial velocity of the shell fired by “Bertha” was only about 1,300 metres a second. For movements so slow as this the Relativist contraction is entirely negligible. Hence, as the classical mechanics had never observed such contraction, it regarded the shapes and dimensions of rigid objects as independent of systems of reference.

It was very nearly true; and that makes all the difference between true and false. To say that 999,990 + 9 = 1,000,000, is to say something that is very nearly true, and is therefore false. When it was discovered that the earth was round no change was made in their procedure by architects. They continued to build as if the direction indicated by the plumb-line was always parallel to itself. In the same way those who make our locomotives and aeroplanes will not have to consider the forms of the machines as dependent on their velocities. What does it matter? The practical point of view is not, and cannot be, that of science except indirectly. So much the worse if there is no indirect influence, or if it is slow in coming.

Some years ago, however, we discovered things which move at speeds, relatively to us, of tens or hundreds of thousands of kilometres a second; the projectiles of the cathode rays and of radium. In this case the Relativist contraction is very considerable. We shall see how it has been observed.

But let us first recapitulate what we have seen. Objects seem to alter their shape in the direction of their movement and not in the direction perpendicular to this. Therefore their forms, even if they be composed of an ideal and perfectly rigid material, depend on their velocity relatively to the observer. This is the essentially new point of view which Einstein’s “Special Relativity” superimposes upon the Relativity of classical mechanics and philosophers. For these the absolute dimensions of a rigid object or a geometrical figure were not absolute; it was only the relations of these dimensions which were real.

The new point of view is that these relations are themselves relative, because they are a function of the velocity of the observer. It is a sort of Relativity in the second degree, of which neither the philosophers nor the classic physicists had dreamed.

Spatial relations themselves are relative, in a space which is already relative.

In the case of our Pullman car and the two pegs which mark its length when it is stationary, an observer situated in the carriage would find the distance between the two pegs shortened as he passes them. The coach would seem to him longer than the distance between the pegs. I who remain beside the pegs observe the contrary. Yet I have no means of proving to the passenger that he is wrong. I see quite plainly that the ray of light which comes from the back peg runs behind the coach, and has therefore, relatively to it, a speed of less than 186,000 miles a second. I know that this is the reason for the passenger’s error, but I have no means of convincing him that he is wrong. He will always say, and rightly: “I have measured the speed at which this ray reaches me, and I have found it 186,000 miles a second.” Each of us is really right.

In very rapid motion a square would seem to the observer a rectangle; a circle would appear to be an ellipse. If the earth travelled some thousands of times faster round the sun, we should see it elongated, like a giant lemon suspended in the heavens. If an aviator could fly at a fantastic speed over Trafalgar Square, in the direction of the Strand—and if the impressions on his retina were instantaneous—he would see the Square as a very flattened rectangle. If he flew in a diagonal line about it, he would find it shaped like a lozenge. If the same aviator flew across a road on which fat cattle were being driven to the slaughter-house, he would be astonished, for the beasts would seem to him extraordinarily lean, while there would be no change in their length.

The fact that these alterations of shape owing to velocity are reciprocal is one of the most curious consequences of all this. A man who could pass in every direction amongst his fellows at the fantastic speed of one of Shakespeare’s spirits—let us put it at about 170,000 miles an hour, though there would be no limit—would find that his fellows had become dwarfs only half as large as himself. Would he have become a giant, a sort of Gulliver amongst the Lilliputians? Not in the least. Such is the justice of the scheme of earthly things that he himself would seem a dwarf to the people whom he thought smaller than himself, and who are quite sure of the contrary.

Which is right, and which wrong? Both. Each point of view is accurate, but there are only personal points of view.

Again, any observer whatever will only see things that are not connected with him as smaller—never larger—than the things which are connected with his movement. If I might venture to relieve this sober exposition by a reflexion rather less austere than is usual in physics, I would say that the new system affords a supreme justification of egoism, or, rather, of egocentricism.

It is the same with time as with space. By similar reasoning to that which has shown us how the distance of things in space is connected with their velocity relatively to the observer, it can be shown that their distance in time likewise depends upon this.

It would be useless to reproduce here the whole of the Einsteinian argument as to duration. It is analogous to that which we have used in regard to length, and even simpler. The result is as follows. The time expressed in seconds which a train takes to pass from one station to another is shorter for the passengers on the train than for us who watch it pass, though our watches may be just the same as theirs.[5] Similarly, all the gestures of men who are on moving vehicles will seem to a stationary observer slowed down, and therefore prolonged, and vice versa. But the velocity would, as in the case of variation in length, have to be fantastic to make these variations in time perceptible.

It is not less true that the time between the birth and the death of any creature, its life, will seem longer if the creature moves rapidly and fantastically relatively to the observer. In this world, where appearance is almost everything, this is not without importance, and it follows that, philosophically speaking, to move on is to last longer; but for others, not for oneself; just as others may seem to me to last longer. A striking, a profound, an unforeseen justification of the words of the sage: immobility is death!

Formerly, before the Einsteinian hegira, 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”:

“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.”

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 algebraical expressions of the first degree, surfaces or spaces of two dimensions by formulæ of the second degree, and volumes or spaces of three dimensions by expressions of the third degree, it was natural to ask oneself if formulæ of the fourth and higher degrees are not also the algebraical 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 two 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 relatively to the things observed, we must always remember) acts in a sense as a double brake 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 visualise 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 to-day we have no direct sense-impression of them. They exist none the less. 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 paralysed 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 theatre. 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 skilful 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 italicised 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. “The translator is a traitor.” 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 had 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 colouring 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 under foot, 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.[6]

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, “space and time are mere phantoms. All that exists in reality is a sort of intimate union of these entities.”

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 some day 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.

CHAPTER IV

EINSTEIN’S MECHANICS

The mechanical foundation of all the sciences—Ascending the stream of time—The speed of light an impassable limit—The addition of speeds and Fizeau’s experiment—Variability of mass—The ballistics of electrons—Gravitation and light as atomic microcosms—Matter and energy—The death of the sun.

When Baudelaire wrote:

I hate the movement that displaces lines,

he thought only, like the physicists of his time, of the static deformations which have been known as long as there have been men to observe them. What we have seen about Einsteinian time and space has taught us that there must be, in addition to these, kinematic deformations, to which every material object, however rigid it seems, is liable.

Movement, therefore, displaces lines much more than Baudelaire supposed, even the lines of the hardest of marble statues. This kind of deformation, which is pleasant rather than hateful, since it brings us nearer to the heart of things, has upset the whole of mechanics.

Mechanics is at the foundation of all the experimental sciences, because it is the simplest, and because the phenomena it studies are always present—if not exclusively present—amongst the phenomenal objects of the other sciences, such as physics, chemistry, and biology.

The converse of this is not true. For instance, there is not a single phenomenon in chemistry or biology in which one has not to study bodies in movement, objects endowed with mass and giving out or absorbing energy. On the other hand, the peculiar aspects of a biological, chemical, or physical phenomenon, such as the existence of a difference of potential, an oxidation, or an osmotic pressure, are not always found in the study of the movements of a ponderable mass and of the forces which act upon and through it.

Compared with mechanics, the sciences of physics, chemistry, and biology have, in the order in which we name them, objects of increasing complexity and generality, or, to put it better, of decreasing universality. These sciences are mutually dependent in the way that the trunk, branches, leaves, and flowers of a tree are. They are to some extent related to each other as are the various parts of the jointed masts on which military telegraphists fix their antennæ. The lower part of the mast, the larger part, sustains the whole; but it is the upper parts which bear the delicate and complicated organs.

The object of the great synthetists in science has always been, and is, to reduce all phenomena to mechanical phenomena, as Descartes attempted. Whether these attempts are well-grounded or no, whether they will some day succeed or are condemned a priori to failure because physico-biological phenomena involve elements that are essentially incapable of reduction to mechanical elements, is a question that has been, and will continue to be, much discussed. But, however thinkers may differ on that point, they are agreed on this: in all natural phenomena, in all phenomena that are objects of science, there is the mechanical element—exclusive in some, the principal element in others.

All this leads to the conclusion that whatever modifies mechanics, modifies at the same time the whole structure of ideas founded thereon—that is to say, the other sciences, the whole of science, our entire conception of the universe. But we are now going to see that Einstein’s theory, as a direct effect of what it teaches in regard to space and time, completely upsets the classical mechanics. It is in this way, particularly, that it has shaken the rather somnolent frame of traditional science, and the vibration is not yet over.

In approaching the Einsteinian mechanics we shall have the pleasure of passing from ideas of time and space that are rather too exclusively geometrical and psychological to the direct study of material realities, of bodies. Here we can compare theory and reality, the mathematical premises and the substantial verifications; and we shall be pleased to see what the facts, given in experience, have to say on the matter. We shall be able to make our choice, with informed minds and sound criteria, between the old and the new ideas.

In a word, if I may use this illustration, as long as we were dealing with ideas of space and time—which are empty frames in themselves, vases that would interest us chiefly by the liquids they contain—we were rather like the young men who have to choose a fiancée solely by the description of her which has been given them. We are now going to see with our own eyes, and see at work the two aspirants to our affection: classical science and Einstein’s theory. We shall see both of them take up the paste of facts, and we shall be able to compare the delicious dishes which they respectively make from it for the nourishment of the mind.

Theories have no value except as functions of facts. Those which, like so many in metaphysics, have no real criterion by which we may test them, are all of the same value. Experience, the sole source of truth of which Lucretius said long ago:

unde omnia credita pendent,

or the material facts, is going to judge Einstein’s system for us.

The result of the Michelson experiment, the impossibility of proving any velocity of the earth in relation to the medium in which light is propagated, amounts to this: we have no means whatever of detecting a speed higher than that of light. This consequence of the Michelson experiment will be better understood, perhaps, if we put it in a tangible form. Here is an illustration that will serve our purpose.

In some astronomical novel an imaginary observer is supposed to recede from the earth at a speed greater than that of light—at 300,000 miles a second, let us say—yet to keep his eyes (armed with prodigious glasses) steadily fixed on this little globe of ours.

What will happen? Evidently, our observer will see the train of earthly events in inverse order, because in the course of his voyage he will catch up in succession the luminous waves which left the earth before him. The farther away they are, the longer it must be since they left the earth. After a time our man, or our superman, will witness the Battle of the Marne. He will first see the field strewn with the dead. Gradually the dead men will rise and join their regiments, and presently they will be seen in groups in Gallieni’s taxis, which will travel backwards at full speed to Paris, arriving in the midst of a population that is extremely anxious about the issue of the struggle, and the soldiers will, naturally, be unable to give them any news. In a word, our observer will, if he recedes from the earth at a speed greater than that of light, see terrestrial events happening as if he were ascending the stream of time.

It would be very different if the observer remained stationary, and the earth receded from him at a speed of 300,000 miles a second. What would happen then? It is clear that in this case our observer will see terrestrial events, not in inverse order, but as they are: except that they would seem to him to take place with majestic slowness, because the rays of light which leave the earth at the end of some particular event will take a much longer time to reach him than the rays which left the earth at the beginning of the event.

In sum, the phenomena observed by him being essentially different in the two cases, our imaginary observer would be able to say whether it is he who is receding from the earth or the earth that is receding from him; to detect the real movement of the event through space. This means, of course, movement relatively to the medium of the propagation of light, not necessarily, as we saw, movement in relation to absolute space.

The experiment we have imagined could not very well be carried out with the actual resources of our laboratories. We cannot attain these fantastic speeds, and even if we could the observer would not distinguish much. But we have chosen a colossal instance, and the results of it would be colossal, as there would be question of nothing less than a reversal of the order of time.

If we were to use more modest means, the results will be more modest, but according to the older theories they ought to be recorded in our instruments. But the Michelson experiment—a miniature version of what we have just described—shows that the differences we should expect are not observed. Therefore the premise we laid down—that there can be velocities greater than that of light in empty space—does not harmonise with reality. Hence this velocity of light is a wall, a limit that cannot be passed.

Now let us see what follows. There is at the base of classical mechanics, as it was founded by Galileo, Huyghens, and Newton, and as it is taught everywhere, a principle which is in the long run, like all the principles of mechanics, grounded upon experience. It is the principle of the composition of velocities. If a boat, which makes ten miles an hour in smooth water, sails down a river which flows at five miles an hour, the speed of the boat in relation to the bank will be, as we may find by actual measuring, equal to the sum of the two speeds, or fifteen miles an hour. This is the rule of the addition of velocities.

In a more general way, if a body starts from a state of rest, and under the action of some force takes on in a second the velocity V, what will it do if the action of the force is prolonged for another second? According to classical mechanics it will take on the velocity 2V.[7] Let us imagine an observer who is travelling at the velocity V, yet thinks he is at rest. It will seem to him, at the end of the first second, that the body is at rest (because it has the same velocity as the observer). In virtue of the Classical Principle of Relativity, the apparent movement of the body must be the same for our observer as if the rest were real. This means that at the end of the second second the relative velocity of the body in reference to the observer will be V, and, as the observer already has the velocity V, the absolute velocity of the body will be 2V. In the same way it will be 3V at the end of three seconds, 4V at the end of four seconds, and so on. Could it increase indefinitely if the force continues to act long enough? Classical mechanics says “yes.” Einstein says “no,” because there cannot be a greater velocity than that of light.

We have imagined an observer who has the velocity V relatively to us, and who believes that he is at rest. For him the body observed was likewise at rest at the beginning of the second second, because its velocity was the same as that of the observer. From the fact that the apparent movement of the body is for the observer, during the second second, the same as it was for us during the first, classical mechanics concluded that its velocity doubles during the second second. It did not know what Einstein has now taught us: that the time and space of this observer are different from ours.

What is a velocity? It is the space traversed in the course of a second. But the space thus measured by our moving observer, which he believes to be of a certain length, is in reality, for us who are stationary, smaller than he thinks, because the rules he uses are, as Einstein has shown, shortened by velocity without his perceiving it. Therefore the velocities are not added together in equal proportions and indefinitely for a given observer, as classical mechanics maintained.

Under the action of the same force, the old mechanics said, a body will always experience the same acceleration, whatever be the velocity already acquired. Under the action of the same force, the new mechanics says, the motion of the body will be accelerated less and less in proportion to its velocity.

Take, for instance, some movable object having, relatively to me, a velocity of 200,000 kilometres a second. Let us place an observer on this object. The observer will then start, in the same direction and under the same conditions as we have done, a second movable object, which will thus have, relatively to him, a speed of 200,000 kilometres. The Relativist says that the resultant velocity of the second object relatively to us will not be, as the classical addition of velocities would make it, 200,000 + 200,000 = 400,000 kilometres a second. It will be only 277,000 kilometres a second. What the second moving observer took to be 200,000 kilometres (because his measuring rod was shortened owing to velocity) was really only 77,000 of our kilometres. How is it possible to calculate that? Simply by using the formula of Lorentz which I gave in [Chapter II], which gives us the value of the contraction due to velocity. We then easily find that, if we have two velocities, v and v₂, and if we call the resultant w, classical mechanics stated that

w = v₁ + v₂

The Einstein mechanics says that this is not correct, and that what we really have (C being the velocity of light) is

I apologise for again introducing—it shall be the last time—an algebraical formula into my work. But it spares me a large number of words, and it is so simple that every reader who has even a tincture of elementary mathematics will at once see its great significance and the consequences of it.

The formula expresses in the first place the fact that the resultant of the velocities, however great it may be, cannot be greater than the speed of light. It conveys also that, if one of the component velocities is that of light, the resultant velocity must have the same value. It means, in fine, that in the case of the slight velocities we have to do with in actual life (that is to say, when the component velocities are much smaller than that of light) the resultant is very nearly equal to the sum of the two components, as the classical mechanics says.

The classical mechanics was, we must remember, founded upon experience. We understand how, in those circumstances, Galileo and his successors, dealing only with relatively slowly moving bodies, reached a principle which seemed to be true for them, but is only a first approximation.

For instance, the resultant of two velocities, each equal to a hundred kilometres a second (which is far higher than any velocities obtainable by Galileo and Newton), amounts to, not 200 kilometres, but 199·999978 kilometres. The difference is scarcely twenty-two millimetres in 200 kilometres! We can quite understand that the earlier experimenters could not detect differences even less minute than that.

Amongst the verifications of the new law of composition of velocities we may quote one, the outcome of an early experiment of the great Fizeau, which is very striking.

Imagine a pipe full of some liquid, such as water, and a ray of light travelling along it. We know the speed of light in water: it is much lower than in air or in empty space. Suppose, further, that the water is not stationary, but flows through the pipe at a certain speed. What will be the velocity of the ray of light when it leaves the pipe after traversing the moving liquid? That was what Fizeau, with many variations of the conditions of the experiment, tried to ascertain.

The velocity of light in water is about 220,000 kilometres a second. There is question here of so rapid a propagation that there is a great difference between the law of addition of the old classical mechanics and of Einsteinian mechanics. Now the results of Fizeau’s experiment are in complete harmony with Einstein’s formula, and are not in harmony with that of the older mechanics. Many observers, including, recently, the Dutch physicist Zeeman, have repeated Fizeau’s experiment with the greatest care, but the result was the same.

When Fizeau made the experiment in the last century, attempts were made to interpret his results in the light of the older theories. This, however, led to very improbable hypotheses. Fresnel, for instance, trying to explain Fizeau’s results, had been compelled to admit that the ether is partially borne along by the water as it flows, and that this partial displacement varies with the length of the luminous waves sent through, or that it is not the same for the blue as for the red waves! A very startling deduction, and one very difficult to admit.

The new law of composition of velocities given to us by Einstein, on the other hand, immediately and with perfect accuracy explains Fizeau’s results. They are opposed to the classical law.

The facts, the sovereign judges and criteria, show in this case that the new mechanics corresponds to reality; the earlier mechanics does not, at least in its traditional form. Here is something, therefore, which enables us to see at once the profound truth (scientific truth being what is verifiable), the beauty, of the doctrine of Einstein: something which shows us, superbly, how a scientific, a physical, theory differs from an arbitrary and more or less consistent philosophical system.

Experience, the supreme judge, decides in favour of the Einsteinian mechanics against the older mechanics. We shall see further examples; and we shall not find a single case in which the verdict is the other way.

Let us turn now to a different matter. The new law of composition of velocities and the resistance of a velocity-limit equal to that of light may be expressed in a different language from that we have hitherto used. Up to this we have spoken only of velocities and movements. Let us see how these things look when we at the same time examine the particular qualities of the moving objects, of bodies, of matter.

Everybody knows that the characteristic feature of matter is what we call inertia. If matter is at rest, a force is needed to set it in motion. If it is in motion, it needs a force to stop it. It needs one to accelerate the movement and one to alter the direction. This resistance which matter offers to the forces which tend to modify its condition of rest or movement is what we call inertia. But different bodies may offer a different degree of resistance to these forces. If a force is applied to an object, it will give it a certain acceleration. But the same force applied to another object will, as a rule, give it a different acceleration. A race-horse making a supreme effort will get along much more quickly under a small jockey than under a man of fifteen stone. A draught-horse will run more quickly if the cart it draws is empty than if it is full of goods. You can start a perambulator with a push that would be useless in the case of a heavy truck.

When a locomotive with a few coaches suddenly starts, the velocity imparted to the train during the first second is what we call its acceleration. If the same locomotive starts, in the same conditions, with a much longer train, we see that the acceleration is less. Hence the idea, introduced into science by Newton, of the mass of bodies, which is the measure of their inertia.

If in our example the locomotive produces in the second case an acceleration only half as great, we express this by saying that the mass of the second train is double that of the first. If we find that the acceleration produced by the locomotive is the same for three trucks loaded with wheat as for a single truck loaded with metal, we see that the two trains are equal in mass.

In a word, the masses of bodies are conventional data defined by the fact that they are proportional to the accelerations caused by one and the same force. To put it differently, the mass of a body is the quotient of the force which acts upon it by the acceleration given to it. Poincaré used to say picturesquely: “Masses are coefficients which it is convenient to use in calculations.”

If there is one property of bodies which comes within the range of our senses, a property of which every man has some sort of instinct or intuition, it is mass. Yet careful analysis shows us that we are unable to define it otherwise than by disguised conventions. Poincaré’s definition seems paradoxical in its admission of powerlessness. But it is correct. Mass is only a “coefficient,” a conventional outcome of our weakness!

Nevertheless, something remained upon which we thought we could base, if not our craving for certainty—genuine men of science gave up the idea of certainty long ago—at least our desire for accuracy of deduction in our classification of phenomena. We believed in the constancy of mass, of this convenient and so clearly defined coefficient.

Here again, unfortunately, we have to recant—or, perhaps, we should say fortunately, as there is no pleasure like that of novelty.

The older mechanics taught us that mass is constant in one and the same body, and is therefore independent of the velocity which the body acquires. From which it followed, as we have already explained, that, if a force continues to act, the velocity acquired at the end of a second will be doubled at the end of two seconds, tripled at the end of three seconds, and so on indefinitely.

But we have just seen that the velocity increases less during the second second than during the first, and so on, continuously diminishing until, when the velocity of light is attained, that of the moving body can increase no further, whatever force may act upon it.