CHAPTER IVEINSTEIN’S MECHANICS
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,
I hate the movement that displaces lines,
I hate the movement that displaces lines,
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 condemneda priorito 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, ofbodies. 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 afiancéesolely 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 tocompare 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,
unde omnia credita pendent,
unde omnia credita pendent,
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 theBattle 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 wereascendingthe 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 thesefantastic 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 velocityV, what will it do if the action of the force is prolonged foranother second? According to classical mechanics it will take on the velocity2V.[7]Let us imagine an observer who is travelling at the velocityV, 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 beV, and, as the observer already has the velocityV, the absolute velocity of the body will be2V. In the same way it will be3Vat the end of three seconds,4Vat 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 inChapter II, which gives us the value of the contraction due to velocity. We then easily find that, if wehave two velocities,vandv₂, and if we call the resultantw, 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 havehitherto 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 callinertia. 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 themassof 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 themass 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 ismass. 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 definedcoefficient.
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 bodyacquires. 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.
What does that mean? If the velocity of a body increases less during the second second, it must be because it offers an increasing resistance to the accelerating force. Everything happens as if its inertia, its mass, had changed! Which amounts to saying thatthe mass of bodies is not constant: it depends upon their velocity, and increases with an increase of velocity.
In the case of feeble velocities this influence is imperceptible. It was because the founders of classical mechanics, an experimental science, had experience only of relatively feeble velocities that they found that mass wasperceptiblyconstant, and believed they might conclude that it wasabsolutelyconstant. In the case of greater velocities that is not so.
Similarly, in the case of feeble velocities, in the new mechanics as well as the old, bodies perceptibly oppose the same resistance of inertia to the forces which tend to accelerate their movement as to those which tend to alter the direction, to give a curve to their trajectories. In the case of great velocities that is not so.
Mass, therefore, increases rapidly with velocity. It becomes infinite when the velocity equals that of light. No body whatever can attain orsurpass the velocity of light, because, in order to pass that limit, it would need to overcome an infinite resistance.
In order to make it quite clear, let us give certain figures which show how mass varies with velocity. The calculation is easy, thanks to the formula which we have previously seen, giving the values of the Fitzgerald-Lorentz construction.
A mass of 1,000 grammes will weigh an additional two grammes at the velocity of 1,000 kilometres a second. It will weigh 1,060 grammes at the velocity of 100,000 kilometres a second; 1,341 grammes at the velocity of 200,000 kilometres a second; 2,000 grammes (or double) at the velocity of 259,806 kilometres a second; 3,905 grammes at the velocity of 290,000 kilometres a second.
That is what the new theory tells us. But how can we verify it? It would have been impossible only fifty years ago, when the only velocities known were those of our vehicles and projectiles, which then did not rise, even in the case of shells, above one kilometre a second. The planets themselves are far too slow for the purpose of verification. Mercury, for instance, the swiftest of them, travels at a speed of only a hundred kilometres a second, which is not enough.
If we had at our disposal no higher velocities than these, we should have no means of settling which was right, the classical mechanics with its constancy of mass or the new mechanics with its assertion of variability.
It is the cathode rays and the Beta rays of radium which have provided us with velocities great enough for the purpose of verification. These rays consist of an uninterrupted bombardment by small and very rapidprojectiles, each of a mass less than the two-thousandth part that of an atom of hydrogen, and charged with negative electricity. They are theelectrons.
The cathode tubes of radium give out a continuous bombardment of these minute projectiles, charged, not with melinite, but electricity: far smaller than the shells of our artillery, but animated with infinitely greater initial speeds. The velocity of “Bertha’s” shells is contemptible in comparison.
But how was it possible to measure the speed of these projectiles?
We know that electrified bodies act upon each other. They attract or repel each other. Now our electrons are charged with electricity. If, therefore, we put them in an electric field, between two plates connected at the edges by an electrical machine or an induction coil, they will be subjected to a force that will cause them to change their direction. The cathode rays, in other words, will change their direction under the influence of an electric field. The amount of diversion will depend upon the speed of the projectiles and upon their mass; that is to say, upon the resistance of inertia which the mass opposes to the causes which tend to divert it.
But this is not all. The electric charges borne by the projectiles are in movement, even rapid movement. Now, electricity in movement is an electric current, and we know that currents are diverted by magnets or magnetic fields. Therefore the cathode rays will be diverted by the magnet. This diversion will, like the former, depend upon the velocity and the mass of the projectile; but not quite in the same way. Otherthings being equal, the magnetic diversion will be greater than the electrical diversion, if the velocity is high. As a matter of fact, the magnetic diversion is due to the action of the magnet on the current. It will be greater in proportion to the intensity of the current; and the current will be more intense in proportion to the height of the velocity, since it is the movement of the projectile which causes the current. On the other hand, the trajectory of our little projectiles will be less influenced by the electrical attraction in proportion as the velocity of the projectile is great.
Hence it is easy to see that when we subject a cathode ray to the action of an electric field, then to that of a magnetic field, we may, by comparing the two deviations, measure at one and the same time the velocity of the projectile and its mass (related to the known electric charge of the electron).
In this way we find enormous velocities, rising from a few tens of kilometres to 150,000 kilometres a second, and even more. As to the Beta rays of radium, they are still more rapid. In cases they attain velocities not far short of that of light, and higher than 290,000 kilometres a second. Here are just the velocities we need in order to test whether or no mass increases with them.
In order to understand clearly the progress of the experiments, it remains to say a few words about the curious phenomenon of electrical inertia which is calledself-induction. When we want to set up an electric current, we find a certain initial resistance which ceases as soon as the current begins. If afterwards we want to break the current, it tends to maintain itself, and we have just the same trouble to stop it as to stop a vehicle in motion. It is a matter of dailyexperience. Sometimes the trolley of a tramcar leaves for a moment the wire which conducts the current, and we then see sparks. Why? There was a current passing from the wire to the trolley, and if the trolley breaks away from the wire for a moment, leaving an interval of air which obstructs the passage of electricity, the current will not stop. It has been set going, as it were, and it leaps the obstacle in the form of a spark. This phenomenon is what we call self-induction.
Self-induction—or “self” as the electrical workers call it—is a real inertia. The surrounding medium offers resistance to the force which tends to establish an electric current, and to that which tends to stop a current already set up; just as matter resists the force which tends to cause it to pass from rest to movement, or from movement to rest. There is, therefore, a real electrical inertia as well as mechanical inertia.
But our cathodic projectiles, our electrons, are charged. When they begin to move, they start an electric current; when they come to rest, the current ceases. Besides mechanical inertia, then, they must also have electrical inertia.They have, so to speak, two inertias; that is to say, two inert masses, a real and mechanical mass, and an apparent mass due to the phenomena of electro-magnetic self-induction.By studying the two deviations, electric and magnetic, of the Beta rays of radium or of the cathode rays, it is possible to determine the respective parts of each of these masses in the total mass of the electron. The electro-magnetic mass due to the causes which we have explained varies with the velocity, according to certain laws which we gather from the theory of electricity. Hence, byobserving the relation between the total mass and the velocity, we can see what part belongs to the real and invariable mass and what to the apparent mass of electro-magnetic origin.
The experiment has been made repeatedly by physicists of distinction. The result of it is surprising: the real mass isnil, and the whole mass of the particle is of electro-magnetic origin. Here is something that is calculated to modify entirely our ideas of the essence of what we call matter. But that is another story.
Physicists then asked themselves—this is what we were coming to, after clearing the way of various difficulties—whether the relation between the mass and the velocity of the cathodic projectiles was the same as that which we found in virtue of the Principle of Relativity.
The result of the experiments is absolutely clear and consistent, and some of them have dealt with Beta rays corresponding to a mass-value ten times greater than the original mass. This result is: mass varies with velocity, and in exact accord with the numerical laws of Einstein’s dynamics.
Here is a new and valuable experimental confirmation. This in turn tends to show that classical mechanics was merely a rough approximation, valid at the most only for the comparatively slight velocities with which we have to deal in the very restricted course of daily life.
Thus the mass of bodies, the Newtonian property which was believed to be the very symbol of constancy, the equivalent of what loyalty to treaties is in the moral order of things, is now merely a small coefficient, variable, undulating, and relative to the point of view. In virtue of the reciprocity which we have described, when there isquestion of contraction due to velocity, the mass of an object increases in the same way, not only if the object is displaced, but if the observer is displaced, and without any other observer, connected with the object, being able to detect the difference.
For instance, a measuring rod that moves at a velocity of about 260,000 kilometres a second will not only have its length shortened by one-half, but will have its mass doubled at the same time. Hence its density, which is the relation of its mass to its volume, will be quadrupled.
The physical ideas which were believed to be most solidly established, most constant, most unshakeable, have been uprooted by the storm of the new mechanics. They have become soft and plastic things moulded by velocity.
Further confirmations of the new formula, quite independent of the one we have just described, have recently been provided by physicists. One of the most astonishing of these is given in spectroscopy.
As is well known, when we cause a ray of sunlight, admitted through a narrow slit, to pass through the edge of a glass prism, the ray expands, as it issues from the prism, like a beautiful fan, the successive blades of which consist of the different colours of the rainbow. When we examine closely this coloured fan, we notice certain fine discontinuities, narrow lines or gaps, in which there is no light. They look like cuts made with a pair of scissors in our polychrome fan. They are the dark lines of the solar spectrum. Each of these lines, or each group of them, corresponds to a special chemical element, and serves to identify this, whether in our laboratories or in the sun and the stars.
It was explained long ago that these lines are due to electrons which revolve rapidly round the nuclei of the atoms. Their sudden changes of velocity give rise to a wave (like those caused in water when you drop a pebble into it) in the surrounding medium, and this is one of the characteristic luminous waves of the atom. It reveals itself in one of the lines of the spectrum. The Danish physicist Bohr has recently developed this theory in detail, and has shown that it accurately explains the various spectral lines of the different chemical elements. These, I may note, differ from each other in the number and arrangement of the electrons which revolve within their atoms.
Now Sommerfeld has argued as follows. The electrons which gravitate near the centre of an atom must have a higher velocity than those which revolve in its outer part; just as the smaller planets, Mercury and Venus, revolve round the sun far more rapidly than the larger planets, Jupiter and Saturn. It follows if Lorentz and Einstein are right that the mass of the interior electrons of the atoms must be greater than that of the exterior electrons: appreciably greater, as the former revolve with enormous velocities. We can calculate that, in those conditions, each line in the spectrum of a chemical element must in reality consist of a number of fine lines joined together. This is precisely what Paschen afterwards (1916) found. He discovered that the structure of the fine lines is strictly such as Sommerfeld had predicted. It was an astonishing confirmation of an hypothesis: a proof of the soundness of the new mechanics.
But that is not all. We know that the X-rays are vibrations analogous to light, the same in origin, but consisting of much shorter waves, orwaves with a far higher frequency. Hence, while light comes from the external electrons of the miniature solar system which we call an atom, the X-rays come from the most rapid electrons—those nearest to the centre. It follows that the special structure of the fine lines, due to the variation of the mass of the electron with its velocity, must be much more marked in the case of the X-rays than in the case of the spectral lines of light. This, again, was confirmed by experiment. The figures expressing the observed facts correspond exactly with the calculations of the new mechanics, as regards the predicted variation of mass with velocity.
It is therefore settled that the phenomena which take place in the microcosm of each atom are subject to the laws of the new mechanics, not the old, and that, in particular, masses in motion vary as the new mechanics demands.
Experience, “sole source of truth,” has given its verdict.
We are now very far from the ideas which were once prevalent. Lavoisier taught us that matter can neither be created nor destroyed. It remains always the same. What he meant was that mass is invariable, as he proved by means of scales. Now it appears that, perhaps, bodies have no mass at all—if it is entirely of electro-magnetic origin—and that, in any case, mass is not invariable. This does not mean that Lavoisier’s law has now no meaning. There remains something that corresponds to mass at low velocities. Our idea of matter is, however, revolutionised. By matter we particularly meant mass, which seemed to us to be at once the most tangible and most enduring of its properties. Now this “mass”has no more reality than the time and space in which we thought we located it! Our solid realities were but phantoms.
The reader must pardon me for whatever difficulties he finds in this exposition. The new mechanics opens out to us such strange new horizons that it is worth far more than a rapid and superficial glance. If you want to see a vast prospect in an unexplored world, you must not hesitate to do some rough climbing, however breathless it may leave you for the time.
There is, in fine, another fundamental idea of mechanics, that ofenergy, which takes on a new aspect in the light of Einstein’s theory: an aspect which, in turn, is largely justified by experiment.
We saw that a body charged with electricity and in motion makes a certain resistance to interference, on account of the electrical inertia which is known as self-induction. Calculation and experiment show that, if we reduce the dimensions of a body that is charged with a certain quantity of electricity, without altering the charge, the electrical inertia increases. As a matter of fact, in our hypotheses, and if the inertia is entirely electro-magnetic in origin, the electrons are now merely a sort of electric trails moving in the propagating medium of electrical and luminous waves which we call ether.
The electrons are no longer anything in themselves. They are merely, in the words of Poincaré, a sort of “holes in ether,” round which the ether presses much as a lake makes eddies which check the progress of a boat.
In that case, however, the smaller the holes in the ether are, the more important will be the agitation of the ether round them; and,consequently, the greater will be the inertia of the “hole in ether” which represents the corpuscle under investigation. What will follow? We know from measurements we have made that the mass of the tiny sun of each atom, thepositive nucleus, round which the planet-electrons revolve, is greater than that of an electron. If this mass and the corresponding inertia are electro-magnetic in origin, it follows that the positive nucleus of the atom is much smaller than the electron.
Let us consider the atom of hydrogen, the lightest and simplest of the gases. We know that it consists of one planet only, one single negative electron revolving round the minute central sun, the positive nucleus. We know also that the mass of the electron is two thousand times as small as that of the hydrogen atom. It follows, as we can calculate, that thepositive nucleusmust have a radius two thousand times smaller than that of the electron. Now, the experiments of the English physicists have proved that the large Alpha particles of the radium emanation can pass through hundreds of thousands of atoms without being appreciably diverted by the positive nucleus. We conclude that the latter is in reality much smaller than the electron, as theory predicted.
All this irresistibly compels us to think that the inertia of the various component parts of atoms—that is to say, of all matter—is exclusively electro-magnetic in origin. There is now no matter. There is only electrical energy, which, by the reactions of the surrounding medium upon it, leads us to the fallacious belief in the existence of this substantial and massive something which hundreds of generations have been wont to call “matter.”
And from all this it also follows, by calculation and by the simple and elegant reasoning of Einstein, of which I here convey only the faintest adumbration, that mass and energy are the same thing, or are at least the two different sides of one and the same coin. There is, then, no longer a material mass. There is nothing but energy in the external universe. A strange—in a sense, an almost spiritual—turn for modern physics to take!
According to all this the greater part of the “mass” of bodies must be due to a considerable and concealed internal energy. It is this energy which we find gradually dissipated in radio-active bodies, the only reservoirs of atomic energy which have as yet opened externally.
If this is true, if energy and mass are synonymous, if mass is merely energy, it follows that free energy must possess the property of mass. As a matter of fact, light, for instance, has mass. Careful experiments have shown that when a ray of light strikes a material object, it exerts upon it a pressure which has been measured. Light has mass; therefore it has weight, like all masses. When we come to consider the new form given by Einstein to the problem of gravitation, we shall see a further and beautiful proof that light has weight.
We can calculate that the light received from the sun by the earth in the space of a year is rather more than 58,000 tons. It seems very little when one thinks of the formidable weight of coal that would be needed to maintain our globe at the temperature at which the sun keeps it—in the event of a sudden extinction of our luminary.
The reason for the difference is that, when we produce heat from a certain amount of coal, we use only a small proportion of its totalenergy, its chemical energy. Its intra-atomic energy is inaccessible to us. It is a pity, as otherwise we should need only a few ounces of coal to supply heat for a whole year to all the towns and workshops of England! How many problems that would simplify! When humanity emerges from the ignorance and the clumsy barbarism in which it lives to-day—that is to say, in some hundreds of centuries—this will be accomplished. Yes, it will one day be done. It will be a glorious spectacle, one in which we may justly rejoice in advance.
Meantime, our sun, like all the other stars, like every incandescent body, loses its weight in proportion as it radiates. But this happens so slowly that we need not fear to see it disappear at some early date, like the ephemeral things which die because they gave themselves too freely.
To finish with Einstein’s mechanics, let me reproduce a very suggestive application of these ideas about the identity of energy and mass.
There is in chemistry a well-known elementary law which is called “Prout’s Law.” It states that the atomic masses of all the elements must be whole multiples of the mass of hydrogen. Since hydrogen has the lightest atoms amongst all known bodies Prout’s Law started from the hypothesis that all the atoms are built up of a fundamental element, the atom of hydrogen. This supposed unity of matter seems to be more and more confirmed by the facts. On the one hand, it is proved that the electrons which come from different chemical elements are identical. On the other hand, in the transformation of radio-active bodies we find heavy atoms simplifying themselves by successively emitting atoms ofhelium gas. Lastly, the great British physicist Sir Ernest Rutherford showed in 1919 that by bombarding the atoms of nitrogen gas, in certain circumstances, by means of radium emanation, we can detach hydrogen atoms from them. This experiment, the importance of which has not been fully realised—it is the first instance of transmutation really effected by man—also tends to prove the soundness of Prout’s hypothesis.
Yet, when we accurately measure and compare the atomic masses of the various chemical elements, we find that they do not strictly conform to Prout’s Law. For instance, while the atomic mass of hydrogen is 1, that of chlorium is 35·46, which is not a whole multiple of 1.
But we can calculate that, if the formation of complex atoms from hydrogen upwards is accompanied, as is probable, by variations of internal energy, as a consequence of the radiation of a certain amount of energy during the combination, it necessarily follows (since the lost energy has weight) that there will be variations in the mass of the body composed, and these will explain the known departures from Prout’s Law.
In our somewhat hurried and informal excursion into the bush of the new facts which confirm the mechanics outlined by Lorentz and completed by Einstein our progress has been rather difficult. It is because, since we could not use terminology and technical formulæ which would be unsuitable in this work, we have had to be content with bold and rapid moves into the districts we wished to reconnoitre. Perhaps they have sufficed to enable the reader to understand what a revolution in the very bases of science, what an explosion amidst its age-oldfoundations, the brilliant synthesis of Einstein has caused. New light now streams upon all who slowly climb the slopes of knowledge: upon all who, wisely renouncing the desire to know “why,” would at least learn the “how” in many things.
A little before his death, foreseeing, with the intuition of genius, that a new era opened in mechanics, Poincaré advised professors not to teach the new truths to the young until they were steeped to the very marrow of their bones in the older mechanics.
“It is,” he added, “with ordinary mechanics that their life is concerned: it is that alone that they will ever have to apply. Whatever speed our motor-cars may attain, they will never reach a speed at which the old mechanics ceases to be true. The new is a luxury, and we must think of luxuries only when it can be done without injury to necessaries.”
I would appeal from Poincaré’s text to the man himself. For him this luxury, the truth, was a necessary. On the day in question, it is true, he thought of the young. But do men ever cease to be children? To that the master, too early taken from us, would have replied, in his grave, smiling manner: “Yes—at all events, it is better to suppose so.”