The Present Crisis of Mathematical Physics
The New Crisis.— Are we now about to enter upon a third period? Are we on the eve of a second crisis? These principles on which we have built all, are they about to crumble1 away in their turn? This has been for some time a pertinent2 question.
When I speak thus, you no doubt think of radium, that grand revolutionist of the present time, and in fact I shall come back to it presently; but there is something else. It is not alone the conservation of energy which is in question; all the other principles are equally in danger, as we shall see in passing them successively in review.
Carnot’s Principle.— Let us commence with the principle of Carnot. This is the only one which does not present itself as an immediate3 consequence of the hypothesis of central forces; more than that, it seems, if not to directly contradict that hypothesis, at least not to be reconciled with it without a certain effort. If physical phenomena4 were due exclusively to the movements of atoms whose mutual5 attraction depended only on the distance, it seems that all these phenomena should be reversible; if all the initial velocities6 were reversed, these atoms, always subjected to the same forces, ought to go over their trajectories7 in the contrary sense, just as the earth would describe in the retrograde sense this same elliptic orbit which it describes in the direct sense, if the initial conditions of its motion had been reversed. On this account, if a physical phenomenon is possible, the inverse8 phenomenon should be equally so, and one should be able to reascend the course of time. Now, it is not so in nature, and this is precisely9 what the principle of Carnot teaches us; heat can pass from the warm body to the cold body; it is impossible afterward10 to make it take the inverse route and to reestablish differences of temperature which have been effaced11. Motion can be wholly dissipated and transformed into heat by friction12; the contrary transformation13 can never be made except partially14.
We have striven to reconcile this apparent contradiction. If the world tends toward uniformity, this is not because its ultimate parts, at first unlike, tend to become less and less different; it is because, shifting at random15, they end by blending. For an eye which should distinguish all the elements, the variety would remain always as great; each grain of this dust preserves its originality16 and does not model itself on its neighbors; but as the blend becomes more and more intimate, our gross senses perceive only the uniformity. This is why, for example, temperatures tend to a level, without the possibility of going backwards17.
A drop of wine falls into a glass of water; whatever may be the law of the internal motion of the liquid, we shall soon see it colored of a uniform rosy18 tint19, and however much from this moment one may shake it afterwards, the wine and the water do not seem capable of again separating. Here we have the type of the irreversible physical phenomenon: to hide a grain of barley20 in a heap of wheat, this is easy; afterwards to find it again and get it out, this is practically impossible. All this Maxwell and Boltzmann have explained; but the one who has seen it most clearly, in a book too little read because it is a little difficult to read, is Gibbs, in his ‘Elementary Principles of Statistical21 Mechanics.’
For those who take this point of view, Carnot’s principle is only an imperfect principle, a sort of concession22 to the infirmity of our senses; it is because our eyes are too gross that we do not distinguish the elements of the blend; it is because our hands are too gross that we can not force them to separate; the imaginary demon23 of Maxwell, who is able to sort the molecules25 one by one, could well constrain26 the world to return backward. Can it return of itself? That is not impossible; that is only infinitely27 improbable. The chances are that we should wait a long time for the concourse of circumstances which would permit a retrogradation; but sooner or later they will occur, after years whose number it would take millions of figures to write. These reservations, however, all remained theoretic; they were not very disquieting28, and Carnot’s principle retained all its practical value. But here the scene changes. The biologist, armed with his microscope, long ago noticed in his preparations irregular movements of little particles in suspension; this is the Brownian movement. He first thought this was a vital phenomenon, but soon he saw that the inanimate bodies danced with no less ardor29 than the others; then he turned the matter over to the physicists30. Unhappily, the physicists remained long uninterested in this question; one concentrates the light to illuminate31 the microscopic32 preparation, thought they; with light goes heat; thence inequalities of temperature and in the liquid interior currents which produce the movements referred to. It occurred to M. Gouy to look more closely, and he saw, or thought he saw, that this explanation is untenable, that the movements become brisker as the particles are smaller, but that they are not influenced by the mode of illumination. If then these movements never cease, or rather are reborn without cease, without borrowing anything from an external source of energy, what ought we to believe? To be sure, we should not on this account renounce33 our belief in the conservation of energy, but we see under our eyes now motion transformed into heat by friction, now inversely34 heat changed into motion, and that without loss since the movement lasts forever. This is the contrary of Carnot’s principle. If this be so, to see the world return backward, we no longer have need of the infinitely keen eye of Maxwell’s demon; our microscope suffices. Bodies too large, those, for example, which are a tenth of a millimeter, are hit from all sides by moving atoms, but they do not budge35, because these shocks are very numerous and the law of chance makes them compensate36 each other; but the smaller particles receive too few shocks for this compensation to take place with certainty and are incessantly37 knocked about. And behold38 already one of our principles in peril39.
The Principle of Relativity.— Let us pass to the principle of relativity; this not only is confirmed by daily experience, not only is it a necessary consequence of the hypothesis of central forces, but it is irresistibly40 imposed upon our good sense, and yet it also is assailed41. Consider two electrified42 bodies; though they seem to us at rest, they are both carried along by the motion of the earth; an electric charge in motion, Rowland has taught us, is equivalent to a current; these two charged bodies are, therefore, equivalent to two parallel currents of the same sense and these two currents should attract each other. In measuring this attraction, we shall measure the velocity43 of the earth; not its velocity in relation to the sun or the fixed44 stars, but its absolute velocity.
I well know what will be said: It is not its absolute velocity that is measured, it is its velocity in relation to the ether. How unsatisfactory that is! Is it not evident that from the principle so understood we could no longer infer anything? It could no longer tell us anything just because it would no longer fear any contradiction. If we succeed in measuring anything, we shall always be free to say that this is not the absolute velocity, and if it is not the velocity in relation to the ether, it might always be the velocity in relation to some new unknown fluid with which we might fill space.
Indeed, experiment has taken upon itself to ruin this interpretation45 of the principle of relativity; all attempts to measure the velocity of the earth in relation to the ether have led to negative results. This time experimental physics has been more faithful to the principle than mathematical physics; the theorists, to put in accord their other general views, would not have spared it; but experiment has been stubborn in confirming it. The means have been varied46; finally Michelson pushed precision to its last limits; nothing came of it. It is precisely to explain this obstinacy47 that the mathematicians48 are forced to-day to employ all their ingenuity49.
Their task was not easy, and if Lorentz has got through it, it is only by accumulating hypotheses.
The most ingenious idea was that of local time. Imagine two observers who wish to adjust their timepieces by optical signals; they exchange signals, but as they know that the transmission of light is not instantaneous, they are careful to cross them. When station B perceives the signal from station A, its clock should not mark the same hour as that of station A at the moment of sending the signal, but this hour augmented50 by a constant representing the duration of the transmission. Suppose, for example, that station A sends its signal when its clock marks the hour O, and that station B perceives it when its clock marks the hour t. The clocks are adjusted if the slowness equal to t represents the duration of the transmission, and to verify it, station B sends in its turn a signal when its clock marks O; then station A should perceive it when its clock marks t. The timepieces are then adjusted.
And in fact they mark the same hour at the same physical instant, but on the one condition, that the two stations are fixed. Otherwise the duration of the transmission will not be the same in the two senses, since the station A, for example, moves forward to meet the optical perturbation emanating51 from B, whereas the station B flees before the perturbation emanating from A. The watches adjusted in that way will not mark, therefore, the true time; they will mark what may be called the local time, so that one of them will be slow of the other. It matters little, since we have no means of perceiving it. All the phenomena which happen at A, for example, will be late, but all will be equally so, and the observer will not perceive it, since his watch is slow; so, as the principle of relativity requires, he will have no means of knowing whether he is at rest or in absolute motion.
Unhappily, that does not suffice, and complementary hypotheses are necessary; it is necessary to admit that bodies in motion undergo a uniform contraction52 in the sense of the motion. One of the diameters of the earth, for example, is shrunk by one two-hundred-millionth in consequence of our planet’s motion, while the other diameter retains its normal length. Thus the last little differences are compensated53. And then, there is still the hypothesis about forces. Forces, whatever be their origin, gravity as well as elasticity54, would be reduced in a certain proportion in a world animated55 by a uniform translation; or, rather, this would happen for the components56 perpendicular57 to the translation; the components parallel would not change. Resume, then, our example of two electrified bodies; these bodies repel58 each other, but at the same time if all is carried along in a uniform translation, they are equivalent to two parallel currents of the same sense which attract each other. This electrodynamic attraction diminishes, therefore, the electrostatic repulsion, and the total repulsion is feebler than if the two bodies were at rest. But since to measure this repulsion we must balance it by another force, and all these other forces are reduced in the same proportion, we perceive nothing. Thus all seems arranged, but are all the doubts dissipated? What would happen if one could communicate by non-luminous signals whose velocity of propagation differed from that of light? If, after having adjusted the watches by the optical procedure, we wished to verify the adjustment by the aid of these new signals, we should observe discrepancies59 which would render evident the common translation of the two stations. And are such signals inconceivable, if we admit with Laplace that universal gravitation is transmitted a million times more rapidly than light?
Thus, the principle of relativity has been valiantly60 defended in these latter times, but the very energy of the defense61 proves how serious was the attack.
Newton’s Principle.— Let us speak now of the principle of Newton, on the equality of action and reaction. This is intimately bound up with the preceding, and it seems indeed that the fall of the one would involve that of the other. Thus we must not be astonished to find here the same difficulties.
Electrical phenomena, according to the theory of Lorentz, are due to the displacements62 of little charged particles, called electrons, immersed in the medium we call ether. The movements of these electrons produce perturbations in the neighboring ether; these perturbations propagate themselves in every direction with the velocity of light, and in turn other electrons, originally at rest, are made to vibrate when the perturbation reaches the parts of the ether which touch them. The electrons, therefore, act on one another, but this action is not direct, it is accomplished63 through the ether as intermediary. Under these conditions can there be compensation between action and reaction, at least for an observer who should take account only of the movements of matter, that is, of the electrons, and who should be ignorant of those of the ether that he could not see? Evidently not. Even if the compensation should be exact, it could not be simultaneous. The perturbation is propagated with a finite velocity; it, therefore, reaches the second electron only when the first has long ago entered upon its rest. This second electron, therefore, will undergo, after a delay, the action of the first, but will certainly not at that moment react upon it, since around this first electron nothing any longer budges64.
The analysis of the facts permits us to be still more precise. Imagine, for example, a Hertzian oscillator, like those used in wireless65 telegraphy; it sends out energy in every direction; but we can provide it with a parabolic mirror, as Hertz did with his smallest oscillators, so as to send all the energy produced in a single direction. What happens then according to the theory? The apparatus66 recoils68, as if it were a cannon69 and the projected energy a ball; and that is contrary to the principle of Newton, since our projectile70 here has no mass, it is not matter, it is energy. The case is still the same, moreover, with a beacon71 light provided with a reflector, since light is nothing but a perturbation of the electromagnetic field. This beacon light should recoil67 as if the light it sends out were a projectile. What is the force that should produce this recoil? It is what is called the Maxwell-Bartholi pressure. It is very minute, and it has been difficult to put it in evidence even with the most sensitive radiometers; but it suffices that it exists.
If all the energy issuing from our oscillator falls on a receiver, this will act as if it had received a mechanical shock, which will represent in a sense the compensation of the oscillator’s recoil; the reaction will be equal to the action, but it will not be simultaneous; the receiver will move on, but not at the moment when the oscillator recoils. If the energy propagates itself indefinitely without encountering a receiver, the compensation will never occur.
Shall we say that the space which separates the oscillator from the receiver and which the perturbation must pass over in going from the one to the other is not void, that it is full not only of ether, but of air, or even in the interplanetary spaces of some fluid subtile but still ponderable; that this matter undergoes the shock like the receiver at the moment when the energy reaches it, and recoils in its turn when the perturbation quits it? That would save Newton’s principle, but that is not true. If energy in its diffusion72 remained always attached to some material substratum, then matter in motion would carry along light with it, and Fizeau has demonstrated that it does nothing of the sort, at least for air. Michelson and Morley have since confirmed this. It might be supposed also that the movements of matter proper are exactly compensated by those of the ether; but that would lead us to the same reflections as before now. The principle so understood will explain everything, since, whatever might be the visible movements, we always could imagine hypothetical movements which compensate them. But if it is able to explain everything, this is because it does not enable us to foresee anything; it does not enable us to decide between the different possible hypotheses, since it explains everything beforehand. It therefore becomes useless.
And then the suppositions that it would be necessary to make on the movements of the ether are not very satisfactory. If the electric charges double, it would be natural to imagine that the velocities of the diverse atoms of ether double also; but, for the compensation, it would be necessary that the mean velocity of the ether quadruple.
This is why I have long thought that these consequences of theory, contrary to Newton’s principle, would end some day by being abandoned, and yet the recent experiments on the movements of the electrons issuing from radium seem rather to confirm them.
Lavoisier’s Principle.— I arrive at the principle of Lavoisier on the conservation of mass. Certainly, this is one not to be touched without unsettling all mechanics. And now certain persons think that it seems true to us only because in mechanics merely moderate velocities are considered, but that it would cease to be true for bodies animated by velocities comparable to that of light. Now these velocities are believed at present to have been realized; the cathode rays and those of radium may be formed of very minute particles or of electrons which are displaced with velocities smaller no doubt than that of light, but which might be its one tenth or one third.
These rays can be deflected73, whether by an electric field, or by a magnetic field, and we are able, by comparing these deflections, to measure at the same time the velocity of the electrons and their mass (or rather the relation of their mass to their charge). But when it was seen that these velocities approached that of light, it was decided75 that a correction was necessary. These molecules, being electrified, can not be displaced without agitating76 the ether; to put them in motion it is necessary to overcome a double inertia77, that of the molecule24 itself and that of the ether. The total or apparent mass that one measures is composed, therefore, of two parts: the real or mechanical mass of the molecule and the electrodynamic mass representing the inertia of the ether.
The calculations of Abraham and the experiments of Kaufmann have then shown that the mechanical mass, properly so called, is null, and that the mass of the electrons, or, at least, of the negative electrons, is of exclusively electrodynamic origin. This is what forces us to change the definition of mass; we can not any longer distinguish mechanical mass and electrodynamic mass, since then the first would vanish; there is no mass other than electrodynamic inertia. But in this case the mass can no longer be constant; it augments78 with the velocity, and it even depends on the direction, and a body animated by a notable velocity will not oppose the same inertia to the forces which tend to deflect74 it from its route, as to those which tend to accelerate or to retard79 its progress.
There is still a resource; the ultimate elements of bodies are electrons, some charged negatively, the others charged positively80. The negative electrons have no mass, this is understood; but the positive electrons, from the little we know of them, seem much greater. Perhaps they have, besides their electrodynamic mass, a true mechanical mass. The real mass of a body would, then, be the sum of the mechanical masses of its positive electrons, the negative electrons not counting; mass so defined might still be constant.
Alas81! this resource also evades us. Recall what we have said of the principle of relativity and of the efforts made to save it. And it is not merely a principle which it is a question of saving, it is the indubitable results of the experiments of Michelson.
Well, as was above seen, Lorentz, to account for these results, was obliged to suppose that all forces, whatever their origin, were reduced in the same proportion in a medium animated by a uniform translation; this is not sufficient; it is not enough that this take place for the real forces, it must also be the same for the forces of inertia; it is therefore necessary, he says, that the masses of all the particles be influenced by a translation to the same degree as the electromagnetic masses of the electrons.
So the mechanical masses must vary in accordance with the same laws as the electrodynamic masses; they can not, therefore, be constant.
Need I point out that the fall of Lavoisier’s principle involves that of Newton’s? This latter signifies that the center of gravity of an isolated82 system moves in a straight line; but if there is no longer a constant mass, there is no longer a center of gravity, we no longer know even what this is. This is why I said above that the experiments on the cathode rays appeared to justify83 the doubts of Lorentz concerning Newton’s principle.
From all these results, if they were confirmed, would arise an entirely84 new mechanics, which would be, above all, characterized by this fact, that no velocity could surpass that of light,9 any more than any temperature can fall below absolute zero.
9 Because bodies would oppose an increasing inertia to the causes which would tend to accelerate their motion; and this inertia would become infinite when one approached the velocity of light.
No more for an observer, carried along himself in a translation he does not suspect, could any apparent velocity surpass that of light; and this would be then a contradiction, if we did not recall that this observer would not use the same clocks as a fixed observer, but, indeed, clocks marking ‘local time.’
Here we are then facing a question I content myself with stating. If there is no longer any mass, what becomes of Newton’s law? Mass has two aspects: it is at the same time a coefficient of inertia and an attracting mass entering as factor into Newtonian attraction. If the coefficient of inertia is not constant, can the attracting mass be? That is the question.
Mayer’s Principle.— At least, the principle of the conservation of energy yet remained to us, and this seemed more solid. Shall I recall to you how it was in its turn thrown into discredit85? This event has made more noise than the preceding, and it is in all the memoirs86. From the first words of Becquerel, and, above all, when the Curies had discovered radium, it was seen that every radioactive body was an inexhaustible source of radiation. Its activity seemed to subsist87 without alteration88 throughout the months and the years. This was in itself a strain on the principles; these radiations were in fact energy, and from the same morsel89 of radium this issued and forever issued. But these quantities of energy were too slight to be measured; at least that was the belief and we were not much disquieted90.
The scene changed when Curie bethought himself to put radium in a calorimeter; it was then seen that the quantity of heat incessantly created was very notable.
The explanations proposed were numerous; but in such case we can not say, the more the better. In so far as no one of them has prevailed over the others, we can not be sure there is a good one among them. Since some time, however, one of these explanations seems to be getting the upper hand and we may reasonably hope that we hold the key to the mystery.
Sir W. Ramsay has striven to show that radium is in process of transformation, that it contains a store of energy enormous but not inexhaustible. The transformation of radium then would produce a million times more heat than all known transformations91; radium would wear itself out in 1,250 years; this is quite short, and you see that we are at least certain to have this point settled some hundreds of years from now. While waiting, our doubts remain.
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crumble
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| vi.碎裂，崩溃；vt.弄碎，摧毁 | |
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pertinent
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| adj.恰当的；贴切的；中肯的；有关的；相干的 | |
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immediate
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| adj.立即的；直接的，最接近的；紧靠的 | |
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phenomena
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| n.现象 | |
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mutual
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| adj.相互的，彼此的；共同的，共有的 | |
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velocities
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| n.速度( velocity的名词复数 )；高速，快速 | |
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trajectories
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| n.弹道( trajectory的名词复数 )；轨道；轨线；常角轨道 | |
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inverse
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| adj.相反的，倒转的，反转的；n.相反之物；v.倒转 | |
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precisely
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| adv.恰好，正好，精确地，细致地 | |
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afterward
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| adv.后来；以后 | |
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effaced
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| v.擦掉( efface的过去式和过去分词 )；抹去；超越；使黯然失色 | |
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friction
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| n.摩擦，摩擦力 | |
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transformation
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| n.变化；改造；转变 | |
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partially
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| adv.部分地，从某些方面讲 | |
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random
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| adj.随机的；任意的；n.偶然的(或随便的)行动 | |
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originality
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| n.创造力，独创性；新颖 | |
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backwards
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| adv.往回地，向原处，倒，相反，前后倒置地 | |
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rosy
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| adj.美好的，乐观的，玫瑰色的 | |
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tint
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| n.淡色，浅色；染发剂；vt.着以淡淡的颜色 | |
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barley
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| n.大麦，大麦粒 | |
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statistical
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| adj.统计的，统计学的 | |
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concession
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| n.让步，妥协；特许(权) | |
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demon
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| n.魔鬼，恶魔 | |
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molecule
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| n.分子，克分子 | |
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molecules
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| 分子( molecule的名词复数 ) | |
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constrain
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| vt.限制，约束；克制，抑制 | |
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infinitely
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| adv.无限地，无穷地 | |
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disquieting
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| adj.令人不安的，令人不平静的v.使不安，使忧虑，使烦恼( disquiet的现在分词 ) | |
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ardor
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| n.热情，狂热 | |
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physicists
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| 物理学家( physicist的名词复数 ) | |
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illuminate
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| vt.照亮，照明；用灯光装饰；说明，阐释 | |
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microscopic
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| adj.微小的，细微的，极小的，显微的 | |
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renounce
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| v.放弃；拒绝承认，宣布与…断绝关系 | |
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inversely
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| adj.相反的 | |
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budge
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| v.移动一点儿；改变立场 | |
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compensate
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| vt.补偿，赔偿；酬报 vi.弥补；补偿；抵消 | |
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incessantly
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| ad.不停地 | |
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behold
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| v.看，注视，看到 | |
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peril
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| n.(严重的)危险；危险的事物 | |
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irresistibly
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| adv.无法抵抗地，不能自持地；极为诱惑人地 | |
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assailed
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| v.攻击( assail的过去式和过去分词 )；困扰；质问；毅然应对 | |
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electrified
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| v.使电气化( electrify的过去式和过去分词 )；使兴奋 | |
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velocity
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| n.速度，速率 | |
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fixed
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| adj.固定的，不变的，准备好的；(计算机)固定的 | |
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interpretation
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| n.解释，说明，描述；艺术处理 | |
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varied
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| adj.多样的，多变化的 | |
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obstinacy
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| n.顽固；（病痛等）难治 | |
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mathematicians
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| 数学家( mathematician的名词复数 ) | |
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ingenuity
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| n.别出心裁；善于发明创造 | |
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Augmented
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| adj.增音的 动词augment的过去式和过去分词形式 | |
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emanating
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| v.从…处传出，传出( emanate的现在分词 )；产生，表现，显示 | |
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contraction
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| n.缩略词，缩写式，害病 | |
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compensated
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| 补偿，报酬( compensate的过去式和过去分词 )； 给（某人）赔偿（或赔款） | |
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elasticity
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| n.弹性，伸缩力 | |
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animated
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| adj.生气勃勃的，活跃的，愉快的 | |
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components
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| （机器、设备等的）构成要素，零件，成分； 成分( component的名词复数 )； [物理化学]组分； [数学]分量； （混合物的）组成部分 | |
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perpendicular
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| adj.垂直的，直立的；n.垂直线，垂直的位置 | |
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repel
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| v.击退，抵制，拒绝，排斥 | |
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discrepancies
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| n.差异，不符合（之处），不一致（之处）( discrepancy的名词复数 ) | |
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valiantly
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| adv.勇敢地，英勇地；雄赳赳 | |
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defense
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| n.防御，保卫；[pl.]防务工事；辩护，答辩 | |
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displacements
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| n.取代( displacement的名词复数 )；替代；移位；免职 | |
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accomplished
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| adj.有才艺的；有造诣的；达到了的 | |
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budges
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| v.（使）稍微移动( budge的第三人称单数 )；（使）改变主意，（使）让步 | |
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wireless
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| adj.无线的；n.无线电 | |
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apparatus
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| n.装置，器械；器具，设备 | |
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recoil
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| vi.退却，退缩，畏缩 | |
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recoils
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| n.（尤指枪炮的）反冲，后坐力( recoil的名词复数 )v.畏缩( recoil的第三人称单数 )；退缩；报应；返回 | |
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cannon
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| n.大炮，火炮；飞机上的机关炮 | |
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projectile
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| n.投射物，发射体；adj.向前开进的；推进的；抛掷的 | |
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beacon
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| n.烽火，（警告用的）闪火灯，灯塔 | |
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diffusion
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| n.流布；普及；散漫 | |
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deflected
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| 偏离的 | |
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deflect
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| v.(使)偏斜，(使)偏离，(使)转向 | |
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decided
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| adj.决定了的，坚决的；明显的，明确的 | |
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agitating
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| 搅动( agitate的现在分词 )； 激怒； 使焦虑不安； （尤指为法律、社会状况的改变而）激烈争论 | |
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inertia
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| adj.惰性，惯性，懒惰，迟钝 | |
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augments
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| 增加，提高，扩大( augment的名词复数 ) | |
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retard
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| n.阻止，延迟；vt.妨碍，延迟，使减速 | |
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positively
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| adv.明确地，断然，坚决地；实在，确实 | |
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alas
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| int.唉(表示悲伤、忧愁、恐惧等) | |
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isolated
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| adj.与世隔绝的 | |
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justify
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| vt.证明…正当(或有理)，为…辩护 | |
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entirely
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| ad.全部地，完整地；完全地，彻底地 | |
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discredit
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| vt.使不可置信；n.丧失信义；不信，怀疑 | |
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memoirs
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| n.回忆录；回忆录传( mem,自oir的名词复数) | |
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subsist
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| vi.生存，存在，供养 | |
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alteration
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| n.变更，改变；蚀变 | |
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morsel
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| n.一口，一点点 | |
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disquieted
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| v.使不安，使忧虑，使烦恼( disquiet的过去式和过去分词 ) | |
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transformations
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| n.变化( transformation的名词复数 )；转换；转换；变换 | |
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