CHAPTER IIIEINSTEIN’S SOLUTION
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 ofour 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 andthe 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 benil.
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 eyesimultaneously.
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 havetwo 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 replyyes. Yet according to Einstein the answer isno.
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 raycoming to me from the blue peg. This front ray reaches my eyeat the same timeas 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 carriageafterit 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 theirprocedure 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 therelationsof 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 manwho 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 whowatch 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 Einsteinianhegira, before the Relativist Era opened, everybody was convinced that the portion ofspaceoccupied by an object was sufficiently and explicitly defined by its dimensions—length, breadth, and height. These are what are called the threedimensionsof 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 apparentspaceoccupied by it, depend upon its velocity: that is to say, upon thetimewhich the observer takes to traverse a certain distance relatively to the object. Here we havespacealready depending upontime. 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 theEncyclopédie, 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 durationsand 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 mentaleye the whole chronological and spatial series of moves that may follow the first move, with all their effects on the board. Hesees 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 theworld 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, whatisreal?
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 oftime 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 theIntervalof 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 EinsteinianIntervalas we have defined it. TheIntervalof 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 theIntervalof 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 theEinsteinian Intervaltells 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.