PHILOSOPHY AND THE NEW PHYSICS
ROUG1ER
PHILOSOPHY AND THE NEW PHYSICS
An Essay on the Relativity Theory and the Theory of Quanta
BY
LOUIS ROUGIER
PROFESSEUR AGREGE DE PHILOSOPHIE, DOCTEUR ES LETTRES
Authorized Translation
From the Author's Corrected Text of
'La Materialisation de I'finergie"
BY MORTON MASIUS, M.A., PH.D.
PROFESSOR OF PHYSICS IN THE WORCESTER POLYTECHNIC INSTITUTE TRANSLATOR OF M. PLANCK'S THEORY OF RADIATION
PHILADELPHIA
P. BLAKISTON'S SON & CO.
1012 WALNUT STREET
COPYRIGHT, 1921, BY P. BLAKISTON'S SON & Co.
THE MAPLE PRESS YORK PA
TRANSLATOR'S PREFACE
The recent remarkable developments of physical theories, especially those concerned with relativity and quanta of energy, cannot fail to have far-reach- ing influences on philosophical thought. Physicists, as a rule, are too much occupied with their special field to give much attention to matters of more general philosophical interest, and few philosophers possess the knowledge of science required for dis- cussing and criticizing fruitfully the work of the physicist. Professor Rougier's very wide reading in mathematical and experimental Physics has en- abled him to present and interpret the new ad- vances in Physics in a way which should prove of great interest to both philosopher and physicist. This book seems to mark a measurable advance toward a confluence of the broad streams of philo- sophical and scientific enquiry.
M. M.
449830
CONTENTS
CHAPTER PAGE
I. THE DUALISM OF MATTER AND ENERGY.
1. Introduction 1
2. The Dualistic Theory 3
3. The Difficulties of the Dualistic Theory and the
Check given to the Monistic Attempts at Reduction 6
4. Ostwald's Energetics 12
5. The Insufficiency of Ostwald's Energetics and the
Experimental Discovery of the Inertia of Energy . 15
II. MASS AND THE RELATIVITY PRINCIPLE.
6. The Idea of Mass; Einstein's Equivalence Principle
and Newton's Principle of Action and Reaction . . 22
7. The Relativity Principle 28
III. ELECTROMAGNETIC DYNAMICS.
8. The Localization of Energy Outside of Matter .... 41
9. Electromagnetic Inertia and the Dynamics of an
Electrified Particle in Motion 47
IV. THE ELECTRONIC THEORY OF MATTER.
10. Lorentz's Synthesis .'.... 57
11. The Dematerialization of Matter 62
V. THE INERTIA OF ENERGY.
12. The Materialization of Energy 73
13. The Evaluation of the Internal Energy of Bodies and
the Variations of Mass 84
VI. THE WEIGHT OF ENERGY.
14. The Weight of Energy; Its Experimental Verifications. 91
15. The Generalized Principle of Relativity and Einstein's
Theory of Gravitation 96
16. Astronomical Verification 106
vii
Vlll CONTENTS
CHAPTER PAGE
VII. THE STRUCTURE OF ENERGY.
17. The Success of the Electronic Theory; the Explanation
of the Relations of Matter and Radiation 1 10
18. Check to the Electronic Theory; Black Radiation and
the Quantum Theory 120
19. The Structure of Radiation 134
20. The Physics of the Discontinuous 143
VIII. CONCLUSION.
21. Conclusion 148
BIBLIOGRAPHY 153
INDEX OF NAMES. 157
INTRODUCTION
It has been customary since the time of Auguste Comte (1795-1857) to distinguish two entirely separate categories of problems: those which may be treated by the methods of science and which are, because of the perfection these methods have reached, capable of being solved sooner or later, and those which, being beyond these methods, lie out- side of the limits of experience and on that account are called metaphysical problems. The former problems, taken together, mark off the field of the exact science; the latter comprise what Herbert Spencer and Du Bois-Reymond have called the unknowable.
We have no criterion by means of which to decide a priori to which of these two categories a given problem belongs. In every case in which a philosophical system has claimed to lay down, dogmatically, the limits to our experience, later scientific discoveries have seemed to make a point of proving it to be wrong. For example, the real nature of phenomena, according to Auguste Comte, the founder of positivism, will never be disclosed; "When a scientific theory," declares Poincare,1
1 H. Poincare, La valeur de la Science, p. 267.
ix
X INTRODUCTION
"claims to tell us what heat, what electricity, or what life really is, it stands convicted at the outset.7' This idea led Ostwald to his Energetics, and Duhem to his Theory of Physics. Who at the present time would bind himself to this prudent agnosticism, to the extent of doubting the objective existence of discontinuous elements, like molecules, atoms, or electrons? "The atoms are no longer a convenient fiction; it seems to us that we can, so to speak, see them, since we know how to count them/'1 as Poin- care had to admit in the later part of his life when confronted by the achievements of the atomic hypotheses.
It occurs much more often that metaphysical problems, supposedly incapable of solution, vanish, simply because the progress of ideas shows that they are fictitious problems or pseudo-problems, or problems which have been badly conceived. We have nothing but pity mingled with boredom for the endless disputes of the Scholastics on the subject of the unity or the plurality of the substantial forms in animal species. We know, in fact, that nothing but individuals really exist, and that, to the static abstractions of our mind regarding these genera and species, nothing invariable and essential corresponds, beyond a mere bond of relationship and a family resemblance between the individuals that we range in the same class. The question of knowing whether a body is at rest or in absolute motion is, according to Einstein, another example
1 H. PoincarS, Dernieres pensees, p. 196.
INTRODUCTION XI
of a pseudo-problem. There exists no absolute space which might serve as a privileged reference system; there exist merely bodies at rest or in motion relatively to one another.
It is a metaphysical problem of the same kind that arises from the fundamental dualism between ponderable matter and imponderable energy, which classical physics uses for the basis of an explanation of the world and which gives place to two principles of invariance, the principle of the conservation of mass and the principle of the conservation of energy. There is a radical difference of character between these two components of all phenomena of nature, matter and energy; matter alone is endowed with mass, with weight in proportion thereto, and with structure; energy has no inertia, no weight, and no structure. Therefore how is it to be conceived that an imponderable noncorpuscular agent that has no distribution in space capable of representation can be applied to a body with both mass and inertia, and possessing a definite shape, and act on it to the point of deforming or moving it? How, for example, can luminous radiation, if it is destitute of mass, exert a repulsion in one direction on the source which emits it and an impulse on the opaque body absorbing it, as if it represented momentum ; how, in other words, if it -is deprived of inertia, can it behave like a material projectile which exerts a recoil pressure on the fire-arm which throws it and a ballistic action on the obstacle which it strikes? And if energy possesses some
Xll INTRODUCTION
inertia how can it accumulate without effect on a body, like electric energy, without the mass of this body increasing? This is the metaphysical problem of the mutual action of energy and matter. If one assimilates mental labor to a kind of energy sui generis, the metaphysical problem of the relations between mind and body presents itself as a special case of the foregoing.
Vain attempts have been made to reduce one of the two terms placed in contrast to the other by the dualistic theory, in order to escape the problem of their mutual action; these monistic attempts have merely served to shift the problem by substituting new difficulties for the one that they attempted to overcome.
All this became quite different as soon as the theory of relativity of Einstein and the theory of quanta of Max Planck led, if not to a complete rejection of the fundamental dualism of matter and energy, at any rate to an approach of the two terms by assigning to them such common properties as to render their relations intelligible. Thus these theories confer on energy inertia, weight in propor- tion thereto, and even a certain kind of structure. Consequently, radiation represents electromagnetic momentum and may be properly likened to a ma- terial projectile. The pressure of radiation ceases to be incomprehensible and to give rise to the vexa- tious metaphysical problem of the action of the imponderable on the ponderable.
Not only does the inertia of energy, which as late
INTRODUCTION Xlll
as 1913 Brillouin1 described as a " paradoxical fancy/7 relieve us from the drawbacks of such a problem, but it satisfies our inherent need of uni- fication by absorbing the principle of the conserva- tion of mass into the more general one of the conservation of energy and by extending the law of Newtonian attraction to radiant energy. More- over the fundamental formula E = mV2 leads to the following important result: it permits the evalua- tion of the internal energy of a gram of matter at the absolute zero for an observer at rest, an energy which is no less than that of 3 million kilo- grams of coal.
The weight of energy, which, according to the experiments of Eotvos, is proportional to its inertia, furnishes, as Langevin has shown, a natural inter- pretation of the discrepancies in Prout's law of atomic weights. It is the origin of the theory of gravitation of Einstein based on the generalized principle of relativity. The practical interest of this theory is that it leads to a successful calculation of the secular anomaly of the perihelion of Mercury, and to the correct interpretation of the shift of Fraunhofer's lines in the solar spectrum as compared with those from a terrestrial source, observed by Fabry and Buisson. But its essential advantage lies in the fact that it excludes from the domain of physics those metaphysical entities that are still encumbering it, such as the absolute space and the
1 Brillouin, Propos sceptiques au sujet du principe de relativite (Scientia, Janvier 1913, p. 23).
XIV INTRODUCTION
privileged axes of Newton and their successors, the body alpha of Neumann and the stationary ether of the opticians. Psycho-physiology teaches us with respect to this point that our senses perceive merely relative variations in the external world without ever detecting an absolute change. Ein- stein has reconciled natural philosophy with the requirements of epistemology by showing that it is possible to put the equations of physics into a form which is generally invariant with respect to all changes of coordinate axes. By making use of the absolute differential calculus he has shown that it is thus possible to substitute for coordinate equa- tions intrinsic equations expressed in terms of tensor equalities in order to have in the terminology of the physical laws the magnitudes characteristic of the gravitational field; or, putting it more exactly, to regard the physical laws as relations between the quantities characteristic of the gravitational field and the quantities characteristic of the special phenomena to be studied.
Lastly, free radiation seems to possess structure. The law of energy distribution in the spectrum of the black body and the study of the specific heats of solids at low temperatures lead to the belief that the energy exchanges between material systems by means of radiation take place in sudden jumps, according to integral multiples of elementary in- divisible quantities, veritable energy atoms, called quanta. The discontinuity of emission and ab- sorption of radiation paves the way for the idea of a
INTRODUCTION XV
discontinuous distribution of radiant energy in the front of transverse light waves. This follows necessarily from the interpretation of certain phe- nomena such as the photo-electric effect and the production of rays by the impulse of secondary cathode rays or X-rays. Consequently radiation appears no longer to be a form of energy propagated after the manner of continuous1 waves through a hypothetical stationary medium, Maxwell's dielec- tric ether, but as expelled into space in the form of discrete units with a uniform velocity in a constant gravity field. The ancient dualism of the ponder- able and the imponderable, of matter and energy, becomes transformed into one of energy stabilized in material structures of definite architecture and free radiation, both of these modes of energy being endowed equally with inertia, weight in proportion thereto, and structure. This may then be called the materialization of energy.
Philosophy and the New Physics
CHAPTER I THE DUALISM OF MATTER AND ENERGY
1. INTRODUCTION.
It is a general truth that the majority of philo- sophical problems are insoluble because the prob- lems do not properly exist. The subjectivism of our senses, the anthropomorphism of our reason- ing by analogy, the substantialistic tendency to realize our ideas and to take purely logical dis- tinctions as objects lead us to conceive fictitious problems, or pseudo-problems, that have no more meaning than the insolubilia on which the eristics of the ancient sophists or the forensic dialectics of the theological schools of the middle ages were exercised. To solve them is always to show that they were problems which have been badly stated.
One category of these pseudo-problems is derived from the mental transformation of a simple analogy into an absolute identity, or of a partial difference into a perfect contrast. Having obtained certain dissimilarities between two classes of phenomena, we deliberately deny to those of one class everything that is shown to belong to those of the other; and in this perfect antithesis the mind derives the satis- faction of symmetry. We hold this radical line of
PHILOSOPHY AND THE NEW PHYSICS
tb l:be the equivalent of reality. If we begin to reflect on the fact that these diametric- ally opposed phenomena show mutual interaction, then the shock of this possibility outrages our understanding; how can absolutely heterogeneous agents be combined, and how can they influence each other? For example, having supposed as a fact that matter alone has mass, proportional weight, and shape; and that force or energy pos- sesses no inertia, no weight, and no structure, then the following problem inevitably arises : How can an imponderable agent be applied to and act on a ponderable one, with mass and inertia, to the point of deforming it or imparting an impulse to it? In accordance with the adage of Leibniz, "causam aequat effectus" our mind refuses to conceive this and our imagination declares itself incapable of visualizing it. Of the same nature and issuing from the same origin is the problem of the relations between mind and body.
The solutions proposed for such problems ine- vitably reiterate the following theme. In a dog- matic period it is declared that one of the two terms of the antithesis exists only apparently, the other one alone being reality; one tries to reduce the real forces either to thrusts of elementary masses or to fictitious forces of connection and inertia, or, in- versely, one sacrifices the notion of material mass in order to avoid contemplating anything but force centers that attract or repel in a straight line, according to a certain power of the distance. In
INTRODUCTION 6
supporting such reductions one perceives that in- surmountable difficulties which merely displace the original antinomy are encountered. The dogmatic period is followed by the prudent agnosticism, of a positive period. It is maintained that science is merely descriptive, and one is content to express the laws that govern the relations of the two classes of agents without seeking to penetrate into their nature or the hidden mechanism of their interaction; this is, in the case in which we are interested, the period of Ostwald's energetics. Finally an experimental and critical period arrives; in it the claims of the two terms under discussion to be placed in antithesis are examined, and it is then discovered that the latter is not well founded. On the contrary, it is shown to be true that the two terms, taken to be diametrically opposite, enjoy such properties in common as explain their interaction; energy appears to be en- dowed with inertia, weight, and structure, like matter. The profound reason for the lack of success of the preceding attempts at reduction is accounted for, and the metaphysical problem, thus removed unexpectedly, vanishes of itself.
2. THE DUALISTIC THEORY.
A superficial view of the external world leads to a classification of the agents found in it into two quite distinct categories. In the first place, there are ponderable bodies, endowed with mass, weight, and structure, the aggregate of which constitutes matter; in the second place, there are imponderable forces,
4 PHILOSOPHY AND THE NEW PHYSICS
divested of inertia, weight, and structure, the aggre- gate of which constitutes energy. Take a piece of matter and make it undergo all the possible physical and chemical transformations, such as motions; sub- divisions and recompositions; expansions and com- pressions; electrification; magnetization; changes of state; and chemical combinations. There exists an invariant with respect to this group of physico- chemical transformations; that is, a certain coef- ficient characteristic of the individuality of this piece of matter which remains invariable and un- changed: this is its mass. The mass, therefore, serves for measuring a portion of matter so well that matter and mass appear to be synonyms. Lavoisier established the indestructibility of matter by showing, by means of accurate weighing, the conservation of mass. A certain amount of energy is, because of the principle that different forms of energy are equivalent, measured by the amount of mechanical work into which it may be converted; and experiment proves that, in an isolated system, the amount of energy is constant. This is summar- ized by a classical passage of Robert Mayer:1 " Nature presents two categories of agents between which experiment shows an insurmountable barrier to exist. The first category comprises agents havi ng the properties of being ponderable and impenetrable : these are forms of matter; the second comprises agents lacking these properties: these are forces
1 Robert Mayer, Annalen der Pharmacie und Chemie, von Lie- big und Wohler, 1842.
DUALISTIC THEORY 5
called imponderables on account of the negative property which characterizes them. Forces are in- destructible, variable and imponderable objects."
Matter and energy being equally indestructible and radically heterogeneous, neither can grow at the expense of the other or decrease to its gain. Matter must return the energy to an amount equiva- lent to what has been given to it in another form. It serves to store it up like a sponge that has been saturated with water and is then in turn pressed dry; but, in contrast with the analogous case, the presence of energy in matter does not increase its mass since energy is imponderable. Matter is the natural receptacle of energy, which does not exist independently of it. Mass of ponderable bodies and imponderable energy are conserved, each on its own account like two distinct worlds, one super- imposed on the other, which completely penetrate without knowing each other and without consenting to reciprocal exchanges. This is what a recent author expresses as follows:1
"The world where we live is in reality a double world, or rather it consists of two distinct worlds, one of which is the world of ^matter, the other the world of energy. Copper, iron, carbon, that is the world of matter. Mechanical work, heat, these are forms of energy. Each of these two worlds is governed by a law of conservation. Matter can not be created nor destroyed ; energy can not be created nor destroyed."
1 Paul Janet, Legons d'Electricite, 2 Edition, pp. 2 and 5.
6 PHILOSOPHY AND THE NEW PHYSICS
" Matter or energy may appear in a large number of forms, without matter ever changing into energy or energy into matter."
"We can no more conceive of energy without matter than of matter without energy."
This dualistic doctrine of the universe rests on the following principle :
Matter alone is endowed with mass, weight, and structure; energy has no mass, no weight, and no structure.
This principle is supposed to be founded on the following propositions which we shall hereafter call the postulates of the theory:
1. Energy is never localized outside of matter.
2. The presence of energy in a body does not increase the inertia of the latter.
3. The presence of kinetic inertia in particular, that is, the state of motion of a body, does not increase its mass.
4. Absorption or emission of energy by radiation neither increases nor diminishes the mass of a body.
5. The mass of bodies being indestructible, the principle of the conservation of mass is distinct from the principle of^the conservation of energy.
3. THE DIFFICULTIES OF THE DUALISTIC THEORY AND THE CHECK GIVEN TO THE MONISTIC ATTEMPTS AT REDUCTION.
The fundamental dualism of matter and energy leads to insurmountable difficulties. How can im- material forces act on inert bodies so as to move
MONISTIC ATTEMPTS
them and verify the ancient formula mens agitat moleml How can luminous radiation if it is devoid of mass, exert a repulsion in one direction on the material source emitting it, and a propulsion on an opaque body absorbing it, as if it represented an amount of momentum; how, in other words, can it, if it is destitute of inertia, behave like a material projectile, which exerts a recoil pressure on the fire-arm which throws it and a ballistic action on the obstacle which it strikes? And if energy pos- sesses some inertia, how can it accumulate like electric energy on a body to a considerable amount, without the mass of the latter being found to have grown in some manner? In the presence of such an antinomy our understanding feels outraged and our imagination declares itself inadequate.
Furthermore there are the attempts at reduction, by the monists, coming one after another and seek- ing to resolve this initial contradiction by reducing one of the two terms of the antithesis to a mere semblance of the other. But Stallo, Hannequin, Duhem, and Meyerson1 have shown that in these attempts the difficulty is simply shifted.
The first attempt, in time, is that of the atomists, whose fate, from Leucippus and Democritus to Huy- gens and the Bernouilli brothers, is especially note- worthy. It consists in reducing the supposedly
1 Stallo, The concepts of modern physics. Hannequin, Essai critique sur 1'hypothese des atomes dans la science contem- poraine. — Duhem, L' Evolution de la Mecanique. — Meyerson, Identite et Realite.
8 PHILOSOPHY AND THE NEW PHYSICS
occult notion of force to the clear and distinct one of mass; and for this purpose seeks to give account of all phenomena by the motion of elemen- tary masses, indivisible and indeformable, impinging on one another. But if the atoms are rigid, trans- mission of motion through impact is impossible; if they are elastic, they are then deformable and com- posed of parts, which is contrary to the hypothesis and implies forces of cohesion and elasticity; this inevitable dilemma leads to shipwreck. Moreover Hertz,1 in his posthumous mechanics, aims at ex- plaining everything, not by atoms, but by articulated systems, by masses subject to firm bonds which must conform to a single law : every isolated system traverses with constant speed a trajectory of least curvature. What we take as real forces are fic- titious forces of connection, due to the presence of bodies that we do not perceive, or fictitious forces of inertia, arising from a motion that we do not suspect. For example, anyone who, pulling a body tied to another by an invisible cord, saw the second move forward, would believe in a mutual at- traction of the two bodies, while it would be the case of a force of connection produced by a hidden mass; and anyone who, not knowing the motion of rotation of a gyroscope, encountered an active re- sistance to an attempt to produce a deviation of its axis, would believe that a real couple tended to
1 Hertz, Die Principien der Mechanik in neuem Zusammenhang dargestellt (Ges. W. Vol. Ill) Johann Ambrosius Earth, Leipzig 1894.
MONISTIC ATTEMPTS 9
maintain this axis in an invariable direction, while it would be a case of a force of inertia produced by a hidden motion.
Thus "what we are accustomed/' says Hertz, "to denote by the names of force and energy is nothing more than the action of mass or motion." As, moreover, nothing limits the motions and the hidden .masses that may thus be introduced, it seems as if it were always possible to give an account of the behaviour of natural phenomena in this man- ner. But this very elasticity is in itself the cause of sterility. In spite of all the interest which is at- tached to attempts of this kind, such as those of Maxwell on electricity, of Lord Kelvin on the gyro- static ether, and of Helmholtz on cyclical systems; the concept of Hertz has not led to a single positive result. Boltzmann1 has shown that, however great the number of masses and hidden motions one im- agines, and, however great, the consequent number of arbitrary variables to which one resorts, it is im- possible to represent the simplest phenomenon in a satisfactory manner. Later, we shall find the reason for this lack of power to be the incompati- bility of the form which the relativity principle imposes on physical laws, with the form of the equations of classical mechanics which govern the displacements of the elementary masses of the atomists and the articulated systems of Hertz.
The inverse reduction, of mass to force, has been
1 Boltzmann, Anfrage die Hertz' sche Mechanik betreffend (Wiedemann's Annalen, Suppl. 1889).
10 PHILOSOPHY AND THE NEW PHYSICS
the work of Boscovich, following Kant. The atoms lose all material existence, and all spatial meaning; they are nothing more than dynamical points, or force centers. These forces are directed in the straight line joining them, whence their name of central forces. They are transformed in the nick of time from repelling to attracting and their in- tensity depends on a certain power of the distance. But if these force centers are mathematical points in space, how can we imagine that a force can apply itself to them in order to repel them or attach itself to them in order to attract them? How can these entities of thought resist motion, and exhibit iner- tia? "No arrangement of centers of force," de- clares Maxwell,1 " however complicated, can account for this fact; no part of this mass can be due to the existence of the supposed centers of force." It seems, moreover, that certain phenomena, such as crystallization and permanent deformations, can not be explained if one is limited to purely central forces.
Lastly Lord Kelvin's2 concept of a gyrostatic non-dynamical ether, in supressing the rebellious concepts of force and mass, has not thereby been more successful. Reviving the ideas of Descartes, Lord Kelvin contemplates a perfect fluid, homogene- ous and incompressible, which fills all space. In this fluid there exist eddy rings, which are eternal
1 Maxwell, Theory of Heat, p. 86.
2 Cf. Tait, Lectures on Some Recent Advances in Physical Science.
MONISTIC ATTEMPTS 11
and can neither be cut nor penetrated and which play the part of vortex atoms. Forces are due to pressure of the medium between these vortices. But, as Maxwell1 remarks, one gets no sight of the invariable element which one would agree to con- sider as the mass of the atom. He contemplates therefore a pure motion in pure space without moving object, kinetic energy being half of the product of the square of a velocity by a zero mass, which is absurd.
Thus the mechanistic theory of the atomists, the kinematic theory of Hertz, the dynamical theory of Boscovich, and the non-dynamical theory of Lord Kelvin, have conclusively failed in their attempt to reduce either force to mass or mass to force. The efforts of this youthful dogmatism are followed by the prudent attitude of reserve of a school of physics that takes no interest in a me- chanism explaining the phenomena, but describes simply, in its equations, the relations which connect the simultaneous variations of directly measurable physical quantities; a kind of physics in which there are many integral quantities, but no atoms.
Rankine and Ostwald have both remarked that these directly measurable physical quantities are always quantities of energy. "To establish such relations between measurable quantities that, some of these quantities being given, the others may be deduced, is the entire task of science. Hereafter there is no need of troubling ourselves about forces,
1 Maxwell, Art. "Atom" in Encyclopedia Britannica.
12 PHILOSOPHY AND THE NEW PHYSICS .
the existence of which we cannot prove, exerted between atoms that we know nothing of, but we are concerned with quantities of energy put into play by the phenomenon under investigation . . . All the equations that connect one phenomenon with one or several of a different kind are neces- sarily equations between quantities of energy; there can be no others, for, apart from time and space, energy is the only quantity common to all kinds of phenomena."1
4. OSTWALD'S ENERGETICS.
From OstwakTs energetics, the genesis of which we have just traced, we may accept the following two propositions:
(1) Of the external reality we know only changes of energy and all physical phenomena may be described in terms of energy.
(2) In particular a body is only a 'complex of indissolubly associated energies, so that the concept of matter becomes included in the more general one of energy, and the principle of conservation of mass is absorbed by the more universal one of conservation of energy.
The first proposition is easy to justify. In the first place by psycho-physiology: The apparatus of our senses is set into action only by energy changes between it and the external world. The energy of the physico-chemical agents that im-
1 Ostwald, Zur modernen Energetik, (Scientia, 1907, number 1).
ENERGETICS 13
presses it is transformed into nervous energy, which itself is transformed, in the cortical centers, into psychic energy sui generis. "If you receive a blow with a stick," says Ostwald1 jocularly, "what do you feel, the stick or its energy?" The physical phenomena appear, on the other hand, as reducible to a condition of rest, transfer or transformation of energies of different forms: kinetic energy; potential energy of position; radiant energy; electric energy; magnetic energy; heat energy; and chemical energy. Every form of energy is the product of two factors, an intensity factor and a quantity factor, a variable of equilibrium and a variable of condition: kinetic energy is half the product of mass and the square of velocity; gravitational energy is the product of height and weight ; volume energy, of pressure and volume; shape energy or elasticity, of force and displacement; electric energy, of potential and charge, or of elctromotive force and quantity; heat energy, of temperature and entropy; chemical energy of thermodynamic po- tential and the masses of the constituents.
The idea of a body becomes reduced to that of an energy complex: volume energy which causes a body to occupy a definite region of space; energy of motion which causes it to possess a definite capacity for kinetic energy or mass; gravitational energy which causes it to have weight. It is neces- sary to add to these three fundamental forms of
1 W. Ostwald, Die Ubcrwindung des wissenschaftlichen Mate- rialismus. Zeitschr. f. phys. Chem,, vol. 18, p. 305, 1895,
14 PHILOSOPHY AND THE NEW PHYSICS
energy another which is essential for solid bodies, namely, energy of shape or elasticity, by virtue of which solids resist agents that tend to penetrate or deform them. Every body possesses, in addition to these energies, others in varying proportion : chemi- cal, heat, electric, and magnetic energy. If we ask why energies of volume, motion, and gravity appear always associated in material systems, the answer is found to be in the very conditions of the external perception. If any one of these three energies were lacking in a body, the latter would escape our notice. A body not possessed of volume energy would not occupy space, it would be imperceptible; in the case of one not possessed of mass, an infinitely small im- pulse would give it an infinitely large speed but it would again be imperceptible; if not possessed of weight it would leave the earth and escape our ob- servation. The objects that constitute the world of our sense experiences must therefore necessarily possess these three indissolubly associated forms of" energy, the complex of which expresses all the positive contents of the idea of matter.
Two general principles govern the states of rest, transfer, and transformation of energy. They are the principle of conservation of energy and the principle of degradation of energy.
The first rests on the discovery of the mechanical equivalent of heat and its generalization for all forms of energy. It states that in an isolated system energy cannot be destroyed or created but can merely pass from one form into an equivalent
INSUFFICIENCY OF ENERGETICS 15
amount of another form. The second enunciates that all forms of energy have a tendency to be con- verted into heat and all temperatures to become equal, and may be also formulated thus: "In a closed system the entropy always increases." Boltzmann, by reducing the entropy to the logar- ithm of the probability of a certain state, has shown that this principle is a law of large numbers which expresses, for very complex systems, the chance of the most probable states being realized. In the following we shall have to refer to the principle of conservation of energy alone.
5. THE INSUFFICIENCY OF OSTWALD'S ENERGETICS AND THE EXPERIMENTAL DISCOVERY OF THE IN- ERTIA OF ENERGY.
In proposing his energetics as a method of exposition, a gnosiological theory, Ostwald is un- assailable. He is merely applying to physical questions the positive method, as formulated by Auguste Comte, who was gratified to see it applied in the work of Fourier on heat ; that is, he establishes the equations that connect the simultaneous varia- tions of measurable physical quantities, without pre- occupying himself with the question whether they can be reduced to each other qualitatively, accord- ing to their real nature. However, by excluding alt explanatory theories, Ostwald does not solve, but eludes, the problem of knowing how an imponderable agent by applying itself to a body with mass and inertia can impart to it an acceleration or determine
16 PHILOSOPHY AND THE NEW PHYSICS
a deformation in it. He condemns us to an intellec- tual asceticism, a position which the human mind is not satisfied to take; and that too at the precise instant when the success of the hypothesis of elec- trolytic ions, advanced by Clausius and Svante Arrhenius, rehabilitated the more concrete theories, by making it possible, through the efforts of the latter, van't Hoff, andOstwald himself, to group into a coherent body of doctrine the aggregate of electro- lytic, osmotic and chemical properties of aqueous solutions. Moreover, physicists have come by a different path to doubt the traditional antithesis between matter and energy and to establish by experiment the inertia of the latter.
The experiments of Faraday have demonstrated the localization of electric and magnetic energy outside of conductors and magnets, contrary to the first postulate of the dualistic doctrine. The self- induction of electric currents in the conductors has revealed to them the existence of a veritable electromagnetic inertia. The discovery of con- vection currents, predicted by Maxwell, realized by Rowland, has led them to foresee that, on the inertia proper of a charged particle in motion, there must be superimposed a supplementary inertia of electromagnetic origin, contrary to the second postulate of the dualistic doctrine. This inertia, for velocities exceeding 30,000 km. per sec., varies with the velocity. The cathode particles issuing from Crookes' tubes and the /3 rays from radium, issuing from the disintegration of the atom, have velocities
INERTIA OF ENERGY 17
of just this order of magnitude. Experimental study of the variations of the electromagnetic inertia as a function of the velocity becomes here- after possible. Undertaken by Kaufmann and Bucherer, it shows the existence of grains of resinous (negative) electricity, or electrons, destitute of material support, the mass of which is solely of electromagnetic origin and varies as a function of velocity. Hence we have here a form of energy, electric energy, that is endowed with mass and struc- ture, contrary to the fundamental principle of the dualistic doctrine. The relativity principle, which results from the failure of all attempts to demonstrate the absolute movement of a system by experience within this system, then intervenes to show that all bodies behave during translation as if their mass were solely electro-magnetic, that is, as if it were no more an invariable scalar quantity, but a tensor quantity, symmetrical and variable, contrary to the third postulate of the dualistic theory.
Absorption and emission of radiant energy by bodies and the pressure of radiation resulting there- from, in accord with the theoretical views of Maxwell and Bartoli and the experimental results of Lebedew, lead, on the other hand, to endowing electromagne- tic radiation with momentum, in order to safeguard the relativity principle and Newton's principle of action and reaction or its corollary, that of the conservation of momentum in an isolated system. But momentum implies moving mass. Radiant energy, therefore, has mass; and all other energy,
18 PHILOSOPHY AND THE NEW PHYSICS
no matter what its form, being convertible* into energy of radiation, possesses a coefficient of inertia. The mass of a material system changes through absorption or radiation of energy, since, in the first case, it is increased by all the inertia of the absorbed radiation, and, in the second case, diminishes by the amount appertaining to the radiation emitted. From the formulae it follows that, if we take the velocity of light as unity, the inertia of a material system is the measure of its internal energy. The mass ceasing to be invariable, the principle of the conservation of mass becomes merged in the princi- ple of the conservation of energy, as previously con- ceived by Ostwald, and contrary to the last postu- late of the dualistic theory.
Matter and energy are endowed equally with inertia and, according to the experiments of Eotvos on the proportionality of inert mass and gravita- tional mass, with weight in proportion thereto. There still persists the opposition between matter, characterized by its structure (the number and na- ture of the electrons, and possibly also the positive remainders, of which it is composed) and the pos- sibility of moving with velocities varying from zero to that of light, and radiation which moves uni- formly with the speed of light. This opposition becomes weaker because of the quantum theory. This theory shows that radiant energy is not uni- formly propagated in a hypothetical medium, but is emitted into space in the form of discrete units.
The metaphysical problem of the action of
INERTIA OF ENERGY 19
an imponderable on a ponderable, of force on mat- ter, disappears as a pseudo-problem, born of an artificial antithesis between matter and energy assumed at the outset. On account of the inertia of the latter, there is no more difficulty in under- standing, for example, the pressure on the matter absorbing it, since, being endowed with momentum, light behaves strictly like a material projectile striking an obstacle.
Thus, as already stated, matter was errone- ously opposed to energy, by denying to the latter everything that had been found to belong to the former. This led to a fundamental antinomy which gave rise to a pseudo-problem. The monistic at- tempts to reduce one of the terms of the antithesis to the other have merely served to shift the diffi- culty without relieving it, and Ostwald's energetics merely to escape the problem without solving it. The recent discoveries, however, by opening up a field of experiment of an extent beyond all expecta- tion, such as the possibility of having at our dis- posal velocities approaching that of light, have led physicists to revise the primitive classification of natural agents into ponderables and imponder- ables. It has been discovered that this Classification was not well founded, that the admitted anti- thesis between matter and energy was artificial, and that these two realities, in spite of differences which forbid their confusion, possess precisely the common characteristics that permit the explanation of their mutual action.
20 PHILOSOPHY AND THE NEW PHYSICS
This result, essential for natural philosophy, is the end of a long line of theoretical reasoning and experimental verification, which unite conclu- sively to establish the following proposition: Energy is endowed with inertia. In order to understand the inner logic of this wonderful organic de- velopment, it must be followed without haste, step by step. The physicists have not found that royal road leading directly to the goal by which Ptolemy I asked Euclid to conduct him to the comprehension of the most transcendent scientific truths. They have not abandoned the old principles under pres- sure of the new facts until they had tried all possible ways of saving them. They have not risked one step forward without summoning an advocatus diaboli against themselves. They have been pru- dent in their innovations. Old mental habits, stabilized through long ages of scholasticism, and the narrow evidences offered by a traditional ra- tionalism are not got rid of in a day; nor does one easily leave the beaten track of an imagination which derives satisfaction only from the contempla- tion of solids. In particular, the consideration of Maxwell's dielectric ether, which comes under the category of Hertz's hidden masses and movements, has for a long time, by misleading physicists into attempts at mechanical explanations, paralyzed efforts, of which the relativity principle finally had to show the futility. Leaving useless details aside, we shall follow in its essential curves the turnings of the winding road which has led the physicists of
INERTIA OF ENERGY 21
the new school to a conception of the universe, the beauty of which is a joy forever. To arrive there more easily we shall at the outset recall a certain number of ideas and principles which will be con- stantly used in the sequel.
CHAPTER II MASS AND THE RELATIVITY PRINCIPLE
6. THE IDEA OF MASS; EINSTEIN 's EQUIVALENCE PRINCIPLE AND NEWTON 's PRINCIPLE OF ACTION AND REACTION.
In order to show how the idea of energy as en- dowed with mass and weight in proportion thereto has been arrived at, it is necessary to define these two ideas and to state two principles which are closely associated with them.
What constitutes the individuality of a piece of matter through all motions, divisions, recomposi- tions, compressions, expansions, changes of state and chemical combinations that it undergoes, is its mass, which presents itself as an invariant of the group of physico-chemical changes.1 It is considered in classical mechanics as an invariable scalar quantity that characterizes every piece of matter and may be defined in three different ways: (1) as a coefficient of inertia; (2) as capacity for momentum; (3) as capa- city for kinetic energy. * ^
As coefficient of inertia, mass measures the re- sistance of a body to any action tending to modify its state of motion. Newton assumes the propor-
1 Cf. F. Enriques, Les Concepts fondamentaux de la Science, p. 136-144.
22
IDEA OF MASS 23
tionality of the force acting on a body to the change in velocity per unit time communicated to it, or the acceleration; therefore the quotient of force by acceleration defines the mass of a body. Moreover, he assumes the principle of the independence of the effects of a force, which leads to the view that the mass of a moving body is independent of the veloc- ity acquired : if a force acts on a body for a second, starting from rest, and communicates to it the veloc- ity v, the same force acting through another second will communicate to it a second increment of veloc- ity equal to the first, so that its velocity will be- come 2v} if the same force continues to act through a third second, the velocity will become 3v and so on. Theoretically a moving body may be given a veloc- ity as large as we please by the application of the same force a sufficiently large number of times. Lastly the mass of a body is independent of the sense in which the force acts, whether it be parallel to the direction of the velocity (tangential accelera- tion) or at right angles thereto (normal acceleration) . Mass may also be defined, as was done by Mauper- tuis, by starting with the idea of an impulse com- municated to a body by a force during the time element dt. This quantity is determined in mag- nitude and direction by the product f-dt; and the impulse of a body at the instant t is the geometric sum of the elementary impulses that, starting from rest, have been successively communicated to it by the various forces exerted on it. In virtue of the law of inertia, matter tends to conserve its state of
24 PHILOSOPHY AND THE NEW PHYSICS
motion; the impulse is equal to the momentum of a body, or the product of its mass by its velocity taken in the same direction as the latter; hence the vector relation
g = mv.
Mass may therefore be defined, as capacity for momentum, by the impulse divided by the velocity according to the relation
Lastly, the kinetic energy of a body may be de- fined by the total work which had to be expended to bring the body from rest into its actual state of motion without deforming it for observers moving with it. This leads in rational mechanics to the relation
w = ^mv2.
Starting from this, mass can be defined, as the capacity coefficient of kinetic energy, by the quotient of twice the kinetic energy and the square of the velocity
2w ™=^
These three definitions are equivalent in classical mechanics and we shall have to refer to each of them in succession. But we shall see that in the new dynamics, the dynamics of the electron and of relativity, they cease to be identical for velocities above 30,000 km. per sec. All three lead to variable values, functions of the velocity, following three
IDEA OF MASS 25
different laws which make them assume an infinitely large value for the limiting speed of light in a vacuum. The following leads to a rejection of Newton's principle of the independence of the effects of a force. If a force acts for another second its effect will be less than that produced during the first; it will be still less during the third and less in general as the velocity already acquired by the body becomes greater, since the inertia increases step by step, until it becomes infinite for the speed of light. Lastly in the new dynamics the acceleration communicated to a body by a force depends on the angle made by the direction of this force with that of the velocity. We must define a tangential mass and a transverse mass, identical with that defined by Maupertuis, which do not behave symmetrically. Briefly, mass ceases to be a constant scalar quantity and becomes a tensor quantity, which is variable and unsymmetrical.1 The idea of mass designates two quite different things: (1) the coefficient of inertia of a body which measures its resistance to acceleration, or inert mass', (2) the coefficient of attraction of a body on another outside of it, according to Newton's law, or gravitational mass. How do these two masses behave with respect to each other? Newton sup- posed that there was a strict proportionality be- tween the two coefficients; but that was not proved
1 On the subject of symmetry of physical quantities see L. Rougier, La symetrie des phcnomenes physiques et le principe de raison suffisante (Rev. de Met. et Mor. mars 1917, p. 165-198).
26 PHILOSOPHY AND THE NEW PHYSICS
experimentally before Eotvos1 experiment with a torsion balance. On the surface of our globe all bodies are immersed in two superimposed fields of force: the field of gravity, which puts into play the gravitational mass of bodies, and the field of the centrifugal force, which puts into play their inert mass. The resultant of these two actions combined is the apparent weight of a body. If the two masses were different, the resultant, that is, the observed direction of vertical, would not be exactly the same for all bodies: there would not be a single vertical at one and the same point. That this is not so has been established by Eotvos by the aid of a torsion balance with an exactness which excludes relative divergencies of the order 10~7? so that the exact coincidence of the two co- efficients may be taken for granted.
Einstein2 has from this deduced the principle of the equivalence of acceleration and gravitation; the effects produced on any system by a gravitational field are not distinguishable from effects produced by a suitable state of acceleration of the same system when removed from the action of the field, or, what comes to the same thing, from those of a system referred to axes having an accelerated motion with respect to it. In order to learn the action of
1 B. Eotvos, Math. u. naturw. Ber. aus Ungarn, Vol. VIII, 1890; Beibl, vol. XV, 1891, p. 688.
2 Einstein, Ann. der Phys., 1911 p. 898 and seq.; 1912, p. 365 and seq.; 1914/p. 341 and seq.; Zeitschr. f. Math. u. Phys., 1913, p. 6 and seq.; Phys. Zeitschr., 1914, p. 176 and seq.; Archives des Sciences phys. et nat., 1914, p. 7-12.
ACTION AND REACTION 27
gravity on physical phenomena it will suffice to study the effects of acceleration on them, that is, to examine the modifications that the physical phe- nomena undergo when passing from a reference sys- tem at rest, or in uniform translatory motion, to one accelerated with respect to the former. From this Einstein has made the deduction that if energy is inert it has weight in proportion. We can no longer imagine an agent like the ether having a certain density and yet absolutely imponderable, that is to say, devoid of weight. Everything that is endowed with mass must possess a certain coefficient of New- tonian attraction in proportion thereto.
The idea of mass is intimately connected with the Newtonian principle of instantaneous equality of action and reaction. Without attempting to fathom the meaning of this principle or seeking to find how it is connected with the definition of equality of masses,1 we shall restrict ourselves to its enuncia- tion. If a material point A acts on another B, the body B reacts on A, and these two actions are two instantaneously equal and oppositely directed forces. For example, the attraction exerted by the earth on the moon is at every instant equal to and oppositely directed to the attraction exerted by the moon on the earth. It follows from the principle of the equality of action and reaction that in the case of a closed system the momentum g — mv remains constant. If a body through an internal action calls
1 Cf. Mach, History of Mechanics, p. 210-216. Enriques, 1. c., p. 144-149.
28 PHILOSOPHY AND THE NEW PHYSICS
forth the appearance of a certain momentum, count- ed as positive, by acting on another body, it will undergo a repulsion on the part of the latter and consequently take a momentum equal to the first and directed in the opposite sense so as to com- pensate it. The center of gravity of the system will remain at rest. The principle of the conservation of momentum in a system which is subject to internal actions only presents itself therefore as a corrollary of the principle of action and reaction and would cease to be exact at the same time as the former.
7. THE RELATIVITY PRINCIPLE.
The relativity principle originated from the nega- tive result of all the attempts to show the ab- solute motion of a material system by experiments within that system.
Classical mechanics assumed that all mechan- ical phenomena in a system in uniform rectilinear motion are produced exactly as if the system were at rest. An observer enclosed in Jules Verne's bomb-shell could detect accelerations or rotations that this shell undergoes by observing phenomena taking place inside; he could in no way detect his uniform translatory motion. This is no longer the case if he has recourse to optical or electrical experi- ments. The electromagnetic theory introduces a medium at rest, the ether, which transmits the transverse waves of light with a definite speed, just as air transmits sound waves. In the case of a source of sound the relative motion of the source -with
RELATIVITY PRINCIPLE 29
respect to the air can be measured without revealing an absolute motion of the source with respect to the ether, since air is dragged along by the motion of bodies. This is not so in the case of the ether. The latter, if it exists, is incapable of motion, so that one might hope to show, by electromagnetic or opti- cal experiments within a system, the motion with respect to the ether of a source of light connected with the system.1
The earth possesses in its annual motion a trans- latory velocity varying constantly by amounts as high as 60 km. per sec. for the relative velocities cor- responding to two diametrically opposite positions in its orbit, in January and July, for example. Let us consider the optical experiment actually performed by Michelson in 1881 and repeated by him and Morley in 1887 the outline of which is as follows. Let three points O, P, and P' be taken at the vertices of an isosceles triangle with a right angle at O. OP corresponds to the direction of the translatory mo- tion of the earth, OP7 is at right angles thereto; at O a glass plate is fixed inclined at 45°; at the points P and P' mirrors are fastened normally to the two directions OP and OP'. Let us assume that a light ray comes from a lens on the line OP' produced, strikes the glass plate O and there is divided into two rays : one will be reflected towards the mirror P,
1 Cf. H. A. Lorentz, A. Einstein, H. Minkowski, Eine Samm- limg von Abhandlungen, Teubner 1913. Laue, Das Relativitats- princip, 1911. P. Langevin, Le temps, 1'espace, et la causalite" dans la Physique moderne (Bull. Soc. fr. de Philos. vol.. xii, 1912).
30 PHILOSOPHY AND THE NEW PHYSICS
return to O and passing through the glass plate will fall on the lens L; the other will pass through the plate O, be reflected from the mirror P' and be superimposed in the lens on the first ray. If the apparatus is at rest, the paths OPO and OP'O being equal by hypothesis, the times t and t' required to traverse them will also be equal, and the light dis- turbances brought by the two rays to the focus of the lens will coincide in such a manner as to give a maximum of light intensity at this point. The aspect of the interference fringes will, by reason of symmetry, remain the same for any rotation what- ever of the apparatus, the locus of the points P cor- responding to equal paths OP + PO being a sphere with its center at O.
If the system has a uniform translatory motion v, the locus of the points P corresponding to equal paths traversed by the light, is no longer a sphere about O but an ellipsoid of revolution flattened in the direction of motion, the axis of which in a direc- tion at right angles to the velocity is to that parallel
/ v* thereto in the proportion 1 : ij 1 — ™ v being the
velocity of the instrument, and V being that of light. If the apparatus remains undeformed while passing from rest to motion and if the instrument is adjusted at the outset by experiment, so that the paths OPO and OP'O are equal for a given initial position, the aspect of the interference fringes will be changed because a rotation through 90° will interchange the directions OP and OP' and this
RELATIVITY PRINCIPLE 31
change will reveal the absolute motion of the system.
For example, if the preceding experiment is made on the earth, in January, and the interchange of the directions OP and OP' does not bring about any change in the aspect of the interference fringes, we might interpret this negative result by saying that the velocity of the earth relative to the ether is relatively insignificant. Six months later it will have increased by 60 km. per sec. and the repetition of the experiment should bring about an appreciable change in the aspect of the fringes. Contrary to this anticipation the Michelson-Morley experiment gives a negative result, and so too do all optical and electromagnetic experiments undertaken by Lord Rayleigh and Brace, Trouton and Noble, Rankine and Trouton.
To account for this result, Fitz-Gerald and Lor- entz assumed simultaneously that all linear dimen- sions of bodies contract in the direction of their
translation in the ratio A| 1 — ™ written more
simply V 1 — |82. This contraction can not be ascertained by observers O, connected with the moving bodies, since all measuring instruments con- tract in the same proportion, but it can be by station- ary observers O0 who see it pass. Let us call the configuration of a moving body as measured by observers connected with it by means of measuring rods having the same translatory motion, its geo- metric configuration; and let us call its kinematic
32 PHILOSOPHY AND THE NEW PHYSICS
configuration the configuration of the same body measured by observers at rest with respect to it, who observe, at the same instant according to their clocks, during its passage, different points of its circumference, and determine the position of these different points by means of measuring rods grad- uated at rest; the kinematic configuration will not be identical with the geometric configuration , it will differ from it by a contraction in the ratio Vl — /32. In the reasoning that has just led us to an- ticipate a change in the aspect of the fringes as a result of the rotation of Michelson's apparatus, we have assumed the distances OP and OPf to be in- variable during this rotation. According to Lor- entz's hypothesis the distance OP', originally at right angles to the direction of motion, will, for observers 00 at rest who see the instrument pass, contract during the rotation and become OP = '
while inversely the distance OP, originally parallel to this direction will expand in the inverse ratio and become
OP' = 2L-
VI - P2'
so that the equality of the original times of travel ti and ~t2 will carry with it the equality of the later times ti and tz.
As a consequence of the Lorentz contraction spherical bodies become flattened ellipsoids in the ratio
RELATIVITY PRINCIPLE 33
We shall have to use this result in the dynamics of the electron.
The negative result of experiments undertaken to discover the absolute motion of a system with re- spect to the ether has led physicists to generalize the relativity principle assumed in rational mechan- ics for mechanical phenomena:
Given different groups of observers, 00 and 0\, one of which is in a state of uniform motion with respect to the other, the laws of nature will be exactly the same for the different groups of observers. This principle may also be stated thus: the form of the equations expressing the laws of phenomena within a system does not change if they are referred successively to a reference system at rest and then to a second system having a uniform translatory motion with respect to the former.
In order to see the meaning of the relativity principle thus extended to all natural phenomena in the case of uniform translatory motion, it will be best to compare it with the principle of relativity used in geometry and mechanics.
In Euclidean geometry we assume that the shape and the dimensions of a figure do not depend on its absolute position in space: they remain unchanged for all displacements that it undergoes and this is what constitutes the relativity of space. We may express this by saying that if the same figure be referred successively to different systems of axes 00 and 0i, the equations of transformation that permit us to pass from the first system to the second,
34 PHILOSOPHY AND THE NEW PHYSICS
form a group,1 called an Euclidean group, in which the form of the equations that translate the proper- ties of geometrical figures is an invariant. Geome- try is nothing but the study of the structure of the Euclidean group, which is identical with the group of orthogonal substitutions.
In rational mechanics we assume that the me- chanical phenomena taking place in an isolated system do not depend on its state of rest or of uniform translatory motion: this constitutes the relativity of motion. We may express this by saying that if we successively refer the same mechanical system to a reference system 00 at rest with respect to it, and then to a system 0\ in uniform translatory motion with respect to the former, the equations of transformation that permit us to pass from 00 to 0\ form a group, called a Galilean group, in which the form of the equations of classical mechanics is an invariant. In the simplest case we would have for such a transformation of the space coordinates of the first system with the axes x,y,z, into those of the second x', yf, zf
x = x' — vt
V = y'y
z = z',
where v is the velocity of translation of 00 with respect to 0\.
1 Given an aggregate of transformations we say they form a group, if the product of any two of these transformations and the inverse transformation of any transformation still form part of this aggregate.
RELATIVITY PRINCIPLE 35
We have just seen that in physics all phenomena whatever taking place within a system are independ- ent of its state of rest or of uniform translatory motion, and this constitutes the relativity of all natural phenomena. This may be expressed by saying that if we refer successively the same phys- ical system to a reference system 00 at rest with respect to it and then to a system Oi in uniform translatory motion with respect to the first, the equations of transformation that permit us to pass from 00 to Oi form a group, called a Lorentz group. Lorentz's equations of electromagnetism are an invariant of this group. In the simplest case of a translatory motion of the reference system 0± parallel to the rr-axis, the transformation of space and time coordinates of the first system with the axes x, y, z, t into the second x', y' ', 2', £', will be expressed by the four relations
x — vt v
x' =
where 0 = y, and where v represents the relative
velocity of Oi with respect to 00 and V the velocity of light, which is the same in all directions.
Minkowski has given a very elegant geometric interpretation of the principles of relativity in a four-dimensional space. He substitutes in place of the Galilean reference systems formed by three axes at right angles in the Euclidean three-dimen- sional space, much more general systems formed by
36 PHILOSOPHY AND THE NEW PHYSICS
four axes at right angles in four dimensional space, in which time plays the part of a fourth imaginary dimension. Because time and the three space co- ordinates are thus strictly assimilated, Lorentz's transformation assumes a strictly symmetrical form. In fact, if the four coordinates are
Xi = x, Xi = y, xz = z, x 4= ict
where i denotes V — 1 and c the velocity of light in a vacuum, Lorentz's transformation becomes
The axes being at right angles, the coefficients Apq satisfy the known relations
= 0, when q ^ r;
At = 1.
The Lorentz transformation corresponds, there- fore, to a displacement of the reference system of four axes (translation and rotation in the four-di- mensional space). It occurs spontaneously in na- ture whenever an electromagnetic system is moved by a translation as a whole with respect to immov- able axes. Hence the relativity principle may be stated as follows : physical laws retain the same form for all systems of rectangular axes in four-dimensional
RELATIVITY PRINCIPLE 37
space, or again, if we call these systems Lorentz systems : systems having Lorentz axes are equivalent.
It follows as a result of the structure of the Lorentz group that the concepts of space and time cease to be independent concepts; they are absorbed in the more general concept of the universe. The dura- tion of phenomena varies according as it is measured by observers relatively at rest or in motion. For every portion of matter there exists a local time, or time proper, which is the interval of time, measured by observers connected with the system, between . two successive events in it which for them coincide in space.
Simultaneity loses all absolute significance and becomes relative : two events in two different regions of space which are simultaneous for certain observers cease to be so for others in motion with respect to the former. Even the order of succession of two events, the distance between which in space exceeds the path traversed by light during their interval, may be reversed for certain groups of observers.
These new ideas of space and time lead to the erection of a new kinematics. In it the idea of a natural Euclidean solid no longer exists, since all bodies undergo the Lorentz contraction in the direc- tion of their translatory motion. If we still desire to talk about solids, it is necessary to substitute for Euclid's geometry that due to Lobatschewsky : the natural solids undergoing the Lorentz con- traction behave therefore like hyperbolic solids in a space of negative curvature, taking for the
38 PHILOSOPHY AND THE NEW PHYSICS
value of the space constant the velocity of light V = 3 X 1010 cm. per sec.1
A second consequence consists in the impos- sibility of observing objects moving with a velocity exceeding that of light. This follows from the Lorentz contraction : at this limiting velocity bodies would become infinitely flattened out and have an infinite inertia. , This may be derived also from the constancy of the velocity of propagation of light in all directions for any observers whatever as follows from Michelson's experiment. It follows from this experiment that a luminous ray takes the same time for the round trip OPO when it is propagated in the direction of the translatory motion of the earth, so that its velocity is combined with the velocity of that motion, and when it is propagated normally or obliquely to that direction, a fact that is explained by the existence of local time. Therefore, Galileo's law for the composition of velocities ceases to be true: two velocities v and v' in the same sense and the same direction may not be added by vector addition according to the class- ical relation v + vf, but must be added according to the new formula (v + v') / (1 + w'),* where the unit^of velocity is taken as the velocity of light: V = 1. All instantaneous action-at-a-distance be- comes thereby impossible.
It is just this impossibility of any instantaneous
1 L. Rougier, De 1'utilisation des ge'ome'tries non-euclidiennes dans la physique de la relativite" (L'Enseignement mathe'matique, 15 Janvier 1914, p. 5-18).
2 This expression is always less than 1. (Tr.)
RELATIVITY PRINCIPLE 39
action-at-a-distance that leads to momentum, and hence mass, being attributed to radiant energy, if it is desired to preserve Newton's principle of the simultaneous equality of action and reaction or the principle of conservation of the total momentum in the case of a closed system.
Classical mechanics is in no better accord with the relativity principle than Galilean kinematics. There is an incompatibility between the Galileo group that conserves without change the form of Lagrange's canonical equations, and the Lorentz group that conserves the form of Maxwell's equa- tions of the electromagnetic field. In order to put them into accord the coefficient of Lorentz
must be introduced in all the equations of dynamics.
It follows from this that all forces of nature must behave, on passing from one reference system to another, as if they were of electromagnetic origin.
In what follows we shall retain the following consequences of the relativity principle :
1. The existence of absolute motion has no physical meaning.
2. All bodies contract in the sense of their translatory motion in the ratio \/l — /32.
3. No velocity exceeding the limiting velocity of light in a vacuum can be observed; consequently, there is no instantaneous action-at-a-distance.
4. All forces behave as if they were of electro- magnetic origin.
40 PHILOSOPHY AND THE NEW PHYSICS
We shall retain one last consequence. The incompatibility of the Lorentz group and the Galileo group shows the futility of attempts starting from the equations of Lagrange and Hamilton to explain mechanically the phenomena of nature, and in particular electric and magnetic phenomena.
On the contrary, what must be done is to reduce Lagrange's equations of mechanics, equations that are correct as a first approximation for low veloc- ities, to Maxwell's equations of electromagnetism ; the problem is to explain mechanical phenomena on tbe basis of electric phenomena.
CHAPTER III ELECTROMAGNETIC DYNAMICS
8. THE LOCALIZATION OF ENERGY OUTSIDE OF MATTER.
The considerations that have led physicists to the idea of the inertia of 'energy are derived from Faraday's and Mosotti's views on the localization of electric energy outside of conductors in the dielectric space surrounding them.
It is known that the electrification of a body determines the appearance of attracting and re- pelling forces in the surrounding space, under the influence of which oppositely electrified bodies approach each other and similarly electrified bodies repel each other in accordance with Coulomb's law. Before Faraday, the appearance of these forces was explained by the statement that there exist two imponderable fluids, vitreous (positive) electricity and resinous (negative) electricity, to be found in the interior of electrified conductors, the quanti- ties of which constitute the charges of these con- ductors. These fluids have the property of acting instantaneously on each other at a distance and this explains the electrostatic actions. The region of space in which a small electrified body may be directly influenced by the presence of electric fluid,
41
42 PHILOSOPHY AND THE NEW PHYSICS
distributed inside of a neighboring conductor, is called the electric field of this conductor. The field is determined at every point of space by the magnitude, the direction and the sense1 of the attracting or repelling force that would be exerted on an electrified body brought there.
It is possible to draw through each point of an electric field a curve whose direction coincides with that of the force capable of thus acting at that point: an infinite set of lines called lines of electric force is thus obtained. These lines have a purely geometric significance : that is, the attracting and repelling forces, the direction of which they symbolize, do not exist unless an electrified body is actually brought to one of the points through which they pass. In this case only do they acquire a physical significance and that merely at the points considered. The intensity of the electric field of a conductor at a point is defined in magni- tude, sense and direction, by the magnitude, sense and direction of the force that acts at this point on the unit of electric mass conventionally chosen. If we call / the force exerted on the unit mass of electricity 77, the intensity of the field is represented by the relation
1A force acts in a certain line (direction) and in a certain sense (i.e., either forward or backward) along the line. In works written in English the word direction usually includes both ideas. (Tr.)
ENERGY OUTSIDE OF MATTER 43
This idea, which was the current one before Fara- day, rests on three postulates: (1) there exist im- ponderable fluids (the positive electric fluid, the negative electric fluid, the positive magnetic fluid, the negative magnetic fluid and the caloric fluid) the presence of which in a body neither increases nor decreases its mass; (2) these fluids are distrib- uted in the interior of bodies; (3) instantaneous action at a distance is possible.
Faraday rejects these postulates. By means of celebrated experiments he establishes the fact that the electric field is zero in the interior of conductors, that is, that their charge does not manifest itself except at their surface and in the surrounding space. He rejects the idea of instantaneous action-at-a- distance and considers the lines of force as having a real physical significance; they correspond, according to him, to certain permanent modifications of a dielectric medium, called the ether, filling all space and existing between the conductors. The mechan- ical state of this medium around an electrified body is what determines the forces to which this body is subject; and the electric charge carried by a con- ductor is determined by the number of lines or tubes of force attached to it. Starting from the concept of forces propagated by contact from point to point with the velocity of light, Maxwell has accounted for the electrostatic actions, in conformity with Faraday's views and in accord with his experiments.
It is possible to get rid of the consideration of the ether, the existence of which is hypothetical, and,
44 PHILOSOPHY AND THE NEW PHYSICS
as we shall see, contradictory, and contemplate nothing but the only positively accessible reality: electrostatic energy localized outside of the con- ductors in the form of the field. The mechanical work of attraction or repulsion, done by the static forces appearing in an electric field, represents a cer- tain expenditure of energy, that is, the change of a certain amount of potential energy into actual energy of motion. Thus potential energy localized in empty space around a conductor exists in con- formity with Faraday's experiments. This elec- trostatic potential energy comes, by virtue of the principle of the conservation of energy, from the work expended in producing the state of electrifica- tion of the conductor. The energy per unit volume is proportional to the square of the intensity of the corresponding field. Let dr be a volume element taken in this space and let h be the vector defining in magnitude, sense and direction the intensity of the field at a point A of this element ; then the density of the energy localized in the volume dr has the value
where KQ represents the specific inductive capacity of space, the numerical value of which depends on the choice of units. To find the energy localized in a portion of space of finite extent it is sufficient to calculate the integral
We =
in this space.
ENERGY OUTSIDE OF MATTER 45
Thus, for the empty space surrounding a con- ductor, Faraday's experiments and the principle of the conservation of energy lead to the assumption of the localisation of a certain amount of electric energy outside of matter.
The same considerations may be applied to a magnetic field produced by a magnet or a current. Similarly the energy localized in a region of empty space is equal to the integral
W. = I ri"- dr,
J ^^
where H denotes the vector defining the intensity of the field at a point A of the volume element dr and MO the permeability of empty space.
The same conclusions are reached by consider- ing radiation freely propagated in empty space with the velocity of light.
This radiation, if taken at a sufficiently large distance from its source, may be regarded as formed by the superposition of an electric and a magnetic field, one at right angles to the other and both normal to the direction of propagation.
This electromagnetic field represents a cer- tain amount of energy outside of any material sub- stratum, the energy density of which in every unit
Tf h 2 IT 2
volume that it occupies is equal to - - + ^ — ,
O7T O7I
the electric energy being equal to the magnetic en- ergy in every freely propagated plane wave, so as to give the relation
46 PHILOSOPHY AND THE NEW PHYSICS
The localisation of electromagnetic energy of free radiation in empty space follows from the principle of the conservation of energy, as. may be readily demonstrated. Let us consider a material source radiating energy in a definite direction and an obstacle absorbing it completely. At start there is a loss of energy of the source and on arrival a recovery of this energy by the obstacle. The velocity of propagation of radiation being finite because equal to that of light, we would not have conservation of energy at every instant between the time of emission and that of absorption, unless ra- diation represented a transfer of energy through empty space proportional to the energy radiated at the start and absorbed on arrival.
Thus the principle of the conservation of energy and Faraday's experiments lead us to assume the localisation of electric and magnetic energy in empty space, outside of matter, around charged conductors and natural or artificial magnets. Nev- ertheless electrostatic potential energy is, in the case of a charged conductor, still closely bound to matter, for the vector representation of the elec- trostatic field shows the lines or tubes of force to be attached to the surface of a conductor and to spread radially roundabout. In the case of an arti- ficial magnet formed by the magnetic field produced by a current flowing in a metallic circuit, the mag- netic lines of force do not abut on the circuit: they embrace it by returning on themselves; but they still proclaim its immediate neighborhood.
ELECTROMAGNETIC INERTIA 47
This is no longer so in the case of radiant energy, which manifests itself to our senses sometimes as light, sometimes as heat, sometimes in the shape of Hertzian waves. Radiation represents a localisa- tion, in empty space, of electromagnetic energy that is independent of the immediate neighborhood of any matter, and is capable of being propagated indefinitely in a given direction without ever meet- ing bodies which absorb it. Hence the first pos- tulate of the dualistic theory must be rejected, namely that energy can not be localized outside of matter.
9. ELECTROMAGNETIC INERTIA AND THE DYNAMICS OF AN ELECTRIFIED PARTICLE IN MOTION.
If the first postulate of the dualistic doctrine has been set at nought as a consequence of Faraday's experiments, its second postulate has, since 1881, been shaken as a consequence of the theoretical researches of J. J. Thomson.1 This young physicist had the merit of being the first one to understand that an electrified body possesses, because of the electrostatic energy of its charge, a supplementary inertia of electromagnetic origin. This results from the self-induction of conduction currents; from the existence of convection currents as a consequence both of the laws of Maxwell and Hertz of the electromagnetic field and Rowland's experiments relating to an electrified body in motion; and lastly
1 J. J. Thomson, Philosophical Magazine, 5 ser. vol. xi, 1887, p. 219.
48 PHILOSOPHY AND THE NEW PHYSICS
from the identity, as regards magnetic actions, of conduction and convection currents.
When the intensity of a current changes, a counter-electromotive force is produced in the cir- cuit and this, in accordance with Lenz's law, tends to oppose the change of the current. This is self- induction which is a veritable electric inertia: it opposes the change in intensity of a current, just as inertia of matter opposes a change in velocity of moving bodies. Just as a certain effort must be expended to put a body in motion, a certain work must be expended to establish a current ; and again, just as a projected body tends to conserve its ve- locity, so does a current once established tend to maintain its intensity. If a current increases or decreases under the action of an external field, a counter-electromotive force is produced which op- poses the change of the current, just as the inertia force of a moving body that is accelerated or re- tarded opposes the external actions that produce this change.
The existence of convection currents was fore- seen theoretically by Maxwell and was verified by experiment by Rowland and his pupils.1 It results from the laws established by Maxwell and Hertz as to the interdependence which, in time and space, connects the change of the two fields, the electric and magnetic. They may be summarized qualitatively as follows: any change in one of the
1 Cre'mieu, Etat actuel de la convection e"lectrique (Bulletin des stances de la Socie"te" frangaise de Physique, 1902, 3 fasc., p. 155).
ELECTROMAGNETIC INERTIA 49
two fields at a point in space as time goes on, gives rise to the other field, the field thus created being distributed in lines of force which encircle the direction in which the first field changes. A change in the magnetic field in a given region of space produces an electric field, the lines of force of which encircle the direction in which the magnetic field changes: this is the phenomenon of static induction dis- covered by Faraday. If the space where this field is thus created happens to be occupied by a conductor, currents that encircle the direction in which the magnetic field changes are induced in it. Inversely, a change of the electric field in a given region of space produces a magnetic field, the lines of force of which encircle the direction in which the electric field changes; this is the convection current foreseen by Maxwell and brought to light by Rowland's celebrated experiment.
Let us consider, according to this law, a point A taken in the electrostatic field of an electrified particle that is being displaced carrying its charge along with it. The intensity at A changes because the particle approaches this point, passes it, and moves away. There is a variation of the electric field in a fixed region of space as time goes on; therefore a magnetic field is produced on account of the law of the convection current. Hence an elec- trified particle in motion behaves like an element of a voltaic current; to put it more exactly, the charge e that it carries with a velocity v is equiva- lent, as regards magnetic actions exerted by it, to an
50 PHILOSOPHY AND THE NEW PHYSICS
element of voltaic current having the intensity i and the length dl, so that idl = ev. The name convection current is given to the current thus produced.
Let us now consider an electrified sphere of radius a having a surface charge e. At rest it is surrounded by radial lines of force, distributed symmetrically all around, which correspond to a localization of electrostatic energy equal to the work that had to be expended to produce the state of electrification of the sphere, and which has the value
(2) *•-&*
If a rectilinear uniform motion with a velocity v, small compared with that of light, be communi- cated to this sphere, it carries along its train of lines of force, radially and symmetrically distributed as when at rest. On account of the variation, as a function of the time, of the electric field at a point A of space, it produces a magnetic field distributed in circular lines of force encircling the direction of motion. The field at the point A in a direction perpendicular to the plane passing through A itself, the instantaneous position 0 of the particle, and the direction of the velocity has the value
ev sin a = r2
where a denotes the angle A Ov and r the distance AO. The field thus produced, which varies inversely
ELECTROMAGNETIC INERTIA 51
as the square of the distance, is determined by the instantaneous position of the sphere. The latter carries along in its motion the system of circular magnetic lines of force superimposed on the radial electric lines of force. This system of lines of force constitutes the electromagnetic field of the moving particle, moving with it like the wave sys- tem following a ship. It remains invariable so long as the velocity remains constant. The mag- netic field thus superimposed on the electric field represents a localization of energy, equivalent to the work expended by external forces in communi- cating the velocity v to the charged sphere. This energy that continues bound to" it in its motion would be recovered when it is arrested, in the form of work done against the retarding actions. An easy calculation based on the expression for the volume
TJ2
density of energy -|— shows that it is equal to
O7T
(4) *--£*.
which expression may be put into the more sugges- tive form
(40 *V-
3a
This energy presents all the characteristics of kinetic energy >^mz>2, including that of being proportional to the square of the velocity. Every- thing takes place as if, contrary to the second pos- tulate of the dualistic doctrine, the sphere possessed, on account of the potential energy of its charge, a
52 PHILOSOPHY AND THE NEW PHYSICS
supplementary inertia, an additional kinetic mass of electromagnetic origin, equal to
r*\ 2»*e2
(5) m0 = -p-
This is a necessary consequence of the iden- tity of conduction and convection currents. The inertia of a charged particle in motion is due to the self-induction of the convection current that it produces; since electric self-induction has the same properties as the inertia 'of matter, it follows that this particle, on account of its electrification, has an increment of its capacity for kinetic and, conse- quently, electromagnetic energy which becomes super-imposed on its inertia proper.
Electromagnetic inertia due to the presence of a charge on a conductor in motion is proportional to the electrostatic potential energy that this charge represents and that the sphere carries along with itself. Any change of this charge or of the radius of the sphere, causing a change of the poten- tial energy stored around it, must imply a correlated change of this inertia. The question might then be raised, what would happen if the sphere had a velocity approaching that of light and its motion remained always quasi-stationary, that is, not sub- ject to any appreciable acceleration. This is the theoretical problem that has been conceived by Max Abraham. l
In order to solve it we must refer to Maxwell's law on the mutual dependence of the electric and
1 Max Abraham, Annalen der Physik, vol. x, 1903, p. 105-179.
ELECTROMAGNETIC INERTIA 53
the magnetic fields. The magnetic field produced at a fixed point in space by the moving particle changes with the time, according as the particle approaches, passes and recedes, carrying along its train of lines of force. This change of the mag- netic field must, in accordance with Maxwell's law, be accompanied by the production of an induced electric field which is superimposed on the electric field given by Coulomb's law and modifies its dis- tribution. The analysis of the phenomenon shows that the electric lines of force tend to place them- selves in a direction transverse to that of the motion, and reach this state completely when the velocity reaches that of light. The limiting distribution of the field thus realized represents an infinite energy and consequently an infinite inertia; so that infinite work, requiring an infinite time, would be necessary to bring an electrified particle to a perma- nent condition in which it would have the velocity of light. This velocity appears as a limit that no electrified body can reach. This is confirmed by the fact that the velocity of cathode particles reaches nine-tenths of that of light without succeed- ing in attaining it; this, moreover, leads to the kinematic law of the composition of velocities, de- manded by the relativity principle.
The modification of the electric field in con- sequence of the induced field superimposed on it carries with it that of the force of inertia that the particle opposes to changes of velocity. The electromagnetic mass ceases to be proportional to
54 PHILOSOPHY AND THE NEW PHYSICS
the velocity with a constant coefficient; it varies with the velocity according to a certain function, in such a way as to become infinite for the limiting velocity of light, and this in a different manner for the three definitions of mass and according to the direction of the acceleration with respect to that of the motion. In the simplest case, that of an undeformable spherical body, two masses must be distinguished; the longitudinal mass, the force divided by the acceleration, if the latter is tan- gential to the motion; the transverse mass if the acceleration is normal to the motion. The first case corresponds to a change in magnitude of the velocity without a change of direction, the second to a change in direction without a change in mag- nitude. We may call the value of m0, which for low velocities is the same for the different definitions of mass, initial mass mQ and reserve the name electro- magnetic mass m for the transverse mass. This mass m is identical with the Maupertuisian mass; it is the only one accessible to the measurements on the electric and magnetic deviations of the cathode particles in a Crookes' tube.
To find what function connects the change of mass with that of velocity, the shape of the par- ticle and the distribution of its charge must be specified. Max Abraham considers the case of an undeformable sphere having a surface charge of uni- form distribution: for the different definitions of mass he arrives at rather complicated formula?, all of which assign the velocity of light as the upper
ELECTROMAGNETIC INERTIA 55
limit of the velocity of electrified bodies. Lorentz,1 however, in order to account for the negative result of the experiments to discover the abso- lute motion of the earth, was led to assume the contraction of all bodies in the direction of their
translatory motion in the ratio \/l — 02 f°r ob- servers who see them pass. A moving sphere be- comes a flattened ellipsoid. Allowing for this deformation, which modifies the distribution of the electric lines of force at the surface of a charged sphere and, consequently, the intensity of the con- vection current produced by it and the laws of the variation of the self-induction of the current as function of the velocity, Lorentz has obtained a formula much simpler than that of Abraham for the Maupertuisian mass or the transverse mass, which becomes equal to
/ r>\ ^0
(6) ra = .
V1-/32
By starting from this formula the laws of varia- tion corresponding to the other definitions of mass are readily obtained.
The fortunate parallelism, which existed at that time between the progress of theoretical physics and the discoveries of experimental physics, re- vealed in the cathode particles of Crookes' tubes, later in the 0 rays issuing from the atomic dis- integration of radioactive bodies, electrified bodies
1 H. A. Lorentz, Versuch einer Theorie der elektrischen und optischen Erscheinungen in bewegten Korpern, Leyden, 1895.
56 PHILOSOPHY AND THE NEW PHYSICS
projected with velocities close to that of light. It became possible to test whether their inertia varied with the velocity, according to the theoretical anticipation of Max Abraham, or according to that of Lorentz, and to distinguish between the formulae proposed by these two authors.
The experiments of Kaufmann showed that the mass actually varies with the velocity according to the formula of Abraham; but the more recent ex- periments of Bucherer and of Hupka, of a higher degree of precision, showed that the change of the electromagnetic mass of 0 corpuscles is better represented by the formula of Lorentz. We shall see that the relativity principle leads to an extension of this formula to all kinds of moving bodies, whe- ther electrified or not, so that they behave as if their inertia was solely of electromagnetic origin. It follows from this that, contrary to the third postu- late of the dualistic theory, the mass of bodies is not an invariable scalar quantity, independent of their state of motion or of rest. It is a quantity having the symmetry of a tensor,1 and dissymmetrical and variable as a function of the velocity.
JIn modern applications of vector analysis, the branch of mathematics dealing with the algebra and calculus of directed quantities or vectors, the concept of tensor plays an important part. A symmetrical tensor is an aggregate of 9 quantities, which, when combined with the 3 components of a vector by 3 symmetrical relations, give the 3 components of a new vector. A non-symmetrical tensor is more complicated. Earlier writers attached a quite different meaning to the word tensor; it was used by them to denote the mere magnitude of a single vector. (Tr.)
CHAPTER IV THE ELECTRONIC THEORY OF MATTER
10. LORENTZ'S SYNTHESIS.
The interest of Kaufmann's and Bucherer's experiments is, however, not limited to the fore- going result. By subjecting cathode particles to the action of an electric field and a magnetic field the comparison of the two deviations thus pro- duced permits of finding the velocity of these particles and the ratio e/m of their charge to their mass. If their charge is held to be constant and to be the same, these experiments then reveal the variation, as a function of the velocity, of the total apparent mass of these particles, consisting of the sum of their material mass and their electromagnetic mass. Comparing, then, the law of the total mass, as derived from the preceding experiments with the law of variation of the electromagnetic mass, theo- retically calculated, the ratio of the material mass to the total apparent mass may be deduced. The result obtained is remarkable: the inertia of cathode particles is solely of electromagnetic origin. These particles constitute elementary electric charges with- out material support, atoms of negative electricity called electrons. Thus we have here, contrary to the dualistic theory, a form of energy, resinous
57
58 PHILOSOPHY AND THE NEW PHYSICS
(negative) electricity, that appears to be endowed with inertia and corpuscular structure without any material substratum.
This result was the starting-point of the electronic theory of matter.1 It consists in a change of every- thing for its opposite, a reversal of the relation tra- ditionally supposed to exist between matter and electricity. In place of seeking to explain electric phenomena mechanically, the mechanical pheno- mena are now to be explained electrically. Instead of seeking, as Maxwell did, to cast the equations of the electromagnetic field in the analytical mould made by Lagrange and Hamilton, which appears no longer valid except as a first approx- imation for low velocities and for quasi-stationary motions, there is substituted for the dynamics of the material point, conforming to Newton's equa- tions, the dynamics of the electron, conforming to Lorentz's equations.2
Matter, the substratum of mechanical phenome- na, is explained by starting from electricity. The elementary charges of electrolytic ions, the cath- ode particles issuing from Crookes' tubes, the |8 rays issuing from the atomic disintegration of radioactive bodies, the negative emissions consti- tuting the Edison and the Hertz phenomena reveal a universal constituent of matter in the electron.
1 Cf. Les ide"es modernes sur la constitution de la matiere, Con- ferences faites en 1912 & la Societe" francaise de Physique. Paris, 1913.
2 Cf. W. Wien, Archives ne"erlandaises, 2. ser. vol. V, p. 96.
LORENTZ'S SYNTHESIS 59
The molecular structures constituting ponderable bodies are nothing but electronic architecture. Electrons are grains of resinous (negative) elec- tricity, all identical with one another. They carry an individual charge of 4.774 X 10~10 electrostatic C. G. S. units, distributed on a circumference of 2 millionths of a MM (2 X 10~12 millimeter) radius. They are endowed with an initial electromagnetic mass equal to 10~17 grams, that is, 1800 times smaller than that of a hydrogen atom. They play different parts in the production of phenomena, according to the position that they occupy in the atomic and molecular structures and according to the degrees of freedom that they enjoy because of this position. Accordingly there are distinguished: perfectly free electrons moving outside of matter, in the form of cathode rays and 0 rays; free electrons moving inside of matter in the intermolecular spaces with a chaotic motion that gives rise to heat radiation, on which may be superimposed a translatory motion as a whole that gives rise to electric current in conduc- tors; electrons weakly bound to atoms revolving on their circumference and capable of being detached easily under the action of shock or an electromagne- tic action, producing thereafter ionization of solu- tions and gases; electrons bound to atoms, moving in closed curves, and determining the spectral rays as well as the chemical valences, the sources of molecu- lar union; lastly, the electrons that compose the central kernel of the atom, inaccessible to external physical agents, in a chaotic state of agitation,
60 PHILOSOPHY AND THE NEW PHYSICS
which do not become revealed except in the atomic disintegration of radioactive bodies.
On this view the atoms are formed by negative electrons, bound more or less to the kernel, and of positive remainders, the charge of which must be equivalent to the total charge of the negative elec- trons in order to maintain in the neutral state the cohesion of the structure by the mutual attraction of the electricities of opposite kind. Of the struc- ture of these positive charges we know next to nothing. Those that we can capture in the canal rays of Goldstein and the a rays of radioactive bodies are atoms of hydrogen or helium, deprived of one or more negative electrons and having, on that account, become positive ions. It may be asked whether there is a vitreous (positive) electricity, having an independent existence and a corpuscular structure like the resinous (negative) electricity, so that there would be positive electrons side by side with the negative ones, as Jean Becquerel believes can be inferred from the study of absorption spectra at low temperatures; or whether positive electrifica- tion results from the subtraction of one or more negative electrons from the normal atomic structure corresponding to the neutral state. In any case the question of finding whether positive electricity has a material support, so that there would be nega- tive electrons but only ions of positive charge, has lost much of its interest. The relativity prin- ciple leads, as Lorentz has shown, to the assumption that the inertia of these positive ions, if they can
LORENTZ'S SYNTHESIS 61
not be decomposed into simpler units, must follow the same laws of change as a function of the veloc- ity as that of the negative electrons, that is, that these ions must behave as if their mass were solely of electromagnetic origin.
The electronic theory has explained, with in- creasing success — which has only reached its limit when confronted by black radiation and the diminu- tion of the specific heats at low temperatures — the phenomena of static electricity, of the electr c current, of induction and magnetism, of the emis- sion, the propagation in different physical media, and the absorption of different kinds of radiation; those of radioactivity, of the ionization of gases and liquids, of chemical valence, etc. It has led to the explanation of new phenomena such as the phe- nomena of Edison and of Hertz, and above all the Zeeman effect. It has led Lorentz1 to a brilliant synthesis carrying his name. Presented for the first time in 1892, this synthesis has been put into accord with the requirements of the relativity prin- ciple by its author. According to it there is no matter, only electrons, positive and negative, in an ocean of uniform ether; all forces are of elec- tromagnetic origin or behave like such; and the measurements effected in a moving system are rela- tive to the dimensions of the instruments, to the forces taken as comparison terms, and to local time.
1 Cf. H. A. Lorentz, Sur la theorie des electrons (Les quantity's elementaires d'61ectricite, ions, electrons, corpuscules, Paris, 1905, vol. i, p. 430-476).
62 PHILOSOPHY AND THE NEW PHYSICS
In this conception matter, conduction currents and magnetism become mere modes of manifestation. The only constitutive principles are the ether and grains of vitreous (positive) and resinous (negative) electricity. We shall see how the English physi- cists have taken hold of this conception and pushed the reduction in the number of things still further: according to certain of them not only is the exist- ence of matter denied but also that of electricity; and nothing but ether and empty space exists.
11. THE DEMATERIALIZATION OF MATTER.
The conception of the dematerialization of matter of the English physicists results from the success of the electronic theory, taken in connection with the concept of an ether endowed with mechanical properties, that is, inertia, elasticity and rigidity, in conformity with the ideas of Faraday and Maxwell. *
It is the wave theory of light, stated at first by Leonardo da Vinci and Galileo, made more pre- cise by Huygens following the discoveries of Grim- aldi, and established finally by Fresnel following Foucault's experiment that has given physical verisimilitude to the idea of an ether. If light and, in general, any form of radiant energy is a wave phenomenon that is propagated with a finite velocity in the interstellar spaces, it is difficult to avoid the assumption that, because there is wave vibration and therefore motion, there is something that moves, a mobile medium that must fill all
DEMATERIALIZATION OF MATTER 63
space. As Lord Salisbury picturesquely said in his Oxford presidential address to the British Associa- tion, the first and principal reason for the existence of the ether is to supply a subject to the verb to undulate.
The second object of the ether is to account for the localization of energy in dielectrics, surrounding conductors or magnets, in accordance with Fara- day's experiments, and for its accumulation in the neighborhood of a moving electric charge, in accordance with the theoretical views of Abraham and Lorentz and the experimental confirmations of Kaufmann and Bucherer. This energy, considered as a mode of manifestation of a substance, requires a substratum of which it is merely an accident. Lastly, actions at a distance, such as gravitational attraction, electric or magnetic attraction and re- pulsion,rand,without doubt, chemical affinity, be- tween bodies separated in space are incapable of mental representation except as pressures or ten- sions exerted by an intermediate medium in which the bodies are immersed. To account for the phe- nomena of gravitation, of optics^ of electricity and magnetism, physicists have thus been led to con- sider a series of media, the discrepancies of which Hannequin1 found pleasure in exposing, but all of which have in the end become fused together in Maxwell's dielectric ether.
It was the consideration of the ether that per-
1 Hannequin, Essai critique sur 1'hypothese des atomes dans la science contemporaine, p. 178-224.
64 PHILOSOPHY AND THE NEW PHYSICS
mitted a marked reduction of the number of pri- mary principles required for an explanation of things. At the beginning of the nineteenth cen- tury the existence of eight energetic imponderable agents outside of matter was commonly assumed: the electric fluids, positive, negative, and neutral ; the magnetic fluids, south, north, and neutral; the caloric, and the luminous fluid. Gradually these agents have disappeared: heat has turned out to be the sensible manifestation of the internal agita- tion of the molecules of bodies. Inspired by Fres- nel's ideas Maxwell tried to explain the electric, magnetic and optical phenomena that ceased to be attributed to particular fluids by starting from the mechanical properties of the ether. Electrostatic phenomena, for example, seemed to him to be explained by the deformations of the ether. Static charges of electrified bodies have merely a fictitious existence : they are the locus of the ends of the lines or tubes of force corresponding to the deformations of the ether. These deformations subject the conductors to pressures and tensions forcing them to approach each other or to move apart, the electro- static energy being nothing but the potential energy of deformation of the ether. The electro- kinetic phenomena are explained by disturbances of the ether. The electric current is not a phenome- non taking place in the conductor but a state of motion of the adjoining ether, the character, direction, and intensity of which are determined by the geometrical and physical properties of the
DEMATERIALIZATION OF MATTER 65
conductor, the energy of the electric current cor- responding to the kinetic energy of the surrounding ether. There exist only -two realities, matter and ether, the former of which acts like a supernumer- ary; the ether, in which, in the form of static defor- mations, the electric energy is stored, and in which, in the form of disturbances, the radiant energy is propagated, appears more and more as the only active medium taking part in the production of the phenomena.
Wherein has the electronic theory strengthened or weakened this idea, which satisfies our taste for simplicity? It seems at first that, in addition to matter and ether, it restores one of the old agents, the negative electricity, no longer conceived, however, as a continuous fluid but as a substance endowed with corpuscular structure, not as an imponderable but as endowed with inertia. In fact, we have the phenomena of electrolysis, which show us a monovalent ion carrying a well defined quantity of electricity e, which is always the same; a bivalent ion carrying a quantity of electricity 2e and so on. Now this elementary charge, which appears to be indivisible like an atom, is precisely that which is found again, but without material support, in the negative electrons, issuing from the cathode dis- integration in Crookes' tubes, or from the atomic disintegration in the 0 rays of radium, or the kinetic agitation in the negative emissions of heated or illuminated conductors (Edison's and Hertz's phe- nomena). These charges can no longer play the
66 PHILOSOPHY AND THE NEW PHYSICS
part of pure fictions, by being placed geometrically at the ends of lines or tubes of force constituted of deformations of the ether: they have a physical existence, a structure and inertia of their own. In these grains of electricity, and not in the mechanical states of the ether, the initial cause of electric and magnetic phenomena must be sought. The elec- tric current, for example, has not its seat in the ether but in the conductor; it consists of a mass motion of the free electrons of the metal, superim- posed on their chaotic kinetic agitation, like the flow of a gas in a pipe: there is no longer a con- duction current due to the ether, but merely a convection current due to electrified particles in motion.
But at the very time when the discoveries of contemporary physics seem of necessity to increase the number of primary principles, by a return to the old agents, through assuming grains of resinous (negative) electricity, their offspring, the electronic theory, works an inverse reduction by destroying the concept of matter, which from this point of view has merely a gross semblance of reality. According to this theory, the molecular and atomic structures constituting bodies are reduced to electronic archi- tecture, the electrons being positive and negative, or simply negative, and their inertia being entirely of electromagnetic origin and due to the self-in- duction of the convection currents that they produce by their displacement. Matter, ceasing to corre- spond to a distinct reality, is resolved into grains of
DEMATERIALIZATION OF MATTER 67
electricity and these, according to the English school, become subtilized into mere ether cavities.
Rearming themselves with the ideas of Faraday and Maxwell, the English physicists, and the most famous of them, J. J. Thomson, regard the inertia of the electron as due to the ether surrounding it. An electron at rest is a surface charge without mate- rial support, and may be considered as a cavity in the ether. This cavity is the center from which the electric lines of force constituting the electro- static field of the electron diverge. The ether ad- heres to these lines of force, so that the electron can not move without displacing it. The inertia of the electron results from the inertia of the entrained ether, which alone opposes its motion. Knowing the volume and the masfe of an electron, the density of the ether adhering to it may be calculated; it is found to be equal to about 2000 million times the density of lead. Matter ceases to exist, since, being composed of electrons, the sole reason we have to believe in it, namely its inertia, does not properly belong to it, but is borrowed from the ether. Elec- tricity likewise disappears as a substance with an existence of its own, since the electrons are reduced to cavities in the ether. Two principles only sur- vive, ether and empty space, of one of which the ancients would have said that it is the non-being and consequently that it does not exist. The world is merely a bubble of ether in the non-thing.
This theory of the disincarnation of matter, leading to a complete etherization, should, in spite
68 PHILOSOPHY AND THE NEW PHYSICS
of its alluring aspect, be taken with considerable caution. It is threatened with downfall on account of the concept that forms its base, that of an ether endowed with mechanical properties, the hypo- thetical existence of which seems, moreover, to be contradictory.
To fulfill its office the ether must accumulate the mutually exclusive properties of solids and fluids. It must behave like an elastic solid, en- dowed with a rigidity surpassing that of steel in order to transmit nearly instantaneously the trans- verse vibrations of light; it must behave like a fluid with a density much less than that of the lightest gas, in order not to retard the translatory motion of the stars and not to rob them of their atmosphere; but in the neighborhood of the elec- trons it must have a density far surpassing that of lead. All this is incomprehensible and no one can, under the plea of thinking in terms of the ether, evade thinking according to the law of contradiction.
If the ether exists, it is incapable of motion, as is proved by the impossibility . of reconciling Fizeau's experiment with Hertz's hypothesis of a complete entrainment of the ether by matter in mo- tion, and the impossibility of reconciling the prin- ciple of action and reaction with FresneFs and Fizeau's hypothesis of a partial entrainment of the ether. But, if the ether is incapable of motion, our laboratories and our instruments are continuously traversed by an ether current, the velocity of which
DEMATERIALIZATION OF MATTER 69
is equal to and opposite to that of the earth and varies as a function of it. Such a current would exert a considerable influence on electromagnetic and optical phenomena. Now the experiments undertaken to show the absolute motion of the earth with respect to the ether, with which a privileged set of reference axes might be connected, show that there is none. The hypothesis of an immovable ether is in its turn contradicted by the relativity principle.
But there is more than that. Neither Max- well nor any one after him has succeeded in giving a clear and distinct mechanical representation of the deformations and disturbances of the ether that would produce the electric and magnetic phe- nomena. In attempting it he arrived at a concep- tion so strange, that of a cellular ether formed of two substances one of which was impregnated with the other like a sponge soaked with water, that he did not make it play a part in his large Treatise on Electricity and Magnetism. Recently Witte1 has shown, by a very complete analysis, that the proper- ties of the electromagnetic field cannot be explained by the aid of classical mechanics with any supple- mentary hypothesis, if we assume the hypothesis of a continuous ether, and he rejects as improbable that of a discontinuous ether. To-day we see the underlying reason of this impossibility in the rela-
1 Cf. Witte, Ueber den gegenwartigen Stand der Frage nach einer mechanischen Erklamng der elektrischen Erscheinungen, Berlin, 1906.
70 PHILOSOPHY AND THE NEW PHYSICS
tivity principle. The Lorentz group, for transfor- mations of which the form of the equations of electromagnetism remains unaltered, excludes the Galileo group, for the transformations of which the form of the equations of classical dynamics is conserved. Mechanical explanations of electric phenomena are, therefore, definitely condemned.
Lastly, we shall see that there are reasons for believing that radiation is not propagated by means of a hypothetical medium, but projected into empty space in the form of discontinuous elemen- tary quantities, energy atoms, called quanta. With this return to an emission theory the most con- vincing argument in favor of the ether, the consideration of which becomes superfluous, disappears. This is expressed by Einstein:1 "The electric and magnetic fields that constitute light no longer appear as states of a hypothetical medium, but as individual realities, which the luminous sources send into space as in Newton's emission theory."
If we renounce the mechanical ether of Faraday, of Maxwell, of Lord Kelvin and of Sir O. Lodge, and consider nothing but an absolutely empty space, at every point of which an electric and a magnetic field may be superimposed, then the idea of the dematerialization of matter gives place to that of a materialization of energy. The only reality that is positively observable and empirically
1 Einstein, Phys. Zeitschr, vol. x, 1909, p. 849. Cf. Campbell, Phil. Mag. 1910, p. 981.
DEMATERIALIZATION OF MATTER 71
demonstrable is the energy that, at any time, is localized in a region of space and corresponds to the two fields, electric and magnetic, which are at the moment superimposed there, its density per unit volume being proportional to the squares of these fields. Nothing prevents us from considering energy as a substance, endowed in itself with existence, without the aid of any sub- stratum whatever. It is therefore convenient, instead of reducing the inertia of the electron to the agitation of the ether, to attribute it to the energy accumulated around it, which forms its electromagnetic wave system. Moreover we need not picture the electron as a spherical cell of two millionths of a w with a surface charge. It is more natural to suppose either that the density of this charge gradually diminishes from the center of the sphere outwards, so that the electron does not possess a definite surface, or that there is a uniform distribution of the electric density in the inside of the volume of the sphere, which implies a definite contour: in the latter case the same for- mulae are obtained as in the case of a surface distribution, with the single difference that all quantities that play a part in the equations are multiplied by the factor %.
To sum up, a moving electron appears to us as a circumscribed region of space, where there is to be found accumulated electromagnetic energy in the form of fields,- the intensity and distribution of which vary as a function of the velocity. The
72 PHILOSOPHY AND THE NEW PHYSICS
mass of an electron can not be attributed to a material support that does not exist, nor to the sur- rounding ether, the existence of which is hypothet- ical, but it is due to its own energy, which consti- tutes its only substantial reality. But if energy is inert it is endowed with mass, consequently with weight in proportion thereto, and, possessing struc- ture, it becomes a materialized body, while matter is refined away.
If now, in conformity with Lorentz's theory, the molecular structures that constitute bodies are reducible to assemblages of electrons, and if the electron is inert only because of the energy which it possesses, material systems are themselves en- dowed with mass only in proportion to the energy that they contain. Mass becomes a quantity that measures their internal energy, and, as foreseen by Ostwald, the concept of matter is subsumed under the more general one of energy.
CHAPTER V THE INERTIA OF ENERGY
12. THE MATERIALIZATION OF ENERGY.
The conclusion stated at the end of the last chapter appeared as almost an immediate consequence of a paper by Henri Poincare1 on the longitudinal contraction of an electron in motion. Taking the older point of view, that of the partisans of the mechanical ether, though the point of view thus taken matters little in this connection, he showed that this contraction is exactly that demanded for the maintenance of its equilibrium on the assumption that the surface charge carried by the moving electron, the elements of which tend to separate because of their natural repulsion, is maintained by a constant pressure of the ether. Assuming this uniform pressure, the equilibrium configuration of an electron at rest is that which makes the potential energy of the actions superimposed on it — electrostatic repulsions and Poincare pressure — a minimum. If its shape is that of a sphere of radius a the total potential energy of an electron at rest in equilibrium is given by
(7) E • * * *
h°
1 H. Poincare, La theorie de Lorentz et le principe de reaction (Archives neerlandaises, 1900, p. 252-278).
73
74 PHILOSOPHY AND, THE NEW PHYSICS
• A comparison of this expression with that of the initial electromagnetic mass m0 (5) leads at once to the relation
(8) Wo = KQHQ EQ.
According to the relation between the two coefficients KQ and ^o deduced by Maxwell from the comparison of the two systems of C. G. S. units, the electrostatic and electromagnetic, namely,
where V represents the velocity of light, the ex- pression (8) becomes
Tjl
(9) Wo = Y*
The initial electromagnetic mass of an electron at rest is equal to its total potential energy divided by the square of the velocity of light. As the potential energy represents the only energy that an electron at rest can have, it is seen immediately that, the velocity of light being taken as unity, the mass of an electron at rest is equal to its total energy, of which it may serve as a measure.
This result may be generalized. If the elec- tronic theory of matter were confirmed, it would be applicable to all material systems at rest, since they could be reduced to assemblages of electrons without material support. However it may be con- cerning this last point and postponing the question of the structure of the positive centers, whether material or not, Lorentz has shown that the rela-
MATERIALIZATION OF ENERGY 75
tivity principle requires: (1) the longitudinal con- traction of all bodies in the direction of their
translation in the ratio Vl — 02; (2) that the change
m0 of mass with the velocity, m = ,r-^=^, established
first of all for electromagnetic inertia, applies in general to all bodies, as if their mass, like the forces of • elasticity and cohesion, were purely of electro- magnetic origin. It follows therefrom that the relation (9) must be applied to all material systems at rest as if the electronic theory were correct. This relation may be extended to all bodies and material systems in motion. The same rela- tion that exists between the masses m0 and m of the same body observed by observers 00 at rest with respect to it and by observers 0i in motion, must exist, by virtue of the relativity principle, between the energies E0 and E of the same system observed simultaneously by 00 and Oi, so that we have
By virtue of (6)and (10), the relation (9) leads
A
to the generalization already stated, namely, (11) M = ~
The quantity Vl — j32 being always less than one, it follows from (10) that the energy of a body set into motion without deformation is, for observers connected with it, greater than when it is at rest.
76 PHILOSOPHY AND THE NEW PHYSICS '
The difference represents, by definition, the kinetic
EQ
energy; and the two terms "^ and mQV2, of
which it is the difference, are nothing but two meas- ures of -the total energy of the same body, made suc- cessively by observers in motion and at rest.
To sum up, the existence of convection cur- rents, foreseen theoretically by Maxwell and realized experimentally by Rowland, led J. J. Thomson, Max Abraham, and Lorentz to study theoretically the laws of variation of the inertia of an elec- trified particle in motion. The discovery of cath- ode particles permitted Kaufmann and Bucherer to verify these theoretical anticipations and to reveal the existence of grains of resinous (negative) electricity or electrons, the inertia of which is solely of electromagnetic origin and obeys the preceding theoretical laws. Starting from this result Pom- care has shown that the initial mass of an electron in equilibrium and at rest is equal to its total potential energy divided by the square of the veloc- ity of light. The principle of relativity requires, then, that this result be extended to all systems of bodies at rest and in motion, so that the formula M = E/V2, thereafter fundamental, shall hold for them. Taking the velocity of light as unity, this formula states that the mass o/ a body is equal to its total energy, of which it may serve as a measure, and consequently energy is inert.
The decisive arguments in favor of the inertia of energy are, however, drawn from considerations
MATERIALIZATION OF ENERGY 77
of another kind. They result from the necessity of reconciling Maxwell's pressure of radiation with the principle of relativity and with that of action and reaction or of the conservation of momentum in a closed system.1
Maxwell, by starting from the electromagnetic theory of light, and Bartoli, by starting from the principles of thermodynamics, foresaw theoretically, and Lebedew verified experimentally, that all radi- ation exerts a backward pressure on the source that emits it in a single direction and a forward pressure on an obstacle that absorbs it : this is what is called the pressure of radiation. The inertia of energy is a consequence of the existence of this pressure, and brings up the metaphysical problem of the action of an imponderable on a ponderable in a particularly acute form.
Let us consider first a material system the motions of which are due solely to internal actions, such as a firearm and its projectile. When the shot takes place the gun undergoes a recoil, that is to say, takes up a certain momentum which, counted negatively, represents a loss; the projectile is shot forward and acquires a momentum which, counted positively, is equal to that lost by the gun. The conservation of the total momentum of the system holds at every instant, and consequently there is no motion of the center of gravity of the system. The conservation
1 Cf. A. Einstein, Ann. der Phys. vol. xviii, 1905, p. 639. P. Langevin, L'inertie de Tenergie et ses consequences (Journal de Physique, juillet 1913, p. 553 et seq).
78 PHILOSOPHY AND THE NEW PHYSICS
of momentum is but a natural consequence of the instantaneous equality of action and reaction. The gun recoils because, while it acts on the projectile, the latter reacts in turn equally on it.
Let us now consider a material source that radi- ates unsymmetrically in a single direction, such as a lamp provided with a reflector or a Hertzian exciter at the center of a parabolic mirror. At the instant of the emission the source recoils as a consequence of the pressure of radiation : there is a loss of momen- tum. If the radiation encounters an obstacle that absorbs it, it will communicate to it an impulse, that is, a momentum equal to that lost at departure by the source. The action experienced by the obstacle will be equal in magnitude to the reaction undergone by the source.
Does this mean that the principle of the conserva- tion, of momentum or of action and reaction is safe- guarded? Certainly not, for it is not so at every instant from the start. A certain interval of pro- pagation elapses between the time when the radia- tion is emitted and that when it is absorbed, during which the momentum lost by the source and the reaction that the latter undergoes remain uncompen- sated. This compensation would never take place if the radiation were propagated to infinity without encountering matter absorbing it. In this case there would be a definite loss of momentum and the center of gravity of the system formed by the source and the radiation would take an absolute motion, which is contrary to the relativity principle.
MATERIALIZATION OF ENERGY 79
If this principle is to be safeguarded and at the same time those of the conservation of momentum and of action and reaction, the system source- radiation must be assimilated to the material system of the fire-arm and its proj ectile. The radia- tion must be treated as a material projectile, that is, it must be regarded as representing a certain mo- mentum equal to that lost by the source, so that the reaction that the source undergoes is the natural effect of the action exerted on it; only in this case will the center of gravity of the system remain fixed and the relativity principle be satisfied. This is what Henri Poincare has not hesitated to do. He was the first one to introduce the idea of electro-magne- tic momentum of radiation to save Newton's principle of action and reaction, which is sacrificed in Lorentz 's theory.
But momentum means, by definition, mass in motion, in virtue of the vectorial relation
g = mv.
The projectile carries away in its motion a part of the initial mass of the loaded fire-arm, which is diminished by just so much: that is why we have instantaneous equality of action and reaction, conservation of momentum and a stationary center of gravity of the system. Hence, if radiation is a vehicle of momentum, it must carry away with it a part of the initial mass of the radiating material source. It must be possible to assimilate literally the sdurce-radiation system to that of the firearm-
80 PHILOSOPHY AND THE NEW PHYSICS
projectile, as also in the case of a radium atom dur- ing its transmutation when it spontaneously divides into a helium atom and a niton atom by a sudden explosion that projects the helium atom and the niton atom with equal momentum in opposite directions. After we have discussed the structure of radiant energy, which tends to make the radiation assume the characteristics of a corpuscular emission, the preceding assimilation will appear less bold. Hence if radiation carries away momentum it must, just like electric energy, possess electromagnetic mass apart from any material substratum.
Let us determine quantitatively the momentum and the Maupertuisian mass that must be thus attributed to radiant energy, to safeguard the principle of action and reaction and the relativity principle.
In the case of a plane wave, the electromagnetic momentum is in the direction of propagation and at right angles to the plane of the wave containing the two fields, the electric and the magnetic. By taking into account the fact that in this case the two fields are mutually perpendicular, it may be shown that the density g, per unit volume, of the momentum has the value
where V represents the velocity of propagation of the plane wave, which is equal to that of light. Since on the other hand, the density, per unit
MATERIALIZATION OF ENERGY 81
volume, of the electromagnetic energy has, by virtue of (7), the value
# -^ +*£-'- 3*
the result for the density of momentum becomes
(14) g = ~ •
The electromagnetic momentum per unit volume is equal to the electromagnetic energy divided by the velocity of light.
To this momentum there corresponds a Mau- pertuisian mass by virtue of the vectorial relation g = mv.
The magnitude of this mass is, by definition, the momentum divided by the velocity, which, in this case, is that of light,
If we replace g by its value E/V from the relation (14), we get
(15) M=y-*
Free radiation has a mass equal to its energy divided by the square of the velocity of light. Since all forms of energy may be transformed into radiant energy, the preceding result may be generalized as follows: every form of energy E possesses a certain coefficient of inertia determined by the preceding formula. Contrary to the fundamental principle
82 PHILOSOPHY AND THE NEW PHYSICS
of the dualistic theory energy of any form whatever is inert.
The identity of the formulae (11) and (15) shows that the mass of a body and radiant energy are equivalent quantities, capable of being converted one into the other, as heat and mechanical work are. If a body radiates energy, the radiation emitted carries away a part AE/V2 of its initial mass, and when an obstacle absorbs this radiation its previous mass is increased by the entire Maupertuisian mass AE/V2 of the absorbed radiation, whence:
Any. change AE of the internal energy of a material system, due to emission or absorption of radia- tion, is accompanied by a proportional variation of its mass in accordance with the relation:
(16) AM =
V
If we examine closely this enunciation of the proposition as to the variation of the mass of bodies as function of their velocity, it follows, contrary to the third and fourth postulates of the dualistic doctrine, that: the mass of a body is not invariable; it increases or it decreases according as the body ab- sorbs or radiates energy, and as it is in motion or at rest with respect to the system to which it is referred.
The last postulate of the dualistic theory is therefore necessarily unsound. In an isolated sys- tem, the different parts of which exchange energy, the individual masses of the bodies present are not conserved; it is only the total inertia of the system
MATERIALIZATION OF ENERGY 83
(the constant sum of the variable inertia of the bodies and the variable inertia of the radiation) that is conserved, provided the system does not per- mit any exchange with the outside. The principle of the conservation of mass of material bodies is no longer valid by itself: it is replaced by the more general principle, which alone is valid, of the con- servation of the total inertia of an isolated system.
How are we to evaluate this total inertia? An inspection of (9) and (10) shows that the total energy of a body, at rest as well as in motion, is equal to the product of its mass by V2. If the velocity of light in a vacuum is taken as the funda- mental unit, it follows that the mass of a body is equal to its total energy, a statement that translates the identity of the nature of mass and energy into a nu- merical equality; or, better still, the mass of a body measures its internal energy.
The total inertia of an isolated system, which neither loses nor receives energy, is, therefore, equal to its total energy: the internal energy of the bodies present, the kinetic energy of these bodies, the energy of free radiation. The principle of the conservation of the total inertia of a system goes back, therefore, to the principle of the conservation of energy, into which the principle of the conservation of mass is henceforth absorbed. In the new dynamics of relativity, there exist only two fundamental laws of invariance : that of the conservation of energy and that of the conservation of momentum. The first leads to the assumption of energy localized outside
84 PHILOSOPHY AND THE NEW PHYSICS
of bodies; the second leads to the assumption of the existence of electromagnetic momentum in free radiation and consequently of the inertia of energy. These two laws are not independent. If we were to adopt Minkowki's terminology of the " world' ' where the phenomena are referred to four interchangeable axes, implying four homogeneous coordinates, three of space and one of time, these two laws would appear as two different aspects of one single law, that of the conservation of the world impulse.
Lastly, as a consequence of the exact proportion- ality between mass and weight shown by Eotvos, from the inertia of energy follows its weight in proportion thereto: a change of internal energy would at the same time be accompanied by a change of mass and a change of weight. Since it is easier to measure the weight than the mass of a body, the weight of a body might serve to measure its internal energy.
13. THE EVALUATION OF THE INTERNAL ENERGY OF BODIES AND THE VARIATIONS OF MASS.
From the inertia of energy the abandonment of the dualistic theory and numerous special conse- quences result.
1. Evaluation of the internal energy of bodies.
Whenever a body loses heat or changes its dimen- sions under the influence of internal actions alone, it performs thermal or mechanical work that corresponds to a loss of some of its energy. The
INTERNAL ENERGY 85
internal energy of a body was formerly defined as the total work it could perform as a consequence of cooling without limits, or of extension or con- traction without limits, according as the molecular forces are attractive or repulsive. There was no way of evaluating such a quantity of energy; changes only of internal energy could be measured by the work performed in starting from an initial condition.
On the other hand the formula ra0 = E/V2 pro- vides a very simple method of evaluating this energy of intermolecular and intra-atomic nature. It is equal to
for a body at rest for those who observe it. This relation shows that a gram of matter at rest and at the temperature of the absolute zero corresponds to the presence of an internal energy equal to 9 X 1020 ergs, that is, an energy equivalent to the heat furnished by the combustion of 3 X 109 grams or 3,000,000 kilograms of anthracite.
Let us call the energy of a body thus evaluated for the temperature T = 0 and the state of rest latent energy. This enormous energy is nearly entirely of intra-atomic nature. In fact, at the absolute zero the degrees of freedom of the mole- cules are, as it were, anchylosed by the frost. On the other hand the physical molecular forces and the chemical atomic forces put in action a quantity of energy only very small compared with this
86 PHILOSOPHY AND THE NEW PHYSICS
enormous reserve of latent energy, as can easily be calculated. The changes of mass 'that result from the presence or absence of heat or kinetic energy or the presence of radiation within a body are practi- cally imperceptible, except in the case of radioactive transformations, where the intra-atomic energy again comes into play.
2. Change of mass with temperature. — The same portion of matter, taken at twro different tempera- tures, may pass from one to the other by emission or absorption of radiant heat. The change of mass resulting therefrom may be evaluated by dividing the heat exchanged with the outside by V2. To calculate the order of magnitude of the effect anticipated, water may be taken, the heat capacity of which is especially large. A mass of water, having at 0° an inertia equal to 1 g., will have a larger inertia at 100°. The difference is obtained by dividing the heat absorbed, 100 gram degree calories or 4.18 X 109 ergs, by F2, which in the same system of units is equal to 9 X 1020, which gives about 5 X 10~12, that is to say a quite imperceptible change.
Nevertheless this example shows that, from the theoretical point of view, the idea of mass must not any longer be confused with that of quantity of matter, as was done by Newton. Two masses of water of equal inertia, one taken at 100° and the other at 0°, do not contain the same quantity of matter, since they cease to be equal when reduced to the same temperature; to put it differently, two
CHANGES OF MASS 87
masses of water containing the same number of molecules do not have the same inertia unless they are taken at the same temperature, for then their energies are equal.
3. Change of mass with velocity.- — The mass of a body depends on its state of rest or of motion with respect to given observers. In fact, when a body of initial mass m0 acquires the velocity v, its mass increases on account of the kinetic energy acquired according to (6) :
Its total energy E then becomes
or
v4 E =
From this formula it is seen that the quantity J^WoV2, called ordinarily the kinetic energy of the moving body, constitutes only a very small part of the energy corresponding to the passage from the reference system O0, in which the body is at rest, to the system Oi, in which it is in motion. The quantity of latent energy mQV2 remains hidden to our senses, which perceive only its exceedingly fee- ble variation. This formula shows, moreover, that kinetic energy loses its significance as a special form of energy, which is inevitable, for if one reduces inertia to energy one may not in turn reduce
88 PHILOSOPHY AND THE NEW PHYSICS
part of energy to inertia. This is why all forms of energy are functions of the velocity and grow with it.
4. Change of mass with radiant energy. — If the space comprised within a material enclosure is filled with black radiation corresponding to a given temperature, the mass M of the system will be, by virtue of the inertia of the radiation, larger than it would be without the latter: this excess mass is proportional to the total energy of the radiation.
5. Change of mass in chemical reactions.- — We have already seen that the principle of the con- servation of mass, if applied to the individual masses of the bodies present in a closed system, is generally erroneous. In particular, there is no conservation of the mass of the bodies in chemical reactions or in radioactive transmutations.
Chemical reactions being all exothermic or endo- thermic, it follows that, by virtue of the relation deduced from (16),
(17) A#o = Aw0F2
the sum of the masses of the elements combined does not remain constant.
Take, for example, the formation of water start- ing from its elements taken in the gaseous state. The combination of 2 grams of hydrogen with 16 grams of oxygen sets free 69000 gram-degree calories, equivalent to about 3 X 10 12 ergs. We would not obtain 18 grams of water, because the heat liberated in the form of radiation carries with it
CHANGES OF MASS 89
a loss of mass equal to J£ X 10~8 grams, this being a difference of one-fifth of a billionth (.2 X 1Q-9) between the mass of the detonating gas and that of the water that it can form at the same temperature. 6. Change of mass in radioactive transformations. — The same will hold true of the transformations of radioactive bodies. The initial mass of one of these bodies and the total mass of its disintegration products at the end of a certain time will not be equivalent, the transformation being accompanied by radiation. It is known that one gram of metallic radium sets free 130 calories per hour while it is transformed into radium D and helium, through the successive forms of emanation, radium A, B, C. Taking into account the fact that the mean life of an atom of radium is about 2600 years, it may be computed that the complete transformation of one gram of radium into helium and radium D would liberate an energy equal to 1.1 X 1017 ergs. The emission of this energy would correspond to a difference between the initial mass of the radium and that of the radium D and the helium, equal, per
gram, to
i i v in17
A • ^> 1 o vx i r\ 4
Am° = -9x16*-
The disintegration of radium into helium and radium D represents merely one step of the trans- formations that start from uranium and end with he- lium and lead. The complete disintegration of a given quantity of uranium into helium and lead would represent a loss of mass exceeding one-ten-
90 PHILOSOPHY AND THE NEW PHYSICS
thousandth of the original uranium. The fraction of the mass thus transformed into radiant energy is of an order of magnitude much greater than in the case of chemical reactions. It may be presumed that it comes from the latent energy of the uranium, that is, from its intra-atomic energy. If we could succeed in establishing exactly, to quantities of the order of magnitude 10~4, the relation of the masses in the case of radioactive transformations, it would be possible to verify the identity of mass and energy. To sum up, energy is inert and the mass of a body is equal to its internal energy which it serves to measure. This internal energy represents at the absolute zero an enormous accumulation of intra- atomic energy. According as a body acquires or gives up energy, its mass increases or diminishes. It is greater when the body is in motion than when it is at rest, greater when hot than when cold, when electrified than when discharged, it changes in chem- ical reactions and in a more perceptible way in ra- dioactive transformations. The principle of the conservation of mass formulated by Lavoisier is true only as a first approximation : it becomes merged in that of the conservation of energy.
CHAPTER VI THE WEIGHT OF ENERGY
14. THE WEIGHT OF ENERGY; ITS EXPERIMENTAL VERIFICATIONS.
The experiments of Eotvos show that if energy is inert it must have weight in proportion thereto. If this were not so, a certain quantity of uranium and its disintegration products, helium and lead, would have equal weights but different masses, and consequently, would not be given the same acceleration under the action of gravity. There would have to exist at one and the same spot diff- erences equal to at least one-tenthousandth in the values corresponding to the acceleration of gravity, and this seems to be capable of measurement. Thus energy possesses not only an inert mass, but it possesses also a ponderable mass /* = E/V2. In accordance with what is true for inertia, a change in internal energy is accompanied by a simultaneous change of mass and weight. A body is heavier when in motion than when at rest, when hot than when cold, when in a state of electrification than when neutral, detonating gas than the water it produces, uranium than its disintegration products.
Langevin1 sees an experimental proof of the
1 P. Langevin, Journal de Physique, juillet 1913, p. 584.
91
92 PHILOSOPHY AND THE NEW PHYSICS
inertia and of the weight of internal- energy in the departures from Prout's law. This law states that the atomic weights are integral multiples of the same quantity. While it is reasonably exact, nevertheless the weights do present slight irregulari- ties with respect to this law. These departures would be caused by changes of internal energy through emission or absorption of radiation ac- companying the formation of atoms from the dis- integration of primordial elements, as seen in radioactivity or the inverse process of integration, not as yet observed, with the formation of heavy atoms. The sum of the weights of the atoms thus formed would differ from that of the atoms trans- formed, by a quantity equal to the change in energy divided by the square of the velocity of light. These departures are such that the energy put into play would be of the same order of magni- tude as that actually observed in the course of radioactive transformations. If, for example, the atom of oxygen resulted from the condensation of 16 atoms of hydrogen or four atoms of helium, it would be sufficient for an explanation of the atomic weight 15.87 being less than 16, to assume that this condensation is accompanied by a loss of energy only five times greater than that set free during the transformation of one atom of radium into radium D. The interest of such an explanation of the departures from Prout's law is to make the hypothe- sis of the unity of matter possible, that is, the hypo- thesis that all atoms are composed of one primordial
WEIGHT OF ENERGY 93
element or several such, and this can not be recon- ciled with these departures so long as the principle of the conservation of mass in chemical reactions is assumed.
Energy possessing a ponderable mass and the inertia of a body being nothing but the inertia of its internal energy, Newton's law expresses in reality the law of attraction of energy by energy. We shall see what follows from this for free radiation and consequently for luminous radiation.
Radiation propagated freely in a vacuum repre- sents, per unit volume, a certain energy density E and a certain electromagnetic momentum equal to E/V. It follows therefrom that it possesses an inert mass, defined as the momentum divided by the velocity, E/V2. If every inert mass implies the existence of a ponderable mass in proportion thereto, a light ray will have weight; it will be attracted by a mass situated in its neighborhood by virtue of Newton's law. It will be deviated in a gravitational field in proportion to the angle be- tween the direction of the ray and that of the force of attraction. Einstein1 has calculated the magni- tude of this deviation and, in 1911, arrived at the formula
2KM a =
where a is the deviation of the ray passing by a spherical mass M (for example the mass of a star),
1 Einstein, Ann. der Phys. vol. xxxv, 1911, p. 898.
94 PHILOSOPHY AND THE NEW PHYSICS
K the constant of gravitation, R the distance from the center of the sphere to the ray. For a ray passing in the neighborhood of the surface of the Sun, a becomes equal to .S3", which constitutes a quantity that can be measured, by observing, for example, the position of a star near the edge of the Sun at the instant of a total eclipse.
By virtue of Einstein's equivalence principle, luminous radiation in the interior of a system that undergoes an acceleration must behave like a pro- jectile; instead of describing a straight line it will describe a parabola. Observers enclosed in Jules Verne's cannon ball, which would fall with accelera- tion, could therefore by the aid of optical experi- ments detect the state of acceleration of the system, without otherwise knowing whether they should attribute it to the presence of a gravitational field or to the state of acceleration of their cannon ball.
The equivalence between the inert mass and the ponderable mass carries with it the equiva- lence of the effects produced on physical phenomena by a field of gravitation and those due to a suitable state of acceleration of the reference system to which they are referred. It must follow therefrom that the potential of gravitation acts on the passage of time and the dimensions of bodies in the same manner as acceleration. Now it follows from the Lorentz group that a body is the more contracted in the direction of its translation for observers 00 who see it pass and that the velocity of the phenomena that take place in it, measured by these same observers,
WEIGHT OF ENERGY 95
is the more retarded, in proportion as its velocity is more accelerated relatively to them. In the same way, a body will be the more contracted and the march of the phenomena, of which it is the seat, the more retarded, in proportion as the potential of gravi- tation of the place where it is found is increased. Two equal chronometers placed at unequal distances from the Sun will go at different rates, and the more distant one will run ahead of the nearer. Now a chemically defined molecule that, by virtue of its oscillations, emits a given spectral light constitutes a chronometer of atomic dimensions. If, therefore two identical molecules at positions of different gravitational potential are observed with the aid of a spectroscope, one on the surface of the Sun and the other on the surface of the Earth, the os- cillations of the second being more rapid than those of the first and consequently the frequency of the light that it emits being greater, it ought to be found that the ray emitted by the second is displaced in the spectrum, with respect to the ray emitted by the first, in the direction of the violet. Knowing the difference of the potential of gravitation at the surface of the Sun and at the surface of the Earth, it is easy to calculate that the difference in the wave lengths of the spectral rays ought to reach about Koo Angstrom unit, that is, one-millionth of a micron, which is a quantity accessible to experiment. It is remarkable that displacements of this order have actually been observed by Fabry and Buisson by comparing the Fraunhofer rays of the solar
96 PHILOSOPHY AND THE NEW PHYSICS
spectrum with the corresponding rays of a ter- restrial source.1
15. THE GENERALIZED PRINCIPLE OF RELATIVITY AND ELNSTEIN'S THEORY OF GRAVITATION.
The relation /* = E/V2 shows that the laws of the conservation of weight are the same as those of the conservation of energy. Now the weight of a body in a gravitational field changes when it is displaced in the field: it increases when the body is raised. To this increase in weight there must be a corresponding increase in energy equivalent, to the work expended against the weight in raising the body. This change of energy carries with it, in virtue of the formula M — E/V2, either a change of mass or a change ^of the velocity of light in the gravitational field at the point considered. This is the starting point of the new theories of gravi- tation according to the relativistic ideas.
G. Mie2 has developed a theory in which he , makes the mass but not the velocity of light depend on the potential of gravitation. He thus safe- guards the postulate of the constancy of the veloci- ty of light required by the relativity principle, but he is obliged to renounce the equivalence be- tween ponderable mass and inert mass established by Eotvos. G. Nordstrom3 has sought to maintain
1 Cf. Freundlich, Phys. Zeitschr. vol. xv, 1914, p. 369.
2 G. Mie, Ann. der Phys., vol. xc, 1913, p. 25; Phys. Zeitschr., 1914, p. 115 and 169.
3G. Nordstrom, Phys. Zeitschr., 1912, p. 1126; 1914, p. 375, 604; Ann. der Phys., 1913, p. 533, 856; 1914, p. 1101.
THEORY OF GRAVITATION 97
the postulate of the constancy of the velocity of light and the proportionality between ponderable mass and inert mass over the widest range, but he is compelled to assume a change in the length of bodies and in the rate of phenomena, as a function of the potential of gravitation, which does not agree with that predicted by the Lorentz group. Einstein also has resigned himself to abandoning the con- stancy of the velocity of light in a field with varying potential and to assuming that it varies with posi- tion according to the formula
(19) V m 70(l + Y
where <j> represents the magnitude of the Newtonian potential at the point considered. The Lorentz group is then applicable only to regions where the potential of gravitation is constant, or, what in virtue of the equivalence principle comes to the same thing, to systems in uniform translatory motion.
Einstein1 has sought a group of transforma- tions, having that of Lorentz as a special case, and such that the equations of the field of gravi- tation can be reduced to the form that those of a system without gravitation have, when referred to a reference system in a state of acceleration. The discovery of this group, for which he was indebted to the absolute differential calculus, created by
1 Einstein, Die formalen Grundlagen der allgemeinen Relativi- tats theorie (Ber. Berl. Ak., vol. XLI, 1914, p. 1030).
7
98 PHILOSOPHY AND THE NEW PHYSICS
Christoffel and developed by Ricci and Levi-Civita, has enabled him to generalize the relativity prin- ciple by extending it to any arbitrary motion of the reference axes whatever, thus bridging the fol- lowing serious epistemological gap, pointed out by Mach.
Psycho-physiology teaches us that our senses make known to us only relative states and changes of bodies, without ever revealing to us any absolute state or change. Thus our sight does not reveal to us the absolute shape and dimensions of a body, but simply the fact that a figure has such a shape and such a size relatively to another taken as com- parison term, this purely relative shape and size not being altered in any dilatation or any continu- ous transformation whatever of the universe. In the same way, we do not perceive Newton's absolute space, to which we might refer all moving objects, but merely the relative state of rest or motion of bodies. Similarly our thermal sensibility does not teach us anything about the absolute tempera- ture of bodies, but only about the changes in the states of affairs as regards heat losses between our skin and the surrounding medium. If, therefore, we wish to confine ourselves to the data of sense perception, we must not in natural philosophy speak of absolute velocity, acceleration or inertia.
Classical mechanics does not satisfy this require- ment. It succeeds in preserving the relativity of velocities,^but it concedes an absolute sense to the idea of acceleration and to that of inertia, con-
GENERALIZED RELATIVITY 90
sidered as capacity of resistance to acceleration of a body. According to these principles the motion of two masses isolated in space and sufficiently close to be able to exert actions on each other, would be governed by Newton's law of attraction, independently of the system of fixed stars to which it is referred; for: "It would be," says Euler, "a very strange proposition and contrary to a host of other dogmas of metaphysics, to say that the fixed stars influence the inertia of bodies."1 But, in fact, we perceive merely relative distances of bodies: hence we can only observe and define the relative velocities and accelerations of bodies, that is to say, the first and second derivatives of their distances. Consequently the inertia of a body can only be defined as its resistance to the relative accelerations which it experiences with respect to other bodies that do not participate in its state of motion. The inert mass of a body appears, therefore, as a relative quantity, which depends on the distribution of the masses about this body and on their state of rest or motion with respect to it: it will be greater, the greater the number of other masses in its neighborhood, which do not participate in its state of acceleration; it will disappear in the opposite case. A body is inert because it is surrounded by other bodies. Its inertia results from the mean action of all masses distributed in the universe, so that, con- trary to Euler's assertion, the fixed stars do deter-
1 Euler, Reflexions sur 1'espace et le temps, p. 328.
100 PHILOSOPHY AND THE NEW PHYSICS
mine in part the inertia and the motion of the Earth. The principle of inertia thus loses all absolute sense: it becomes a relative and statistical principle.
To put our physical conceptions into accord with the data of our sense perception, and to set natur- al philosophy free from metaphysical entities that encumber it, such as the ideas of absolute space and privileged axes, it is desirable to state the laws of physics in an intrinsic language, independent of any reference system, just as Euclid's geometry as compared with Descartes' is an intrinsic language free of the consideration of coordinates. For this purpose it does not suffice to make the form of the physical laws independent of the state of uniform translation of the coordinate axes; it must be made independent of any motion whatever of these axes; or, in Minkowski's geometrical language, it must be possible to refer them in four dimensional space to a system of oblique axes or a system of curvi- linear coordinates, as well as to a rectangular system formed by rectangular coordinates.
Einstein's will be the glory of having succeeded in satisfying this last condition, by giving to the physical laws a universally invariant form, or, as we say nowadays, co variant for any change of coordi- nate axes. To accomplish this generalization of the relativity principle, Einstein started from the equivalence principle, deduced by him from Eotvos' experiments on the strict proportionality of the inert mass and the gravitational mass of bodies, which
GENERALIZED RELATmiS: 101
states the impossibility of distinguishing the effects of a field of gravitation from those of a field of mo- tion. The effects produced by a field of gravitation can always be interpreted by a state of acceleration of a body removed from any field of force, and con- versely. From this it follows that the existence of a field of gravitation in empty space is purely re- lative; it depends on the problem, insoluble by ex- periment, of finding whether the system from which it is observed is at rest or in accelerated motion; the real field for a system considered at rest will be fictitious for other systems in motion which the gen- eralized relativity principle declares equivalent to the first. Einstein then shows that the form of the physical laws may be rendered independent of any system of privileged axes, provided the quanti- ties characteristic of the field of gravitation are made to appear in the physical laws; or, more pre- cisely, provided these laws be considered as relations between the quantities characteristic of material phenomena and the quantities characteristic of the field of gravitation, these quantities being tensors, and matter designating everything that is super- imposed on the field of gravitation. These rela- tions, being invariant for every transformation of the reference system, are intrinsic equations, ex- pressed with the aid of tensor equalities, from which every coordinate system has disappeared.
This theory involves remarkable consequences.
It appears at first as a universal relativism ex- pressed with the aid of an absolute calculus. The
102 I'HJLOSOPHi' AND THE NEW PHYSICS
metrical properties of space, the kinematical and dynamical properties of mechanical systems, the physical properties of any region whatever of space vary according to the point of view of the observer. These properties depend, in fact, on the field of gravitation and the state of motion of the system from which they are observed.
The influence of gravitation is exerted on every physical process, on all matter including the electro- magnetic and luminous field. Conversely, it has its origin in every region of the universe where the material tensor is different from zero. Since the material tensor corresponds to a reality, gravitation also involves a real element with which a fictitious field of gravitation, equivalent to an arbitrary motion in empty space, may always be 'combined.
Empty space is not the absolute and infinite void of Newton's followers: it is the pure field of gravitation on which no matter is superimposed. From the physical point of view there is no amor- phous void, endowed with pure receptivity, in which material points could be imagined, attract- ing or repelling one another according to certain laws, like Newton's law, and thus communicating absolute accelerations to one another. We can not speak of empty space, except where there is a field of gravitation, not merely coexistent with this space, but veritably the creator of space, of its metrical properties, and, we may say, its extent. The world must be thought of no more as an as- semblage of bodies lost in an infinite void, but as
GENERALIZED RELATIVITY 103
systems of bodies and of electromagnetic or lumin- ous fields, superimposed on gravitational fields of finite dimensions. The absolute, void, amorphous, and infinite space of Newton vanishes like other idola fori; and Kant's antinomies advanced in regard to it are abolished as referring to a pseudo-problem.
To fathom the meaning of this universal rela- tivism, it will be well, at this stage, to contrast the ideas of Einstein and those of Lorentz.
For Einstein there is no infinite void, no motion- less ether, no uniform course of time, and conse- quently there are no privileged reference systems and clocks; there is no region of space enjoying absolute physical properties. The Lorentz con- traction is not true in the sense that it corresponds to an absolute deformation, a body not having any shape except relatively to another. It is a recipro- cal semblance of reality, arising from local time, from the fact that the clocks of observers connected with one body and those of observers in motion with respect to it, do not go at the same rate. The distinction between reality and semblance vanishes: there are only relative truths, science being unable to establish anything but comparisons, by ascertaining coincidences and by comparing colors (frequencies of radiating sources serving as clocks). The only absolute reality that it can attain consists of the laws of physical phenomena, expressed by intrinsic equations by the aid of tensor equalities, the form of which is independent of every system of space and time coordinates. If
104 PHILOSOPHY AND THE NEW PHYSICS
God exists and it should please him to recount the history of the world in the extrinsic and artificial language of time and space, it would be merely by an arbitrary decree of his free will in a complete state of intellectual indifference that would make him choose, once for all, a system of space coordinates and a clock (that is, a radiating source connected with this system and consequently regarded as at rest). Having thus fixed arbitrarily the meaning of simultaneity and order of sequence of events, he could establish their universal chronology.
The ordinary distinction between semblance and reality would, on the contrary, exist for an infinite intelligence, if there be, as Lorentz is in- clined to believe, an ether which is motionless but otherwise of a nature unknown to us, but it would be very different from the mechanical ether of Faraday, Lord Kelvin, and Sir Oliver Lodge. There would, therefore, be for an omniscient intelli- gence a privileged reference system and clock, whatever the system of axes, provided it be con- nected with the ether, and whatever the clock at rest, provided it be free from the influence of gravitation. This omniscient mind could with propriety speak of absolute motion, order of succes- sion, and synchronism. The Lorentz contraction and the retardation of clocks would appear to him as physical phenomena, due to the connections between matter and ether, by which an action is exerted by the latter on the former. But this privileged set of three reference axes and this course
GENERALIZED RELATIVITY 105
of absolute time would be concealed from us forever as a consequence of the same action of the ether on the bodies, which would be precisely such as to prevent us from discovering their absolute motion. The observers 00 would have no better ground for saying they were at rest and recording true time than the observers Oi. Any experimenter whatever could always explain what he observes by supposing either that he is at rest in the ether or that his laboratory is traversed by an ether current which produces the effect of shortening his instruments and retarding his clocks, or that there is no ether at all but it is the motion of a rod or a clock in his laboratory that produces the shortening of the one and the retardation of the other.
According to the pragmatist theory of truth the two concepts, that of Einstein and that of Lo- rentz, are equivalent, since they are equally in accord with phenomena.
On one hand, Einstein's concept, being in accord with the data of psycho-physiology, by respecting the conditions of knowledge imposed by our sense perceptions, by practising the scholastic adage non sunt multiplicanda entia sine causa, by reliev- ing physics of metaphysical entities that encumber it, by delivering us from a swarm of pseudo-prob- lems, is strictly positive, more economic than that of Lorentz, and unassailable in itself; on the other hand, that of Lorentz, by maintaining the ether, safeguards our old habits of thought, satisfies our craving for the absolute, sets our minds at rest,
106 PHILOSOPHY AND THE NEW PHYSICS
establishes a bond of union between the physics of yesterday and that of today, and falls into the category of explanatory theories, since the shorten- ing of bodies, the retardation of clocks, the con- stancy of the velocity of light in a gravitational field of constant potential are explained by the connec- tions of matter and ether. But the existence of the ether is forever rendered problematic by virtue of the relativity principle. Its nature is unknown to us, since the mechanical properties with which Faraday, Helmholtz, Lord Kelvin, Sir Oliver Lodge endowed it have already condemned it. It does not seem to represent necessarily the anticipation of future observations and experiments, like Pasteur's theory of microbes or the atomic theory; it is merely a method of exposition, a figurative hypothesis to uphold the spirit of abstraction, which, useful for the minds that Duhem called broad and weak, be- comes superfluous, cumbersome and tiresome for the minds that he names narrow but profound.
Whatever the outcome may be, Einstein's re- sults as to inertia, the weight of energy, and the relativity of phenomena seem permanent acquisi- tions. The relativity principle represents one of the norms of physical research which limits the field of our investigations and determines, in part, the form of the equations of physics.
16. ASTRONOMICAL VERIFICATION.
The philosophical importance presented by Ein- stein's theory of gravitation, based on the general-
ASTRONOMICAL VERIFICATION 107
ized relativity principle, should not lead us to forget the interest attached to its practical results, the most sensational of which, independently of those already indicated, is the calculation of the secular anomaly of the perihelion of Mercury.1
To account for it, Tisserand,2 as is well known, proposed to replace Newton's law by a more general formula, similar to Weber's electrodynamical law. Of all the theories proposed since, none satisfies the mind like Einstein's theory of gravitation.
According to this theory, there exists no absolute invariant appertaining to a material point, that is, the exact equivalent of inert mass or of gravitational mass in classical mechanics. We have here to do with a tensorial theory in which the numbers Gab, characteristic of the field of gravitation, and the numbers Tab, characteristic of the field of inertia (ordinary matter and electromagnetic field), form a covariant tensor. It is possible to attach to these tensors two relative scalar invariants, G and T, which may, if desired, be considered by extension as measures of a gravitational mass and an inert mass.
Let us now consider what follows from this, tak- ing account of the inertia of energy, in the case of the attraction of Mercury by the Sun.
In the study of the motion of a material point attracted by gravitational masses, a quite natural simplification consists in treating of this point as "a
1 Einstein, Erklarung der Perihelbewegung des Merkur (Ber. Berl. Ak., vol. xlvii, 1915, p. 830).
2 Tisserand, Traite de Mecanique celeste, vol. IV, p. 500.
108 PHILOSOPHY AND THE NEW PHYSICS
test body" placed in a field of gravitation due to other masses and not modifying perceptibly the field of gravitation of these masses. Given two bodies, an attracting body and an attracted body, the inert mass of the attracting body (and the same holds for the attracted body) is increased in pro- portion with the increase of the potential of gravi- tation due to other masses. Now the Sun, con- sidered as an attracting mass occupying a sphere of radius a, is surrounded with a field of gravitation, so that it is immersed in gravitational energy. This energy, always positive so far as concerns physical reality, is negative in the sense that in order to produce an attracting mass by condensation of ele- ments coming from infinity, negative work must be expended. It results from this that there is less energy in the neighborhood of an attracting mass than in the absence of all mass. If, therefore, we attribute inertia to the energy of gravitation, as to all the other forms of energy, it follows that, for a point situated at a very great distance from the Sun, the mass of the latter will be
M = Mo - AMo,
AM0 representing the correction term due to the in- ertia of the total field of gravitation. For a point situated on the orbit of Mercury the effective solar mass will be equal to
M + AM,
if we denote by AAf the supplementary mass due to the inertia of gravitation outside of the orbit of Mercury. It will thus be larger than the principal mass M.
ASTRONOMICAL VERIFICATION 109
Taking account of these corrections, it is found that the elliptical motion is transformed into a pseudo-elliptic motion with a progressive advance of the perihelion. Applied to the case of the planet Mercury, the calculation indicates a secular motion of the perihelion equal to 43". Since the observa- tions give 45", this aggreement must be regarded as the more remarkable, because it is obtained, with- out any supplementary hypothesis, by the applica- tion and simplification of the general equations furnished by the theory.
A hardly less remarkable confirmation of the theory of inertia of energy and of Einstein's theory of gravitation is obtained from the deviation of light rays in the neighborhood of the Sun. On account of both the Newtonian attraction exerted by the Sun on a light ray propagated in its neighborhood and of the curvature of space produced by the mass of the Sun according to Einstein's gravitational equations, the path of the ray should not be recti- linear but curved towards the center of attraction, with a total deviation given by the expression
4KM
whic,h is just twice the value given by equation (18), p. 93. In consequence of this it is possible to predict that a star seen near the Sun should suffer an outward deviation equal to 1.74" and varying inversely as the distance from the center of the Sun for the more distant stars, as compared with the position that it occupies on the celestial sphere in the absence of the Sun, since the star is seen in the line of its ray, i.e., in the direction of the tangent to the path of the ray. This deduction has been success- fully verified by photographing the same region of the sky in the absence of the Sun and during a total solar eclipse. The negatives obtained by the two British expeditions undertaken for this purpose in the zone of the total solar eclipse of May 29, 1919, have given a mean deviation of 1.79" + .03". This agrees remarkably well with the value predicted by Einstein.
CHAPTER VII ' THE STRUCTURE OF ENERGY
17. THE SUCCESS OF THE ELECTRONIC THEORY; THE EXPLANATION OF THE RELATIONS OF MATTER AND RADIATION.
We have just seen that a body is inert and has weight in proportion to the energy it contains, so that the concept of matter is subsumed under the more general one of energy, and that the princi- ple of the conservation of mass becomes merged in that of the conservation of energy. Energy, as Ostwald would have it, becomes the only existing reality, into which are absorbed the ether and the numerous imponderable agents to which the physics of the beginning of the nineteenth century was partial. Nevertheless it appears that energy pre- sents itself essentially in a double aspect: in the form of resinous (negative) electricity endowed with a corpuscular structure and in the form of free radiation. In its first aspect it is made up of grains of electricity, capable of moving with veloci- ties ranging from 0 to 7, its aggregates constituting atomic and molecular structures, relatively stable and with astonishing vacant spaces, and appearing to our senses in the form of continuous bodies. In its second aspect it appears as made up of transverse
no
ELECTRONIC THEORY HI
waves, infinitely expansible and divisible, sweeping through all space with the uniform velocity of light. In the first case it takes the name of matter; in the second that of radiant energy.
The specific character of a portion of matter must no longer be sought in its mass and its weight, since radiant energy is likewise inert and endowed with weight, and mass is no longer an invariable scalar quantity, but takes the character of a tensorial quantity, which is unsymmetric and variable as a function of the velocity and the internal energy of bodies. That character must be sought in the number and nature of the primordial elements which constitute matter.
These elements, revealed in atomic changes be- yond the domain of chemistry, are the electrons and the positive remainders, the latter having a struc- ture as yet unknown and presented to us in the form of the positive kernels of helium atoms. These elements alone remain invariable throughout the changes that matter undergoes and can serve to define it.
What are the relations between matter thus characterized and radiation, and what is the mechanism of their exchanges of energy? It was on this point that the metaphysical pseudo-problem of the action of imponderable on ponderable arose, and it is to it that the electronic theory of matter claims to give a satisfactory answer.
Let us, for this purpose, consider again the case of a charged particle in quasi-stationary motion
112 PHILOSOPHY AND THE NEW PHYSICS
and let us see what happens when it undergoes an acceleration.
The wave system of the particle in quasi-station- ary motion is formed by the system of radial electric lines of force and of circular magnetic lines of force that it carries with it. It may be, moreover, considered as the aggregate of the electromagnetic waves of velocity, emitted at differ- ent instants in its course, and centered on its former positions, which envelop one another, so that the electromagnetic field produced by the displacement of the particle is determined, not by the instantane- ous state of the particle, but by all its former states. The energy of the field thus created is localized nearly entirely in the immediate neighborhood of the particle, because the intensity of the field varies inversely as the square of the distance, and rapidly approaches zero as the distance from the particle becomes greater. The waves of velocity which constitute the wave system of an electron do not correspond to any energy radiated to a great dis- tance, since they vanish at infinity. They represent kinetic energy which accompanies the electron in its displacement, preserving for a constant velocity a fixed distribution around it. As space exerts^no viscous action, no external influence is requiredj to conserve this energy in motion, and the projected electron moves indefinitely with the same velocity in conformity with the principle of inertia, so long as no external cause modifies its state of motion and produces acceleration.
ELECTRONIC THEORY 113
Let us examine what happens in the latter case. In the first place, the magnetic energy of the
e2 wave system, in conformity with the relation ^- v2
ott
which is valid for small velocities, increases or diminishes by a quantity called energy of change, which corresponds to a reorganization of the lines of force of the wave system. In the second place the change of velocity determines the appearance of a spherical wave of acceleration, the radius of which increases with the velocity of light and which remains centered about the point where the electron was at the instant of the emission. This wave corresponds at every point to the passage of an electric field and of a magnetic field, situated in the plane tangent to the wave and perpendicular to each other in this plane. These fields represent a localization of electric and magnetic energy equal per unit volume, and this has the effect of giving to the wave of acceleration all the characteristics of free radiation.
The fields present in the wave of acceleration are superimposed on those of the velocity waves. Since the latter, varying inversely as the square of the distance, diminish much more rapidly than the former, which vary simply inversely as the distance, at a sufficient distance from the electron only the acceleration wave will exist. The energy thus radiated to infinity with the velocity of light, by the radiation which the electron emits during the time dt, is proportional to the square of the
114 PHILOSOPHY AND THE NEW PHYSICS
charge and the acceleration according to the relation :
9 /p2-,2
(20) \ *± dt
where, in E. M. U., v denotes the acceleration and V the velocity of light.
This radiated energy represents the intrinsic energy of the acceleration wave. It is borrowed from the external actions which modify the velocity of the electron. But it represents only a small part of the energy which the acceleration wave car- ries at the start when leaving the center. The latter is the agency by which the electron reorgan- izes its wave system with the velocity of light, that is to say, by which the field of the electron receives the additional magnetic energy necessary for the increase of the kinetic energy associated with an increase of velocity, or restores the excess magnetic energy when its velocity diminishes. It is through the acceleration wave that the magne- tic energy of change corresponding to an increase or a decrease of velocity is distributed to each part of the wave system or restored in the form of work, this magnetic energy being in the first case bor- rowed from external actions and in the second case restored in the form of work done against the retarding actions. The energy radiated from the acceleration wave represents as it were a necessary shrinkage, a tribute paid to the auxiliary wave for the service rendered.
ELECTRONIC THEORY 115
The emission of radiation is always connected with acceleration of electrified particles. It is the difference in the conditions of electrons present in matter that enables us to explain the emission of different kinds of radiation and the phenomena traceable to them such as mutual induction and self-induction.
Let us consider first these last two phenomena. Conduction currents are reducible to convection currents: they correspond to the mass motion of the free electrons of the metallic or electrolytic conductors under the influence of a potential dif- ference, the positive electrons tending to go in one direction, the negative in the other. If two wires are placed side by side, such as the two neigh- boring windings of a transformer, the passage of the inducing current corresponds to the circulation, in the first wire, of free electrons of which it is the conductor. The intensity of the current is proportional to the ensemble velocity of these electrons, and any variation of intensity will cor- respond to an acceleration experienced by the elec- trons. At the moment when the current varies, there will be an emission of acceleration waves by the electrons of the wire. The superposition of these waves at a point of the neighboring wire will give rise to. the appearance of an electric field parallel to the wire and in the opposite direction to the current if the intensity increases, in the same direction if it diminishes. In the case when this point is situated inside of the neighboring wire, the electric field thus created will determine a
116 PHILOSOPHY AND THE NEW PHYSICS
current, corresponding to the existence of an elec- tromotive force, the direction of which is given by Lenz's law.
The phenomena of self-induction are explained in the same manner. If the intensity of the current increases, the acceleration waves emitted by the electrons when their velocity varies will add them- selves together in the circuit as well as outside, and produce there an electric field, directed oppositely to the current, by the agency of which the energy necessary for the growth of the magnetic field encircling the circuit is borrowed from the electro- motive source that produces the current. If the current diminishes, the accelerations of the positive particles being opposed to the direction of the cur- rent, the radiated waves produce in the conductor an electromotive field in the same direction, by the agency of which the excess of energy of the magnetic field, which encircles it, will be returned to the circuit.
Conduction currents are at basis only con- vection currents; the explanation of their ' self- induction must account for the inertia of an elec- trified particle in motion. This inertia is due to the production of a magnetic field having its source in the displacement of the particle, and to the fact that the energy present in this field must vary with the velocity by the agency of the acceleration wave. The force of inertia that the particle opposes to the change of velocity comes from the action, on every element of its charge, of the
ELECTRONIC THEORY 117
electric field present in the acceleration waves emitted by the other elements of this charge.
The character of the free radiation, which mani- fests itself in the form of Hertzian waves, of light, of Rontgen rays, and of thermal radiations, depends on the nature of the electrons that are accelerated and on the circumstances of their acceleration, sudden, continuous, or periodic.
A sudden acceleration takes place on the arrest b^- an obstacle of the cathode particles or the ]8 particles, projected with velocities comprised between 20.000 km. and 290,000 km. per second. The radiation consists in a sudden pulsation, a kind of electromagnetic noise, emitted at the moment of the arrest of the particle, its thickness being equal to the product of the velocity of light and the dura- tion of the shock, that is to say, it is of atomic dimensions. This extreme thinness explains the penetrating power as well as the absence of refrac- tion of X-rays.
The acceleration is periodic in the case of the electrons that move in closed curves around a posi- tive center inside the atoms. From it there results a continual emission of regular waves, the period of which is equal to the time of revolution of the electrons. This emission corresponds to light of a definite wave-length, like that which constitutes discontinuous spectra.
If the acceleration is due to the chaotic agitation of the free electrons of incandescent sources, there result from it radiations of every wave-length,
118 PHILOSOPHY AND THE NEW PHYSICS
which form the continuous spectra of incandescent bodies.
If the radiation is due to the accelerations which the free electrons of an opaque metallic enclosure undergo as a consequence of the thermal agitation of the molecules that strike them, it constitutes the heat radiation inside of this enclosure, called black radiation.
The presence of moving electrons in matter ex- plains not merely the emission of radiation but also the laws of its propagation through material media and those of its absorption.
The absorption of radiation by matter is due to the fact that the electrons present in the matter are, under the action of the alternating fields of the incident electromagnetic waves, set into vibra- tion in agreement with them. In consequence of the motions thus" produced, they strike the neigh- boring material molecules, the kinetic agitation of which they increase at the expense of their vibra- tory energy. It is by this mechanism that the electromagnetic energy of the absorbed radiation is changed, through the agency of the shocks, into thermal energy, that is, into kinetic energy of cha- otic agitation of the molecules of the absorbing body. It is because of this that a body which ab- sorbs radiation is heated at the expense of the in- cident electromagnetic energy, and that its mass is increased by all the Maupertuisian mass of the absorbed radiation. In the case of selective ab- sorption it is not the free electrons but the elec-
ELECTRONIC THEORY 119
trons in a regular periodic motion within the atoms which absorb, by a resonance phenomenon, the radiations of a period equal to that of their motions.
In conclusion, the electronic theory of matter appears to give an account of the mechanism of the relations existing between matter and radiation. By the success of this step, the electronic theory is correspondingly strengthened. It reinforces the view that at basis there exists only electro- magnetic energy, which presents itself in two different aspects: narrowly circumscribed and concen- trated in space in the form of elementary charges in motion, the velocity of which may vary from 0 to V; and in the form of radiation sweeping through all space with the rapidity of light — in the form of grains of electricity and in the form of transverse waves indefinitely expansible and divisible — stabi- lized in structures of varied and definite architec- ture, or free to lose itself by radiation to infinity.
The success of the electronic theory and of the mechanics of relativity reduces finally the prim- itive pluralism of ponderable matter and imponder- able agents to the dualism of negative electricity (possibly also positive), endowed with a corpuscular structure and of electromagnetic radiation formed of continuous waves, both of them having inertia and weight.
120 PHILOSOPHY AND THE NEW PHYSICS
18. CHECK TO THE ELECTRONIC THEORY: BLACK RADIATION AND THE QUANTUM THEORY.
The electronic theory has nevertheless reached its limits: it has been revealed as incapable of giving an account of the law of the distribution of energy in the spectrum of a black body and of the dimi- nution of the specific heats of solids at low tempera- tures, while the classical theory of radiation, due to Maxwell and Hertz, suffered shipwreck from certain phenomena of physical optics. To bring theory and experiment into accord new supple- mentary hypotheses had to be introduced. Are they going to modify the previously acquired results, to give support to a dualism between matter and radi- ation? Far from it, on the contrary they weaken it, for they amount to nothing less than endowing radiant energy with structure, after it has already been endowed with inertia and weight. In fact they finally lead to the view that radiation is not a system of waves infinitely expansible and divisible, propagated by a hypothetical medium, the ether, but a projection of matter in empty space with the velocity of light, in discrete units, emitted and ab- sorbed by bodies in a discontinuous manner.1
In order to understand the point at issue, we must begin by defining what black radiation is.
Any body that is not at the absolute zero emits energy in the form of thermal radiation. When
1 Cf. La thSorie du rayonnement et les quanta (Rapports et dis- cussions de la reunion de Bruxelles, publics par P. Langevin et M. de Broglie, Paris, 1912).
BLACK RADIATION 121
unequally hot bodies are placed in an enclosure observation shows that they finally come to thermal equilibrium, all bodies in the enclosure reaching the same temperature. This equilibrium can have been attained only by exchange of radiation between the bodies. When it is attained the bodies do not radiate less, but each of them acquires by absorption as much as it expends in emission. This result has led KirchhofT to state the following law on the sub- ject of thermal radiation: the emissive power of a body for every kind of radiation and at any tem- perature is equal to its absorbing power. In the case of an ideal black body, which completely ab- sorbs the radiations that strike it, the absorbing power is equal to unity. We are thus led to seek for the law of the distribution of radiant energy in the spectrum emitted by a black body at a given temperature.
In nature there exists no rigorously black body, no body that does not reflect or diffuse in part the radiation that it receives. Kirchhoff has enabled us^ to get over the difficulty by supplying the means of realizing a black body artificially. Let us consider an opaque enclosure and any radiation whatever that is propagated inside it: it will strike the wall a first time and will be partly absorbed; the remaining radiation, reflected or diffused, will again strike other parts of the wall and be there absorbed in the same proportion, so that the radia- tion which escapes absorption will tend rapidly towards zero. Such an enclosure possesses there-
122 PHILOSOPHY AND THE NEW PHYSICS
fore an absorbing power equal to unity for any radiation: it realizes the black body.
Let us plunge this enclosure, after having evacu- ated it, into a bath at constant temperature, so as to maintain its walls at constant temperature. Ex- periment proves that the space inside this enclosure is isothermal, that is, that a thermometer placed at any point whatever inside of this enclosure, will finally indicate the same temperature. Any action on the thermometer placed in 'the vacuum must be exerted by radiation. In the region where the thermometer is placed, undulations arriving inces- santly from different points of the enclosure become superimposed and form systems of stationary waves of definite frequency, adapted to the distances ex- isting between two walls and realizing a permanent state of extremely rapid changes, the details of which are beyond our scale of time measurement. This is what is expressed by saying that the thermal equilibrium realized in the interior of the enclosure is a statistical equilibrium. This equilibrium is real- ized for every kind of radiation separately, and. it is characterized by the amount of energy that is contained per unit volume in the space within the enclosure and by the distribution of this energy between the different wave lengths of the systems of stationary waves. It is precisely this energy density and this distribution of energy in the spec- trum that has to be determined for black radiation.
This we succeed in doing by the following method. Experiment proves that the temperature indicated
BLACK RADIATION 123
by the thermometer is independent of the nature, the shape and the dimensions of the walls of the enclosure. It follows therefrom that all the direc- tions are equivalent and that each cubic centimeter of the enclosure contains the same density of radiant energy. If we consider in the enclosure a plane closed contour of one square centimeter surface, the amount of radiation that passes through this contour in one second will have a definite value, proportional to the density of the energy of radi- ation in equilibrium at this temperature. To study the composition of the radiation of the isothermal enclosure, which is identical with that of a black body and on that account is called black radiation, it will be sufficient to make a small opening in the enclosure, so as to collect the radiation that comes out of it and that we know to be identical with the radiation which in the isothermal enclosure traver- ses at every instant a section of equal contour. By this device we can deduce Stefan's law for the density of radiated energy: the density of the total energy radiated per unit time is proportional to the fourth power of the absolute temperature. By receiving the radiation in a dispersing apparatus it has been found, as regards the distribution of energy in the spectrum, that at a given temperature the energy radiated shows a maximum for a certain definite radiation on both sides of which it decreases rapidly. When the temperature rises the energy density of each simple radiation always increases, but the maximum of intensity is displaced towards
124 PHILOSOPHY AND THE NEW PHYSICS
the shorter wave-lengths, that is, for the visible part of the emitted spectrum, from the infra-red towards the ultra-violet. The curve of intensities becomes displaced along the spectrum, mounting more and more, at first slowly, then with an extreme rapidity the higher the temperature rises. Wien has discovered the law of this displacement as a function of the wave length for a given temperature, and this determines the distribution curve of the total energy of the spectrum of a black body: the wave length corresponding to the intensity maxi- mum decreases inversely as the absolute temper- ature, while the intensity of the maximum increases in proportion to the fifth power of the absolute temperature.
It remained to express in a single empirical formula the distribution in the spectrum of the intensities of the. black radiation for all the tem- peratures observed: this is what Max Planck has succeeded in doing, and it is the physical interpreta- tion of this formula that is the starting point of the quantum theory.
In the isothermal enclosure thermal equilibrium exists between the matter of the walls and the empty space (or ether) in which there is black radi- ation. The temperature of the walls is due to the internal agitation of the molecules that constitute them; the temperature of the empty space is due to the stationary waves that exist between the walls. This thermal equilibrium between matter and radiation is realized by the mechanism previously
BLACK RADIATION 125
described. The molecules of the walls of the enclosure in their kinetic agitation hit the free electrons existing in the walls. As a consequence of these shocks the electrons undergo accelerations that determine a radiation of all wave lengths, the electromagnetic energy of which is borrowed from the thermal energy of the molecular agita- tion: the temperature of the empty space is increa- sed at the expense of that of the matter.
Reciprocally, and by a mechanism inversely symmetrical, the electrons, being subjected to the action of the radiations, vibrate in resonance with them. In consequence of the oscillations thus star- ted they strike the neighboring molecules, the ther- mal agitation of which increases : the temperature of the matter is increased at the expense of that of the empty space. There is thermal equilibrium be- tween the matter and the radiation, when there is equality between the exchanges of energy brought about by means of the electrons.
The aggregate of the material molecules, the electrons, and the various stationary waves that exist in the empty space, form a system in statistical equilibrium to which can be applied the laws of statistical mechanics, in particular the theorem, discovered by Maxwell, correctly demonstrated by Boltzmann, which siims up the properties of such systems. This is the law of the equipartition of energy. It states that in a thermal system the mean kinetic energy is divided equally among all the degrees of freedom of the system.
126 PHILOSOPHY AND THE NEW PHYSICS
By degrees of freedom of a system are meant the different motions that it can assume consistently with the connections to which it is subject. For example, a material point can move along three axes, it has three degrees of freedom; a sphere can undergo a translation parallel to each of these three axes and a rotation about these axes, it has six de- grees of freedom. The molecule of a monatomic gas, like argon, is supposed to have three degrees of freedom; a molecule of oxygen five; a triatomic mole- cule six, three degrees of translation and three de- grees of rotation. If the law of equipartition is applied to a gas in thermal equilibrium, the vis viva which, on the average, each molecule" will have, is proportional to the number of its degrees of freedom : if, in the statistical equilibrium, a molecule of argon possesses at a certain temperature the vis viva 3, a molecule of oxygen must possess the vis viva 5.
This law, which results from the Hamiltonian form of the equations of dynamics, must be applicable to the statistical system constituted by the material walls of the enclosure and the black radiation. It will be sufficient to count the number of the degrees of freedom of the system to predict, by this law, the most probable spectral composition of radiation in the state of thermal equilibrium, the realization of which is a necessary physical consequence of Carnot's principle. We have to consider on one hand the molecules of the material walls, on the other hand the various systems of stationary waves possible in the empty space. Let N be the number of the
BLACK RADIATION 127
material molecules contained in the walls of the enclosure; the number of their degrees of freedom being equal to 6, there will be, for the matter, as a consequence of its discontinuity, only a finite num- ber of degrees of freedom, namely 67V. Let us con- sider on the other hand the empty space. It has an infinite number of degrees of freedom, for there is an infinite number of systems of possible stationary waves, the wave-lengths of which lie between oo and 0. Hence, if the law of equipartition among all the degrees of freedom is applied, the energy will be found entirely in .the empty space and none would remain for the matter : equilibrium would cease to be possible, or rather a single state of equilibrium only would be possible, that in which the matter is at the absolute zero. Moreover, the energy received by the space must be equally apportioned among its degrees of freedom, infinite in number. Whatever might be the amount of the total energy, that assign- ed to each degree of freedom would be zero, at least if the quantity of energy at our disposal is not in- finite, which has no physical meaning. This would be no longer the case if it were assumed that the length of the luminous oscillations can not descend below a certain limit X0. This would bring us back to assuming an ether and endowing it with structure. In fact, what limits the periodic motions that can be propagated in a given medium is the necessity that their wave-lengths be appreciably larger than the scale of the structure of the medium. The wave- length of sounds given by a vibrating cord must be
128 PHILOSOPHY AND THE NEW PHYSICS
greater than the mutual distance of the molecules that constitute the cord or the sounds would not exist. In the same way the seismic waves that traverse a continent have reality only for an observer whose horizon extends beyond the region of local variations. Similarly, if a structure of the ether ex- ists, we can no longer speak of infinitely small wave-lengths in the mathematical sense of the word : the smallest wave-lengths that it would be possible to assume are those for which the magnitude X is near the number that measures the distance between two ether molecules. The number of degrees of free- dom of the ether would be limited in number, and thermal equilibrium between matter and ether then becomes possible.
This idea has the disadvantage of being based on the hypothetical existence of the ether, and of an ether endowed with discontinuous structure, which is not readily conceived. But, moreover, it is ineffective, for it leads to predictions at variance with experiment. In fact, the number of systems of stationary waves the wave-length of which lies between the limits X and X + d\ is greater the smaller X is. The result would be that the degrees of freedom of the smallest wave-lengths would tend to appropriate all the disposable energy, which would be dissipated in extremely short radiations, and this is contrary to experiment. Thus the preceding hypothesis leads to a wrong law, formu- lated first by Lord Rayleigh and Jeans as a conse- quence of the equipartition of energy, that the
QUANTUM THEORY 129
energy radiated for a given wave-length is propor- tional to the absolute temperature and varies inversely as the fourth power of the wave-length.
To escape from these difficulties, Planck has put forth a radical hypothesis, which has the merit of being free from the consideration of any hypothe- tical medium and considers nothing but the only positively accessible reality, energy. The formula that he has proposed for representing the distribu- tion of energy in a black spectrum amounts to sub- stituting discontinuous series of elements, the sum of which remains always finite, for an integral which occurs in the mathematical expression of Lord Rayleigh's law and which has the disadvantage of becoming infinite. He interprets this discontinuity, not by hypotheses on the structure of the medium in which the radiant energy moves, but by hypo- theses on the absorption and emission of this energy v According to him the exchanges of energy between radiation and matter cannot take place in a con- tinuous manner in any proportion whatever, for this introduces an infinite number of degrees of freedom, but they must take place in a discontinu- ous manner in definite proportions. The electric resonators (free electrons) through the agency of which these exchanges are realized can only absorb or emit radiant energy in a discontinuous manner, by sudden jumps, according to integral multiples of elementary quantities, indivisible energy atoms called quanta. These quanta, which fix the lower limit of intake and output of energy, are not the
130 PHILOSOPHY AND THE NEW PHYSICS
same for all the resonators: they are inversely as the wave-length (or the period of their oscillation) and connected with the frequency v according to the relation
(21) q = hv
where h represents a universal constant. As a con- sequence of this hypothesis the resonators of short period can absorb and emit energy only in large mouthfuls, while the resonators of long period can swallow and give it up in small morsels. It re- quires a large amount of disposable energy to rouse a resonator of short period, so that the resonators of this kind will have a chance to remain at rest, especially if the temperature is low. By this means the noxious role of the wave-lengths near zero, the presence of which made the equilibrium impossible, is eliminated. Thereby also the fact is explained that there is relatively little light of short wave- length in black radiation, which is in accord with Planck's empirical formula. On the other hand, the discontinuity in the intake and output of the energy, which is progressively accentuated and be- comes enormous for short wave-lengths, tends to disappear in the region of large wave-lengths, where, as the result, Lord Rayleigh's law is found again to hold.
Let us consider our isothermal enclosure and see what happens when the temperature is progres- sively raised, starting from the absolute zero. At first the molecules are motionless and, as it were,
QUANTUM THEORY 131
anchylosed by the frost, the thermal energy is zero, the resonators are mute. If the temperature is raised, the resonators whose quantum is the smallest will, because of the incipient molecular agitation, begin to vibrate, and the first radiations will appear in the infra-red which is^far removed from the visi- ble spectrum. Step by step the other resonators will begin to be agitated and the spectrum will be extended towards the luminous radiations, then towards the ultra-violet, in conformity with Planck's law and with what experiments reveal to us. The quantum increasing indefinitely as X tends towards 0, the radiations corresponding to the limit will never appear.
Planck's theory seems thus to imply that the Structure of energy is discontinuous.; Every reso- nator can emit or absorb only a whole number of grains of energy. The value of this grain depends uniquely on the frequency v of this resonator and is proportional to it. There are atoms of energy, as there are atoms of electricity and atoms of matter formed of the first. However, while a hydrogen atom conserves its mass, whatever be the compound into which it enters, while the atom of electricity conserves its individuality through any vicissitude that it undergoes, it is not the same with the quanta of energy. If we have, for example, three quanta of energy on a resonator the wave-length of which is 3, and if this energy passes to a second reso- nator the wave-length of which is 5, it represents
132 PHILOSOPHY AND THE NEW PHYSICS
no longer 3 but 5 quanta of energy. This is one of the numerous difficulties of the theory.
One of the confirmations of the quantum theory is drawn from the determination of Avogadro's constant (the number of molecules contained in a gram-molecule of a gas) which it provides. Two constants figure in Planck's empirical formula: one which expresses the discontinuity of the energy of oscillation of resonators, called the universal constant h, another which expresses the molecular disconti- nuity, called Avogadro's constant N. To determine the numbers h and TV it suffices to use two good measurements of the emissive power (the quantity of energy that leaves the opening of the enclosure per second) for different values of the wave length X and the temperature T. The value of h is then found to be
h = 6.2 X 10-27
and this leads to the value of N N = 64 X 1022
This number agrees quite closely with the mean of the numbers obtained by the methods considered most reliable, and this coincidence is the more sur- prising as the number of molecules is deduced from measurements made on radiant heat.
A second confirmation, due to Einstein and to Nernst, is drawn from the study of the specific heats of solids at low temperature. It is known that the specific heats of solids decrease rapidly when the temperature is lowered. Thus, for diamond
QUANTUM THEORY 133
at the temperature of liquid hydrogen, the specific heat is reduced to about one-seventh of what it is at ordinary temperatures. Everything takes place as if the molecules lost degrees of freedom in cool- ing, as if their joints became anchylosed from the effect of the frost. This is contrary to the the- orem of the equipartition of energy, which provides a method of calculating the atomic heats of solids and of deducing Dulong and Petit's law, accord- ing to which the atomic heats are equal for all bodies and independent of the temperature.
Planck defined the specific heat of a single resonator as the increase, for one degree centigrade, of the mean energy, reduced to calories, that a resonator of given frequency must have at a definite temperature to be in equilibrium with black radi- ation. Einstein introduces the following simpli- fying hypotheses. He considers the solid bodies as possessing only a single kind of resonators and only one resonator per molecule; the product of Avoga- dro's constant by the specific heat of a single resonator then gives him the specific heat, referred to a gram-molecule, of the solid considered. Accord- ing to this formula the specific heat of a solid hardly varies with the change of temperature at high temperatures; but at low temperatures it decreases rapidly and tends towards zero when the absolute zero is approached. Nernst complicates Einstein's hypothesis a little by adding to the lat- ter's unique system of resonators other resonators tuned to the octave and so succeeds in obtaining
134 PHILOSOPHY AND THE NEW PHYSICS
a surprising coincidence with the numbers obtained experimental!}' by himself and his pupils, for a very large number of bodies at temperatures that extend from the ordinary temperatures down to the low- est temperatures that Kammerlingh-Onnes has been able to obtain in his cryogenic laboratory at Ley den. This theory amounts to the following: if a solid is regarded as an aggregate of atoms or of molecules oscillating about an equilibrium position, the en- ergy of each oscillator thus realized must be, as in the case of Planck's electric oscillators, an inte- gral multiple of hv. The diminution of the speci- fic heat of the solid at low temperatures is then readily explained. When the temperature de- creases, the supply of disposable energy offered to each of the material oscillators falls below the quantum of a large number of them; instead of vibrating a little they cease to vibrate at all, so that the total energy diminishes more rapidly than in the old theories. On the contrary, at high tem- peratures, the quantum hv becomes so small that we again get Dulong and Petit'slaw, obtained by start- ing from the law of the equipartition of energy.
19. THE STRUCTURE OF RADIATION.
The study of black radiation and of specific heats at low temperatures has led Planck, and after him, Einstein to the idea that the exchanges between radiation and matter do not take place in a continu- ous manner, but by discrete elements, by quanta of energy. What becomes of these elements when once
STRUCTURE OF RADIATION 135
set free by matter? For the old theory of free radiation considered as formed by indefinitely di- visible and expansible spherical waves propagated by a continuous hypothetical medium, there is now substituted the idea of a radiation projected in space void of matter in the form of distinct ele- mentary units, which implies a discontinuous dis- tribution of energy in the front of the luminous transverse waves. This is an unexpected return to the emission theory, rendering the consideration of an ether serving as the "I" to the verb "undulate" superfluous.
Among the phenomena, the interpretation of which suggests this point of view, we have Hertz's phenomenon and the production of rays by shocks of secondary cathode rays and by X-rays.
Hertz's phenomenon, or the photo-electric effect, denotes the property that bodies have, in particular metals, of emitting cathode rays under the action of light. The electric forces present in the absorbed luminous waves set the electrons stored in the metal in motion, and some of them are, because of the accelerations that they undergo, projected out. The photo-electric current, which results there- from, depends on two factors, the number of the electrons emitted per unit time and the velocity with which these electrons are emitted. If the intensity of the light is increased without a change in its wave-length, the number of the electrons emitted increases proportionally, but the velocity of emission remains the same. If, on the other
136 PHILOSOPHY AND THE NEW PHYSICS
hand, at constant luminous intensity a change is made in the frequency v of the exciting light, the velocity of emission seems to increase propor- tionally to the frequency, and consequently in- versely as the wave length. The velocity with which the electrons are emitted certainly did not belong to them before they had undergone the action of the light, as experiment shows that this velocity is very much higher than that of the thermal agita- tion. If the velocity of the electrons is borrowed from the light, how are we then to understand that it is the same for a very intense light and a very fee- ble light? How, above all, explain that it varies inversely as the wave length?
Stark1 has remarked that these difficulties dis- appear if the quantum hypothesis is assumed. The light that meets the metal is not of homo- geneous structure, it consists of isolated elements each possessing the energy hv. When one of these elements strikes the metal the energy that it con- tains can pass to one of the electrons present in the metal. On account of this fact the electron acquires a velocity that can much surpass the velocity of thermal agitation and leads to its projection from the metal. As the elements of energy are greater in proportion as the frequency v is higher, ultra-violet light will be much more effective than visible light. Thus not only is light absorbed and emitted solely in the form of quanta, but, even when it is freely pro-
1 Stark, Phys. Zeitschr., 1909; Principien der Atomdynamik. Leipzig, 1910.
STRUCTURE OF RADIATION 137
pagated in empty space, it seems to consist of the projection of discrete units.
J. J. Thomson1 has arrived at similar conclu- sions by studying the photo-electric effect on gase- ous bodies. Ultra-violet light of very short wave- length ionizes gases, that is, it breaks up the gaseous molecules by setting electrons free. The work necessary for this rupture can be determined. It is thus found that if, in conformity with Maxwell's ideas, ultra-violet light is thought of as formed by a train of homogeneous waves propagated with- out inequalities of structure, the energy carried through a unit section is not sufficient to account for the work of ionization produced by this pencil. To do this a discontinuous front must be assumed, so that there are spots of light and darkness. At places where the energy is accumulated its density is sufficient to ionize the molecules. The wave front possesses a structure, and the wave train, taken as a whole, is not formed of a regular and uninter- rupted flow of energy, but of discrete elements of energy separated by considerable gaps. This is the old emission theory reappearing in a very modi- fied form.
Einstein has supplemented Stark's ideas in a way that has made a new experimental verification of the quantum theory possible. He assumes that an electron set free by a luminous ray of frequency v can bring into action only an amount of energy equal to Planck's energy element. Consequently
1 J. J. Thomson, Proceed. Camb. Phil. Soc., 1908.
138 PHILOSOPHY AND THE NEW PHYSICS
we have, according to the principle of the conserva- tion of energy, the equality
%mv2 = hv — V
where ^ denotes the work necessary to separate the electron from the atom from which it comes. This formula is capable of direct verification, if we take into account the fact that the velocity v of the photo-electric electrons can be determined ex- perimentally by the measurement of the condenser potential, V + Fo, capable of reducing the current to zero, by reducing the velocity v to zero. We have
e(V + Fo) = Mm*;2
where V denotes the potential difference applied, and V0 the contact potential difference that always exists between two different metals, and these two differences have to be determined separately. Among the most satisfactory verifications of this formula, the recent experiments of Millikan1 must be quoted. Working in a vacuum on freshly prepared surfaces of sodium, potassium, and lith- ium, this physicist has employed the entire range of wave-lengths from the middle of the visible spectrum to the ultra-violet of the mercury arc. His results confirm Einstein's theory. In parti- cular, the linear relations that he obtains between V and v make possible the determination of a con- stant angular coefficient practically equal to Planck's constant.
1 R. A. Millikan, A Direct Photoelectric Determination of Phanck's "h." Phys. Rev. Ser. 2, vol. 7, p. 355.
STRUCTURE OF RADIATION 139
Lorentz1 has shown, not only that the corpuscular idea of light results from Hertz's phenomenon, but that it alone permits the explanation of the absorp- tion of radiation by fixed quanta, multiples of hv, in conformity with the views of Planck. When an incident wave, encountering the molecules of an isotropic body, sets the electrons in it into vibration, the absorption of light is the more intense the nearer its frequency is to the natural frequency of these electrons. In the case of perfect resonance the electron borrows from the luminous wave the maxi- mum amount of energy, and, by taking the case of the solar light and allowing for the forces of damp- ing and friction to which the electron is subject, this maximum energy is found by calculation to be a little more than twice the atom of energy required by Planck. A similar amount of light, therefore, can be absorbed by the electrons. But the ordinary sources have intensities a million times weaker than that of the Sun. If the energy that they emit were uniformly distributed over the surface of a sphere having the source as center, it could com- municate to the electron only a very small fraction of its absorbable quantum. This energy, therefore, must be unequally concentrated in different direc- tions, and the sphere must be traversed by the flow of energy in a certain number of discontinuous spots, very widely spaced, to provide condensations of energy compatible with the mechanism of absorp- tion conceived by Planck.
1 H. A. Lorentz, Phys. Zeitschr., 1910.
140 PHILOSOPHY AND THE NEW PHYSICS
The quantum theory, it is hardly necessary to say, encounters otherwise many difficulties. One is the existence of interferences at large differ- ences of path, which seems to imply that the lum- inous quanta extend over lengths equal to several thousand times their wave-length. There are others which, however, yield to a deeper analysis. The idea that a point source does not radiate symmetrically in all directions may seem incom- patible with experiment. But the physical point sources contain a number of oscillators sufficiently great to wipe out any trace of individual discontin- uity. The physical symmetry, which experiment leads us to attribute to luminous waves, results from a mean effect and is not in contradiction to the possibility of an elementary radiation, endowed with structure. This has been shown by J. J. Thomson1 by supposing that the electrified parti- cles producing the light do not emit lines of force uniformly in all directions, but only, for example, in two solid angles opposed at the vertex and of relatively restricted aperture. Starting from this hypothesis, he has demonstrated that the principal properties of the electron would still belong to those particles, the electromagnetic field of which is concentrated in privileged directions. They would have inertia of electromagnetic origin and would emit radiation on every modification of their state of motion; but the radiated waves of acceleration, light, if we are concerned with periodic waves,
1 J. J. Thomson, Phil. Mag., 1910, p. 301.
STRUCTURE OF RADIATION 141
would be concentrated in certain directions and extraordinarily rarified or absent in others. The radiation emitted by an oscillator would possess a particular structure, which would reappear in part in the emission of real sources and produce the irregularities of the wave front suggested by Hertz's phenomenon. Nothing essential would be changed in the usual laws of electromagnetism, and they would continue applicable to material complexes enclosing a large number of particles distributed according to the law of chance. • V'The quantum theory is far from having the dogmatic certainty of the atomic theory or the electronic theory. The atoms, for example, have become a physical reality: they are counted, they are weighed, their radius of action and their mean velocity are determined, they are seen in the form of scintillations with the aid of the spinthariscope of Crookes, and C. T. R. Wilson, by using their property of becoming the centers of condensation in supersaturated water vapor, has succeeded in photographing the atomic trajectory of the a particles, which are ions of helium, and even the corpuscular trajectory of the 0 rays, which are electrons. No device, up to the present, has enabled us to subject the quanta of radiant energy to a control capable of transforming them from theoretical concepts to experimental concepts. One, however, can be suggested, drawn from the marvelous properties of selenium.1
1 Cf. Fournier d'Albe, The Future of Selenium (Scientia, 1917, p. 165-191).
142 PHILOSOPHY AND THE NEW PHYSICS
When one of the conductors in an electric cir- cuit consists of selenium, a current is produced in this circuit, as soon as it is subjected to a luminous flash. The conductivity thus produced from an instantaneous exposure of the selenium to light is proportional to the incident energy. This property makes it possible to detect, electrically, flashes that are invisible to the naked eye. By applying, for example, an electromotive force of 1 volt, an easily perceptible current of 10~12 ampere could be ob- tained with a cell of selenium of 100 sq. cm., with a short flash of 10~9 lux, corresponding to a star of 8th or 9th magnitude, which is quite invisible to the naked eye. With a selenium cell of the same surface as the pupil of the eye, a star of the 6th order, which is the extreme limit of our vision, could be detected electrically under the same conditions. But, since we are able to measure currents of 10~15 ampere, the sensitiveness of selenium obviously surpasses that of the eye. Now the quantum theory permits the calculation of the size and number of the grains of energy absorbed by a given surface, subject to a radiation of definite wave-length and intensity. It is found that the human eye absorbs 360 quanta per second, when it receives light coming from the visible star of least luminosity. These quanta are, in the most favorable case, 20 times too numerous to be per- ceived as distinct flashes. We can then hope to count them electrically, aided by the selenium, if we can succeed in detecting sufficiently feeble cur-
PHYSICS OF THE DISCONTINUOUS 143
rents. The selenium cell would thus render to us the same service as the zinc sulfide screen of Crookes' spinthariscope; it would enable us to place the existence of the quanta on an experimental basis.
20. THE PHYSICS OF THE DISCONTINUOUS.
Modern discoveries lead to the assumption of the discontinuity of matter, of electricity, of radiant energy, and of exchanges of energy. Henri Poin- care has shown the consequences that result there- from for mathematical physics and natural philo- sophy. We must give up expressing the laws of the phenomena in the form of differential equations in all cases in which the large number of elements which come into play does not suffice to wipe out entirely the influence of the individual discontinui- ties. Contrary to the ancient adage natura non facit saltus, it becomes apparent that the universe varies by sudden jumps and not be imperceptible degrees. A physical system is capable of only a finite number of distinct states, and this introduces discontinuity into the distribution even of probabil- ities. Since between two different and immediately consecutive states the world remains motionless, time is suspended so that time itself is discon- tinuous: there is an atom of time.
The controversy between infinitists and finitists, idealists and empiricists, cantorians and pragmat- ists, to follow current terminology, is, on this basis, settled in favor of the latter. The world does not glide smoothly down a gentle slope, imperceptibly
144 PHILOSOPHY AND THE NEW PHYSICS
inclined, of the course of events according to Leibniz, but it descends by steps of events as conceived by Evellin.1 In the external reality there are no ag- gregates of elements having the attributes of a continuum ,but only aggregates that can be counted, the attributes of which are only intelligible for the mind, because it can define all their elements. Nature following mathematics is reduced to arithmetic. It becomes comprehensible, for if our imagination does not show us anything except in the form of spatial intuition, which is that of a physical contin- uum, our mind, by virtue of the minimum percepti- ble increment of excitation, can comprehend only the countable and the discrete. Just as continuity in mathematics appears more and more as a transi- tory tool, the utility of which at present is not neg- ligible, but which must be regarded as a means of studying the countable aggregates that constitute the only analytical reality that is accessible, so too physical continuity, which lends itself well to appli- cations of the calculus of partial derivatives, will always appear legitimate, as a first approximation, for the order of magnitude of ensemble effects of systems the elements of which are finite in number but sufficiently numerous to wipe out the individual discontinuities by virtue of perfect mixing and the laws of chance. Nevertheless it should not make us forget that the simple elements of things manifest an essential discontinuity, which must reappear in the equations that -translate their individual behavior
*A contemporary French philosopher. (Tr.)
PHYSICS OF THE DISCONTINUOUS 145
and which alone can account, in the hypothesis of the absolute determinism, for the fluctuations of the ensemble phenomena to which they give rise.
On mathematical physics there is hereafter imposed a new task, namely, to establish a bond :—
Between the ensemble phenomena which our observations reveal, and the elementary phenomena of which they are a statistical resultant;
Between the physical quantities directly acces- sible to our measuring instruments, which take in at once, as the sum or average of the individ- ual quantities, so many elements that they can be practically treated as continuous; these quantities being in themselves, or individually, essentially discontinuous;
Between the properties of the discrete grains (molecules, atoms and electrons) and the struc- tures they form by aggregation.
The differential and integral calculus, used to translate analytically the idea of continuity, is appropriate for the study of the systems which alone are directly perceptible, and which are com- posed of a very great number of elements. The calculus of probabilities is appropriate for the study of the relations between the real world of ele- ments of discontinuous structure and the apparent world of continuous phenomena; between the indi- vidual laws, which govern these elements taken separately, and the laws of large numbers, which govern the mixed appearances to which their incessant mixing gives rise.
10
146 PHILOSOPHY AND THE NEW PHYSICS
By correlation, the concept of explanation be- comes transformed in natural philosophy. In former days a physical phenomenon was explained by reducing it to the principles of classical mechanics, by giving to its laws the form impressed by Lagrange on the equations of dynamics. To explain a phen- omenon today is to give a statistical explanation, by regarding it as the resultant of a very large number of underlying phenomena governed by the laws of chance. Maxwell's attempt in his Treatise on Electricity and Magnetism is an example of an explanation of the first type. The statistical interpretation of Carnot's principle, of the expon- ential law of the spontaneous destruction of radio- active substances, and of the law of mass action are examples of the second type.
Thus not only do the most fundamental catego- ries of our mind, those of space, of time, and of causality, pass through an evolution with the pro- gress of science, but the same holds even for the concept of intelligibility. To explain a phenomenon is, for primitive man, to interpret it anthropo- morphically by a supernatural agent endowed with psychological life in his own image ; for a scholastic it is to explain it by ultimate causes; for Bacon to explain it by efficient causes; for Maxwell it fc to deduce it from the principles of mechanics; for Gibbs and Boltzmann it is to account for it by the calculus of probabilities, by starting from a system of elements subject to given conditions. Human reason is not "u'ne et entidre en un chacun" as
PHYSICS OF THE DISCONTINUOUS 147
Descartes taught. It varies with the abstract or concrete nature of our thought, and in proportion as, on contact with experimental facts, the adapta- tion of our mind to nature becomes progressively realized.
CHAPTER VIII CONCLUSION
21. CONCLUSION.
The discoveries of modern physics have led physicists to two quite distinct conceptions of the universe.
The first can be illustrated by the suggestive name of the dematerialization of matter. It con- sists in reducing matter to being only the locus of singular points of torsion, of condensation, or even of destruction of a medium endowed with inertia and mechanical properties, the dielectric ether of Faraday and Maxwell. From this point of view an electron is a simple cell in the ether, behaving like a projectile moving forward in a perfect fluid having no viscosity. On the sides of the projectile a cushion of fluid is formed and behind it a zone of deep eddies: its own inertia is increased by all the inertia of the wave system thus created which follows it. At start a part of the muzzle energy is expended in overcoming the inertia of the fluid displaced at the same time as the inertia of the moving body itself; but, when the motion has been once acquired, it perpetuates itself with- out resistance, since the body carries its wave system with it. The electron has no inertia of its own, but it can not be displaced without entraining
148
CONCLUSION 149
the" surrounding ether coupled to its lines of force, and its inertia results from that of the ether thus disturbed which forms its electromagnetic wave system. Matter resolves itself into cells in the ether and the ether gains in substantiality and in reality all that matter loses.
This idea results from Faraday's work, which has brought to light the importance of the medium that surrounds conductors and magnets in electric and magnetic phenomena. The attention of phy- sicists has been directed to the study of electric and magnetic fields in which energy is concentrated, matter serving only as a support of these fields. But instead of conceiving them as substantial realities existing in an independent manner, it is claimed that they are explained by the mechanical states of this hypothetical medium, the ether, elec- tric energy being merely the potential energy of its deformations and magnetic energy merely the kinetic energy of its displacements. The ether becomes the active medium in which the transfers and the transformations of energy are governed by the equations of the electromagnetic field of Maxwell and Hertz, and thus matter is gradually stripped of all its physical contents and they are referred to the ether, which becomes the only reality that continues to exist.
Going still further Dr. Gustave Le Bon1 has
1 Dr. G. Le Bon, L' evolution de la matiere. On the relations and the differences between this theory and that of Einstein see L. Rougier, L'inertie de 1'energie (Revue scientifique, October 13-20, 1917).
150 PHILOSOPHY AND THE NEW PHYSICS
developed the idea of the evolution of matter. Let us imagine that the electrons, into which in the end the molecular structures that constitute bodies are resolved, are due to vortices in a universal fluid analogous to Lord Kelvin's gyrostatic ether. Instead of regarding these vortices as indestructible in conformity with the hydrodynamic equations of Cauchy and Helmholtz, let us imagine, quite gratui- tously, that they vanish at length in the original fluid, like a waterspout in the ocean, in the form of liquid waves, owing to the gradual retardation of their velocity. The rotatory energy of the electron will be transformed into radiant energy, which, sweeping through all space, will be lost at infinity. Thus matter is resolved into electrons, which themselves vanish in etherized undulations, so that there is a final loss of matter and an uncom- pensated dissipation of energy. For the universal principle of invariance which the Ionic natural philosophers placed at the basis of natural philoso- phy and which assured its intelligibility, namely, " Nothing is created, nothing is lost" one must now substitute the contrary principle: " Nothing is created, everything is lost/' The world marches toward a final bankruptcy, and the ether, of which it has been asserted in vain that it is the matrix of the worlds, is revealed as being their final tomb. Thus Dr. Gustave Le Bon has been, in sumptuous prose, the Zarathustra announcing, after the death of the gods, the twilight of their creation.
These ideas encounter insurmountable difficulties.
CONCLUSION 151
The ether is revealed as endowed with contradictory mechanical properties, and the attempts at an explanation of electromagnetic phenomena start- ing from it have all suffered shipwreck. If it exists it can not be completely entrained by matter, ,as Fizeau's experiment proves; it can not be par- tially entrained by matter, as the principle of action and reaction demonstrates; that it can not be motionless, is implied by the relativity principle. Less tangible than Proteus, it remains only to declare it defunct without estate.
To these antinomies the theory of Dr. Gustave Le Bon adds new difficulties. The hydrodyn mic equations of Cauchy and Helmholtz show that vortices, once started in a perfect, homogeneous, and incompressible fluid, are eternal. The electronic theory of radiation connects its appearance with the presence of electrons, which play the part of agents present in the transformation of different forms of energy into radiation and permit these trans- formations, while themselves remaining, like cataliz- ing agents, unaltered. Their disappearance would entail therefore that of radiation itself, in which they are supposed to vanish. Lastly, if the principle of invariance of the Ionic natural philosophers is abandoned, the very possibility of science is put at stake. Science, being the search for the laws of nature, that is to say, the invariants of all events in the universe, would remain valid, to a first approximation, only for systems in which the disin- tegration of matter and the radiation of energy to
152 PHILOSOPHY AND THE NEW PHYSICS
infinity are practically insignificant; and the meta- physical problem would arise of understanding how the universe, if it has not had a beginning, has not yet finally vanished, milleniums ago, in the "motionless and sleeping" ether.
Abandoning the ether, we are led to an en- tirely different theory: that of the materialization of energy. Energy emerges from the phantom realm of imponderables to take substance, like the shades of the Elysian fields evoked by Ulysses on the Cimmerian river. It appears as endowed with inertia, with weight and structure and manifests itself in two forms: one is called, by virtue of long prescription, matter; the other, radiation.
Matter is characterized by its structure, that is, by the number and nature of the electrons and perhaps the positive remainders that constitute it, and also by its property of moving with veloci- ties, relatively to a reference system, ranging from 0 to V. We know nothing of it, as stated by Ost- wald, except its energetic effects: the electric field of the electron at rest, the magnetic field of the electron in motion, the gravitational field of the molecular structures formed, by electronic architecture, the kinetic effects produced by their vis viva. The electron is revealed as a grain of resi- nous (negative) electricity, so that matter is only a form of energy, enormously accumulated in a narrowly circumscribed region of space. It does not, on that account, as in the preceding theory, lose the reality and the substantial characteristics
CONCLUSION 153
which external perception and common sense have agreed until now to attribute to it, since energy, which is its essence, is endowed with mass, weight, and structure.
Radiation is a form of energy which no longer appears as propagated in the shape of continuous waves by a hypothetical medium, but as expelled in the form of discrete units in space free of matter with the uniform velocity of light. It also is endowed with inertia, weight, and structure. Its possession of fundamental properties in common with matter permits the explanation of its action on the latter. Luminous radiation, representing a cer- tain momentum, can be strictly assimilated to a material projectile. It is because of this that, by virtue of the principle of action and reaction, it exerts a repulsion on the material source that emits it and a propulsion on a material obstacle that ab- sorbs it. The ancient metaphysical problem of the action of the imponderable on the ponderable, of force on matter, which arose in its most modern and most urgent form in connection with the pressure of radiation, disappears henceforth as a pseudo- problem.
BIBLIOGRAPHY 1. The Relativity Principle
H. A. LORENTZ, A. EINSTEIN, H. MINKOWSKI, Eine Sammlung Abhandlungen, Teubner, 1913.
LAUE, Das Relativitatsprinzip, Teubner, 1911.
P. LANGEVIN, Le temps, 1'espace et la causalite dans la Physique moderne (Bulletin de la Societ6 frangaise de Philosophie, t. XII, 1912).
154 PHILOSOPHY AND THE NEW PHYSICS
2. Electromagnetic Dynamics
H. POINCARE, La theorie de Lorentz et le principe de reaction (Archives neerlandaises des Sciences exactes et naturelles, 2e se"rie, t. V, 1900).
H. POINCARE, Sur la dynamique de 1' Electron (Rendiconti del Circolo matematico di Palermo, t. XXI, 1905).
P. LANGEVIN, Les grains d'electricite* et la dynamique electro- magnetique (Les ide"es modefnes sur la constitution de la matiere, Paris, 1913).
3. The Electronic Theory of Matter
P. LANGEVIN ET M. DE BROGLIE, Les quantity's e!6mentaires d'41ectricit4, ions, electrons, corpuscules, 2 vol., Paris, 1905.
Les idees modernes sur la constitution de la matiere (Memoires de la Societe franc, aise de Physique, Paris, 1913).
N. R. CAMPBELL, Modern Electrical Theory, Cambridge, 1913.
4. The Inertia of Energy
A. EINSTEIN, Annalen der Physik, vol. XVIII, 1905. P. LANGEVIN, L'inertie de 1'energie et ses consequences (Journal de Physique, juillet 1913).
5. The Weight of Energy and the Theory of Gravitation
A. EINSTEIN, Annalen der Physik, vol. XXXV, 1911; Ibid., vol. XXXVII, 1912; Physikalische Zeitschrift, vol. XIV, 1913.
A. EINSTEIN ET GROSSMANN, Bases physiques d'une theorie de la gravitation (Archives des Sciences physiques et naturelles, t. XXXVII, 1914).
A. EINSTEIN, Die formalen Grundlagen der allgemeinen Rela- tivitatstheorie (Sitzungsberichte der koniglich preussischen Akade- mie der Wissenschaften, vol. XLI, 1914).
A. EINSTEIN, Erklarung der Perihelbewegung des Merkur aus der allgemeinen Relativitatstheorie (Sitzungsberichte der koniglich preussischen Akademie der Wissenschaften, vol. XLII, 1915).
H. A. LORENTZ, La gravitation (Scientia, 1914).
ED. GUILLAUME, Les bases de la Physique moderne (Archives des Sciences physiques et naturelles, 1916).
L. BLOCK, Relativite et gravitation, d'apres 16s theories recentes (Revue generate des Sciences pures et applique*es, 15 de"cembre 1917).
BIBLIOGRAPHY 155
L. BLOCH, Sur les theories de la gravitation (Annales de Phy- sique, janvier-fevrier 1918).
6. The Structure of Energy
P. LANGEVIN ET M. DE BROGLIE, La the"orie du rayonnement et les quanta, Paris, 1912.
H. POINCARE, Sur la theorie des quanta (Journal de Physique, Janvier 1912).
H. POINCARE, L'hypothese des quanta (Revue rose, 24 fevrier 1912).
P. LANGEVIN, La physique du discontinu (Les progres de la physique moleculaire, Paris, 1914).
M. PLANCK, Die physikalische Struktur des Phasenraumes (Annalen der Physik, vol. L, 1916).
P. S. EPSTEIN, Zur Quantentheorie (Annalen der Physik, vol. LI, 1916).
A. EINSTEIN, Zur Quantentheorie der Strahlung (Physikalische Zeitschrift, vol. XVIII, 1917).
Other Works by L. Rougier :
Les Paralogismes de Rationalisme, Paris, 1920. La philos- ophic ge*ometrique de Henri Poincare, Paris, 1920. En marge de Curie et d'Einstein, Paris, 1920. Theories formelles et logique deductive, Paris, 1921.
INDEX OF NAMES
Abraham, 52, 54, 55, 56, 63, Arrhenius, 16 Avogadro, 132
Bacon, 146 Bartoli, 17, 77 Becquerel, 60 Bernoulli!, 7 Bloch, 154, 155 Boltzmann, 9, 125, 146 Le Bon, 149, 150, 151 Boscovich, 10, 11 Brace, 31 Brillouin, xiii de Broglie, 120 Bucherer, 17, 56, 57, 63, 76 Buisson, xiii, 95
Campbell, 70, 154 Carnot, 126, 146 Cauchy, 150, 151 Christoffel, 98 Clausius, 16 Comte, ix, 14 Coulomb, 41, 53 Cremieu, 48
Crookes, 16, 54, 55, 58, 65, . 143
76 Einstein, x, xii, xiii, xiv, 22, 26, 27, 29, 70, 77, 93, 94, 97, 100, 101, 103, 105, 106, 107, 109, 132, 133, 134, 137, 138, 153, 154, 155
Enriques, 22
Eotvos, xiii, 18, 26, 84, 91, 96, 100
Epstein, 155
Euclid, 20, 37, 100
Euler, 99
Evellin, 144
Fabry, xiii, 95
Faraday, 16, 41, 43, 44, 45, 46,
47, 49, 62, 63, 67, 70, 104,
106, 148, 149 Fizeau, 68, 151 Fitz-Gerald, 31 Foucault, 62 Fournier d'Albe, 141 Fourier, 14 Fraunhofer, xiii, 95 Fresnel, 62, 64, 68 Freundlich, 96
Descartes, 10, 100, 147 Democritus, 7 Du Bois-Reymond, ix Duhem, x, 7, 106 Dulong, 133, 134
Edison, 58, 61, 65
id.1 W* Galileo, 38, 39, 40, 62
Gibbs, 146 Goldstein, 60 Grimaldi, 62 Grossmann, 154 Guillaume, 154
157
Hamilton, 40, 58, 126 Hannequin, 7, 63
158
INDEX OF NAMES
Helmholtz, 9, 106, 150, 151 Hertz, 8, 9, 11, 20, 47, 48, 58,
61, 65, 68, 120, 135, 139, 141,
149
van't Hoff, 16 Hupka, 56 Huygens, 7, 62
Janet, 5 Jeans, 128
Kant, 10, 11, 103 Kaufmann, 17, 56, 57, 63, 76 Kelvin, 9, 10, 70, 104, 106, 150 Kirchhoff, 121
Lagrange, 39, 40, 58, 146 Langevin, xiv, 29, 77, 91, 120,
153
Laue, 29, 153 Lavoisier, 4, 90 Lebedew, 17, 77 Leibniz, 144 Lenz, 48, 116 Leonardo da Vinci, 62 Leukippus, 7 Levi-Civita, 98 Lobatschewsky, 37 Lodge, 70, 104, 106 Lorentz, 29, 31, 32, 35, 36, 37,
38, 39, 40, 55, 56, 57, 58, 60,
61, 63, 72, 73, 74, 76, 79, 103,
104, 105, 139, 153
Mach, 27, 98
Maupertuis, 23, 25
Maxwell, xiv, 9, 10, 11, 16, 17, 20, 39, 40, 43, 47, 48, 49, 52, 53, 58, 62, 63, 64, 67, 69, 70, 74, 76, 77, 120, 125, 137, 146, 148, 149
Mayer, 4
Meyerson, 7
Michelson, 29, 31, 32, 38
Mie, 96
Millikan, 138
Minkowski, 29, 35, 84, 100, 153
Morley, 29, 31
Mosotti, 41
Nernst, 132, 133 Neumann, xiv
Newton, xiv, 17, 22, 25, 39, 58, 70, 79, 86, 93, 98, 99, 102, 107 Noble, 31 Nordstrom, 96
Onnes, 134
Ostwald, x, 3, 11, 12, 13, 14, 16, 18, 19, 72, 110, 152
Pasteur, 106
Petit, 133, 134
Planck, xii, 124, 129, 130, 131,
132, 133, 134, 137, 138, 139,
155 Poincare, ix, x, 73, 76, 79, 154,
155 Prout, xiii, 92
Rankine, xi, 31
Rayleigh, 31, 128, 129, 130
Ricci, 98
Rontgen, 117
Rougier, 25, 38, 149
Rowland, 16, 47, 48, 49, 76
Salisbury, 63 Spencer, 9 Stallo, 7 Stark, 136, 137 Stefan, 123
INDEX OF NAMES 159
Tait, 10 Weber, 107
Tisserand, 107 Wien, 58
Thomson, 47, 67, 76, 137, 140 Wilson, 141
Trouton, 31 Witte, 69
Verne, 28 Zeeman, 61
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