THE EVOLUTION OF THE DOCTRINE OF ENERGY

(1847–1947)

T. P. Kravets

1. THE DISCOVERY OF THE LAW OF CONSERVATION OF ENERGY

Long was humanity’s path toward establishing the concept of energy and the law of its conservation: it would at times approach the truth, at times move away from it; with difficulty it created and refined quantitative ideas about heat, work, temperature, and potential; only gradually did it develop methods for measuring these quantities and, with delay, create the very units of these measurements—mechanical, thermal, electromagnetic; through experience, in the torments of fruitless searches for a “perpetual motion machine,” it came to realize and intuitively foresee the great law of conservation of energy. The beginning of the conservation of motion was proclaimed by Descartes, Newton, and Lomonosov. In Lomonosov, the conservation of motion constituted only a part, only one instance, of a more general and broader physical law of conservation. The concrete expression of the law of conservation of energy belongs, as is known, to J. R. Mayer¹, J. P. Joule², and H. Helmholtz³*).

If we, enriched by the experience and knowledge that have flowed from those times, look closely into the works of the members of this triad, we shall become convinced that there exists a certain difference in the understanding of the law discovered by Mayer, on the one hand, and by Joule and Helmholtz, on the other. The latter two are zealous adherents of the mechanical doctrine, deriving from Huygens; for them all phenomena are, in a hidden form, mechanical in essence, are explained by mechanical forces, and must be treated by mechanical methods; Joule has in mind here chiefly the phenomena of heat, and his numerous statements on this question confirm his conviction of its kinetic character. Helmholtz, already in his first brochure, extends his views and methods of investigation to all phenomena of nature.

In contrast to these two authors, Mayer nowhere mentioned by a single word the mechanical character of all physical

) For the history of this discovery in a very brief form, see our article in Physics at School* for 1947.

phenomena. For him, energy (or “force,” as all three of them wrote) undergoes manifold transformations, appearing now in the form of mechanical energy, now in the form of heat, and now in still other modifications. It would be useless to seek, in these early years of the doctrine of energy, more precise statements—the doctrine of energy itself had not yet taken on more definite forms. But now, a hundred years later, the whole gamut of contradictory views that subsequently developed is readily fitted into the contrast, already outlined in the first years of the doctrine, between what was stated by one side and what was passed over in silence by the other.

2. DIFFICULTIES IN UNDERSTANDING POTENTIAL ENERGY

Let us note that among the “mechanists,” from the very beginning of the doctrine of the conservation of energy, there arose a certain difficulty of a theoretical character. In essence, in mechanics only one kind of energy is perfectly visualizable—kinetic energy (or, in the earliest period, “living force”). As soon as we formulate the concept of potential energy (“tension force” in Helmholtz), we enter upon a train of thought that leads us away from mechanics. In fact, if potential energy is the work expended in giving a system a given configuration, and elementary work is equal to the scalar product of the length of an elementary segment by the forces acting on that segment, then it becomes necessary to justify these forces—and this necessity, even in the simplest case, brings us to questions that are insoluble, or at any rate unresolved: what are the forces of gravitation acting between celestial bodies? gravity, which attracts terrestrial bodies toward the center of the earth? elastic forces, by which one body acts upon another in contact with it, and in general all forces of molecular origin? all forces manifested in the electromagnetic field? and, finally, the “nuclear forces,” as yet still undisclosed, which store in the depths of the atom unheard-of reserves of energy ready for use? Of course, in order to explain anything, we must recognize some one phenomenon as the simplest, and other phenomena reduce to this simplest one. But depending on the level of our knowledge, at different times we recognize now one phenomenon, now another as the simplest, and must, accordingly, reconstruct the logical structure of the entire edifice of the knowledge of our science—and we have indeed repeatedly witnessed such reconstructions during the lifetime of just one generation of scientists. But this means that the very concept of potential energy does not belong to simple concepts. Is it not possible to do without this concept altogether?

3. N. A. UMOV’S WORK “THE THEORY OF SIMPLE MEDIA”

A remarkable attempt of this kind we find in one of the works of our compatriot N. A. Umov[^4]. Written in 1871, i.e., at a comparatively still very early stage of the doctrine of energy, it

...precisely reflects a certain philosophical uneasiness that existed among the scientists of that time, who were pondering the very foundations of this doctrine and, in particular, potential energy. The author rejects this concept and tries to eliminate it from physics by introducing the proposition that the quantity of kinetic (or, as he writes, dynamic) energy in nature remains constant; if, however, it seems to us that its store has diminished (in other words, has passed into potential energy), then this is only an apparent diminution, since in fact, in the opinion of N. A. Umov, potential energy is the kinetic energy of another medium, which remains hidden. P. P. Lazarev^5, expounding Umov’s doctrine, uses the following vivid illustration: a gas under high pressure beneath a piston possesses a certain amount of visible potential energy. But this potential energy is in fact the kinetic energy of the gas molecules—the medium whose internal motions remain imperceptible to us. It is hardly likely that we should now readily follow the path of recognizing hidden media. But let us recall that hidden masses survived until 1891–1894, when they figure in the Principles of Mechanics of H. Hertz, the famous author of Investigations on the Propagation of Electric Force. The very idea of N. A. Umov concerning the reduction of potential energy to kinetic energy still deserves close attention. In his attempt we see clear evidence that the logical structure of our physical world-view cannot make do with the single concept of kinetic energy, but needs additional propositions of a purely physical character—for example, hypotheses about the nature of certain forces, and so on. Here lies the reason that later led to the downfall of the entire mechanical view, despite its enormous successes and conquests *). Let us give due credit to the insight of our remarkable compatriot, who was able, at such early stages of the doctrine of energy, when it was developing on a purely mechanical basis, to discern the weak sides of the latter and to make an attempt to avoid the danger threatening it.

4. THE QUESTION OF THE LOCALIZATION OF ENERGY

The concept of energy passed through an extremely important moment in its development when physicists had to consider the question of the localization of energy. Let us give an example: we took a wire and, contrary to the elastic forces acting between its particles, stretched it with a load \(P\) by a length \(l\). Then one may write an expression for the work we performed in doing this; it is equal to \(P \cdot l\). In this expres—

*) We do not consider it our task to expound the successes of kinetic concepts. The second law of thermodynamics, interpreted statistically in the works of Boltzmann, Smoluchowski, and Einstein—these are the chief conquests in this field.

EVOLUTION OF THE DOCTRINE OF ENERGY

...there figures, alongside the load \(P\) applied to the end of the wire, only the elongation \(l\) of the wire caused by it. However, it is easy to bring this expression to another form, using the equation for the elongation:

\[ l=\frac{PL}{qE} \]

(\(L\) is the initial length of the wire, \(q\) its cross-section, \(E\) a coefficient characteristic of the material of the wire, the so-called Young’s modulus). By substitution we easily obtain:

\[ A=\frac{1}{2E}\left(\frac{P}{q}\right)^2 qL. \]

In this expression \(qL\) is the volume of the wire, and it shows us that the work has been distributed uniformly over the entire volume of the wire; per unit volume there corresponds the amount of energy

\[ \frac{1}{2E}\left(\frac{P}{q}\right)^2, \]

where \(\dfrac{P}{q}\) is the “stress,” i.e. the force acting on a unit area of the cross-section of the wire.

Thus, the energy has been distributed with a certain density in the uniformly stretched wire.

Another example, which we shall need later, is electrostatic energy. It is known that if we wish to charge an isolated metallic sphere of radius \(R\), situated in a vacuum, with a quantity of electricity \(e\), then we must expend the work

\[ \frac{e^2}{2R}. \]

Maxwell\(^6\), guided by Faraday’s fundamental views, showed in his Treatise (1873) that we obtain the same quantity if we imagine the electrostatic energy not as concentrated on the surface of the conductor, but as distributed throughout all infinite space according to the law

\[ w=\frac{E^2}{8\pi}, \]

where \(w\) is the energy density (the amount of energy in a unit volume), and \(E\) is the electric intensity at the given point. It is equal, for all points outside the sphere, to

\[ \frac{e}{r^2}, \]

and inside the sphere—to zero. Calculation shows that, under such an assumption, the total amount of energy in the field of the charged sphere is equal to the above-mentioned quantity

\[ \frac{e^2}{2R}. \]

The energy is arranged in concentric spherical layers of uniform density; beyond a sphere of radius \(nR\) lies the \(n\)-th part of all the energy; the energy is “pressed” toward the surface of the sphere, decreasing rather rapidly with distance from the center of the sphere.

It is useful to point out that around a sphere situated not in a vacuum but in another medium, the energy density will be not \(\frac{E^2}{8\pi}\), but \(\frac{\varepsilon E^2}{8\pi}\), where \(\varepsilon\) is the so-called dielectric constant of the medium—its characteristic constant, discovered by Faraday.^7

Questions concerning the localization of other kinds of energy are posed and solved in a similar way. For example, for magnetic energy we obtain the analogous expression for the energy density:

\[ w_m=\frac{H^2}{8\pi}\quad (H\text{—magnetic intensity}). \]

5. ON THE MOTION OF ENERGY. THE THEOREMS OF UMOV AND POYNTING

The next stage in refining the concept of energy is the posing of the question of its motion. The very emergence of this question is, from the modern point of view, quite understandable. Take, for example, the phenomenon of the Earth’s warming by the Sun. The Sun produces in its depths unimaginably vast stores of thermal energy. Without yet touching on the question of the sources of this energy, let us note that part of it ultimately appears on the surface of the Earth. It is clear that it must have passed through the space separating these celestial bodies, and hence the question arises of the paths and laws of this motion of energy in space and time. Let us repeat that now this line of thought seems even trivial. But it is necessary to bear in mind the enormous difference between modern views of energy and those vague notions of it which prevailed in scientific books in the 1870s, when this problem arose. It should be noted with legitimate national pride that the first to pose and solve the problem was N. A. Umov himself, then still a twenty-eight-year-old docent of Novorossiysk (i.e., Odessa) University; he did this in his famous work on the motion of energy.^8 Already in the work “Theory of Simple Media” cited above and in other works that followed it, he touches on this question in passing, even creating the term “current” of energy. Now he takes it up specially for the important case of an elastically deformed body*) and arrives at the fundamental conclusion: energy flows along the ray, i.e., in the direction of propagation of deformations; the flow of energy, i.e., the amount of it passing per unit time through a unit area, is numerically equal to the pressure

*) When the work was written, the elastic theory of light still reigned supreme in science, and Umov’s formulas found application in optics. Today they retain applicability in acoustics.

in the corresponding point of space, multiplied by the velocity of propagation of deformations in the given medium. We call this elegant theorem Umov’s theorem.

To understand the full significance of Umov’s conclusion and the structure of his thought in general, let us relate several facts connected with this work. He presented it as a doctoral dissertation at Moscow University—and here he barely escaped misfortune; in any case, he heard from his opponents—official and unofficial—many passionate and bitter reproaches. And these opponents were outstanding Moscow professors: A. G. Stoletov, himself a major and subtle scholar, then recognized as the doyen of Russian physics; A. F. Sludskii and V. Ya. Tsinger—mathematicians of great name and authority. In what, then, did these scholars, who respected N. A. Umov himself, reproach the young disputant?—They immediately sensed in his work a new and, as it seemed to them, heretical current). The heresy consisted in the fact that the author was developing an insufficiently, in their opinion, motivated objection that energy represents something, as they then said, substantial. Meanwhile, the opponents said, only one thing had so far been proved: the sum of the energies of different kinds remains constant in all phenomena occurring in an isolated system. But this property belongs to many mathematical functions*, and from it one cannot draw a conclusion about their physical existence. From the point of view of a mathematician such a view is understandable: after all, Lagrange in his analytical mechanics wrote the expression

\[ T+V=H, \]

where in our terms \(T\) is kinetic energy, \(V\) is potential energy, and their sum \(H\) is a constant quantity\(^{11}\). This is the so-called, among mechanicians of the mathematical school, “integral of living forces,” and they were not accustomed to investing it with any physical meaning. Even the term “work,” necessary for the physical grounding of the concept “potential energy,” did not exist before Poncelet, not to mention energy in general; \(V\) is a certain mathematical expression, the “potential function.” And now they, mathematicians accustomed to these terms, are to be assured that these mathematical concepts, these integrals, have physical existence, that they are distributed in space, flow, and so on and so forth. It seems that what was expressed here was the caution characteristic of the scholars of that epoch with respect to new hypotheses and theories, especially in cases when they represented a more or less broad generalization—a reverberation of the protracted struggle of natural scientists with the Schellingian natural philosophers.

*) We know these facts from oral tradition, but we also find them in the biographical sketches of A. I. Bachinskii\(^{9}\) and P. P. Lazarev\(^{10}\).

6. ENERGY—SUBSTANCE

We see, therefore, that precisely in those years—75 years ago—in the minds of the leading physicists there ripened the idea of the substantiality of energy. The course of their reasoning repeated in itself what had happened in the eighteenth century with the concept of heat: that which is distributed among bodies, assimilated by them, passes from one to another, and in doing so is always conserved in an unchanged quantity—this gradually acquires, in the eyes of the interested workers, the “predicate of substantiality,” becomes a physical entity of a special kind. We now know that the adherents of “heat-substance” were wrong, were hasty in their conclusions. As yet, the doctrine of energy as a substance has not been shaken by any later data or arguments*). On the contrary, as we shall see, it has become still stronger with time and has come to reign undividedly. The reasons for this will be revealed in the further exposition.

Let us say a few words about the subsequent fate of Umov’s conception of the flow of energy. It was (apparently, quite independently) applied in 1884 by Poynting[^12] to the electromagnetic field. Poynting showed that through a unit area in a unit time there passes a flow of energy

\[ \mathbf{S}=\frac{c}{4\pi}[\mathbf{E}\times\mathbf{H}], \]

where \(\mathbf{E}\) and \(\mathbf{H}\) are the electric and magnetic field strengths at the place of the flow, \(c\) is the velocity of light. It is not difficult to show that, when recalculated for the simple case of a plane wave, this formula is identical with Umov’s statement. Poynting’s theorem is a subject of school instruction, while the name of Umov, his predecessor, is undeservedly forgotten...

Independently of this, we recommend every lover of the history of our science to reread Poynting in the original. Besides the purely aesthetic pleasure one thereby receives, it is easy, in reading, to become convinced what progress the idea of the substantiality of energy made in 10 years (1874–1884).

In this connection it is necessary to dwell on those vacillations which appeared in the minds of an entire generation of physicists and other scholars who reflected on the new substance that had entered science—energy.

First of all, this new concept was used by the idealist philosophers. They drew attention to the fact that the well-known dualism which had arisen in physics was in some way reminiscent of their dyad—matter and

) Let us note that the concept of energy as a substance is absent from the fundamental works of Mayer, Joule, and Helmholtz. And in the latter’s much later speech (1871), “On the Origin of the Planetary System,” he speaks only of a “store of work in the universe... indestructible, incapable of being increased, eternal and unchanging as matter*” (italics ours—T. K.) and nothing more.

spirit. Matter is an inert, spirit an active substance. And when later in physics there appeared, as it were, a struggle between two coexisting substances for the title of chief and fundamental one, and when victory began to incline toward energy, a downright triumphant cry arose: the physicists have left their traditional materialist position and are passing over to spiritualism!

Of course, physicists cannot be held fully responsible for such a conclusion; there is no doubt that the elucidation of the properties of energy and of the laws relating to it proceeded, above all, along the line of the investigation and accumulation of facts.

However, certain ideological errors did occur among physicists as well; among them belongs the so-called “energetics.” By this name there went in its time—not a physical system in the proper sense, but a natural-philosophical one—a system whose roots went back to Rankine, and whose chief heralds were Helm and W. Ostwald. They repeatedly proclaimed and, one may say, advertised their system; their persistence more than once provoked very authoritative rebuffs, above all from Ludwig Boltzmann. In our country, among others, A. G. Stoletov spoke out against them.^15 Reading now the works of the energetists, one is struck by how small a stock of knowledge they brought to the scientific arena. Even in those cases where their statements are close to modern views, it is easy to convince oneself that with them these statements remain absolutely unsubstantiated; standing in no connection with experimental facts, they remained quite apart from the subsequent successes of physics. A just verdict on energetics is given in V. I. Lenin’s remarkable book Materialism and Empirio-Criticism.^16

The role of Wilh. Ostwald) deserves special attention. Independently of his own scientific works (which left a sufficiently tangible mark on science), he linked his name with a fervent defense of Arrhenius’ theory of electrolytic dissociation, founding especially for this purpose the journal Zeitschrift für physikalische Chemie*. How he reconciled hostility to molecular theory with preaching the decay of molecules is quite impossible to understand. His hatred of mechanical views led him to “energetics.” Incidentally, many chemists were opponents of them—and for perfectly understandable reasons: for chemistry precisely, mechanical theories had been able to give almost nothing.

7. ON THE INERT MASS OF ENERGY

The eighties saw the elucidation of yet another important property of energy: a moving body to which a certain quantity of energy has been imparted (no matter what kind) behaves, when attempts are made

*) He began his scientific career in Dorpat and Riga, but subsequently moved to Germany (Leipzig) and became an ardent Russophobe. He especially showed himself as such during the First World War.

accelerate or slow down its motion, just as if its inertial mass had undergone an increment. This increment is proportional to the body’s store of energy.

At first this remarkable truth was expressed only with respect to electric charge. It had long ago been observed that an electric charge possesses something resembling inertia. This idea had already occurred to Faraday in 1834 in connection with his experiments on self-induction[^13], but he rejected it. Its basis is the fact that when attempts are made to increase the current in the wire carrying it, a current in the opposite direction is excited; conversely, when the current in the wire is weakened, a current in the same direction is induced. Here, however, the matter is masked by the influence of the shape of the wires, which led Faraday astray. In simpler cases later scientists repeatedly returned to the idea of this “electrical inertia.” But the first to dwell on it more attentively and to discern in the phenomenon something deeper than a simple analogy was J. J. Thomson—later the famous head of the Cambridge laboratory, but then, in 1881, still a young scientist. He calculated[^14] what would happen to a metallic sphere of radius \(R\) if it were charged with an amount of electricity \(e\). One may reason as follows: a sphere moving with velocity \(v\) possesses kinetic energy

\[ m\frac{v^{2}}{2}, \]

where \(m\) is the mass of the sphere.

If the sphere carries a charge, then its motion represents a certain current; the magnitude of this current \(i\) is proportional to the magnitude of the transported charge and to the velocity of motion,

\[ i \sim ev \quad \text{or} \quad i = Aev. \]

Around the current there forms a magnetic field, whose intensity \(H\) at each point is proportional to the strength of the current, i.e.,

\[ H \sim ev \quad \text{or} \quad H = Bev. \]

Finally, the density of the magnetic energy at each point is given by the expression (see above, § 4)

\[ w_m = \frac{H^{2}}{8\pi}. \]

Substituting here its expression for \(H\), we obtain

\[ w_m = Ce^{2}v^{2} \]

and a similar expression for the sum of all the magnetic energy surrounding the moving charged sphere,

\[ W_m = De^{2}v^{2}. \]

The sum of the kinetic energy of the sphere and its magnetic energy will be

\[ T'=\frac{m}{2}v^{2}+De^{2}v^{2}, \]

which can be represented as:

\[ T'=\frac{v^{2}}{2}(m+2De^{2})=\frac{v^{2}}{2}(m+m'). \]

The result has the form as though the mass \(m\) had increased, owing to the presence of the charge, by the amount \(m'\).

Thomson calculated the coefficient \(D\) under the simplifying assumption that the velocity of motion \(v\) is small in comparison with the velocity of light \(c\). Then for the additional (so-called electromagnetic) mass one obtains the expression

\[ m'=\frac{2}{3}\frac{e^{2}}{Rc^{2}}. \]

Thomson cautiously calls this quantity an “apparent increase of mass.”

At first sight attention was drawn to the fact that this quantity is very small because of the enormous factor \(c^{2}\) in the denominator; it can be noticed only for very small values of \(R\) and large charges \(e\). But let us write it somewhat differently, namely:

\[ m'=\frac{4}{3}\cdot\frac{1}{c^{2}}\cdot\frac{e^{2}}{2R}, \]

and recall that \(\frac{e^{2}}{2R}\) is the electrostatic energy of the charged sphere. Then we find that the apparent increase in the mass of the sphere is measured by the electrostatic energy of its charge divided by the square of the velocity of light*).

At first only a limited significance was attached to this conclusion: in the course of its derivation it seemed that it related only to electric charge; therefore the additional mass was often called “electromagnetic.” And it was treated precisely as an addition to the principal term of the true inertial mass.

The concept of electromagnetic mass acquired a more general significance at the very end of the nineteenth century, when, on the one hand, in the theoretical works of H. A. Lorentz it was proposed, and, on the other hand, in the experiments of Thomson, Kaufmann, Lenard, Simon, and others

*) We shall not here clarify the origin of the factor \(\frac{4}{3}\) and shall take up this question somewhat later.

it became clear that there exist electrons—particles with a mass that is less than the mass of the hydrogen atom by the round number of 1800 times, and whose charge is equal in absolute value to the charge of the hydrogen ion. With respect to this extremely light particle, a suspicion somehow immediately arose that it does not possess a true inertial mass, and that all its mass is of electromagnetic origin.

We shall not recount the entire historical development of the question. All the theories existing up to that time predicted that the mass of an electric charge would depend on its velocity (J. J. Thomson’s formula, given above, had been derived under the simplifying assumption that the velocity is in general small—small, that is, in comparison with the velocity of light), but different theories predicted different dependences, since their authors made different assumptions about the behavior of the electron in motion. Abraham assumed that the electron is an unchanging rigid body. Lorentz thought that, in rapid motion, the electron contracts in the direction of motion and becomes an ellipsoid compressed in the longitudinal direction.

In experiments with fast electrons they sought confirmation of one or another theory, and, incidentally, to decide a question important for us: is the whole mass of the electron electromagnetic? It was thought that only the electromagnetic mass is variable when the velocity changes, while the true mass remains unchanged. From the experiments, it seemed, one could infer that the electron behaves most nearly in accordance with Lorentz’s theory and that all its mass is of electromagnetic origin. But subsequently, as we shall see, this conclusion lost its probative force.

At first, as we have already indicated, little attention was paid to the fact that in the expression for the electromagnetic mass of an electric charge there stands the energy of the charge. But somewhat later the young Viennese scientist Hasenöhrl found a new case in which the same apparent mass must manifest itself. He imagines a certain cylinder, polished on the inside and—in theory this is possible—completely reflecting all rays incident on its inner walls. Let us imagine that, for a negligible interval of time, we opened a small aperture and let a beam of light enter the cylinder, and then immediately closed the aperture again. The radiant energy caught in the trap can no longer get out, since the walls reflect it completely. But it also cannot be absorbed by the material of the cylinder; it is condemned forever to move between the walls of the cylinder in all possible directions. This experiment, of course, belongs to the class of imaginary ones.

Not long before Hasenöhrl’s arguments, our compatriot, the teacher of many Moscow physicists, Pyotr Nikolaevich Lebedev, made the strongest impression on the whole scientific world by being the first to prove that rays of light, falling on an obstacle barring their path,

produce pressure upon it. Lebedev’s elegant experiments, which were extremely difficult for their time, agreed in their quantitative conclusions with the prediction of Maxwell’s theory; namely, the pressure per unit area of an obstacle should numerically be equal to the density of light energy in the space adjacent to the obstacle. Lebedev obtained results differing from the theory by no more than 20%. To any physicist familiar with experimental technique it was clear that Lebedev’s experiments stood, in general, at the limit of what was possible to carry out, and that the accuracy he attained was the height of perfection achievable at that time. The whole world acknowledged that the existence of light pressure had been proved, and that the value found coincided with Maxwell’s theoretical predictions.

Hasenöhrl, probably, was also under the general impression made by the proof of the forces of light pressure, and made their existence the basis for his reasoning about his imaginary experiment. Thus, let us suppose together with the author that we push the cylinder; it will acquire, as mechanics teaches, a certain acceleration. As a result, at that end of the cylinder which is behind, the radiant energy will be condensed; conversely, at the front end it will be relatively rarefied. But the dense energy, according to Maxwell–Lebedev, will press on the adjacent (rear) wall more strongly, while the rarer energy (at the front wall) will press more weakly. A certain additional force is created, acting in the direction opposite to our push. The acceleration of the cylinder decreases, as if its mass had increased. Hasenöhrl was able to cast these qualitative considerations into the form of mathematically rigorous derivations—rather, it must be admitted, complicated ones. They will not interest us, but the result is of exceptional importance for our subject: according to Hasenöhrl, the additional, or apparent, mass that the cylinder acquires has the following expression:

\[ m'=\frac{4}{3}\frac{\mathcal{E}}{c^{2}}, \]

i.e., in appearance identical with the one we obtained for the case of electrostatic energy.

After this it is no longer possible to dismiss, as something particular, the fact that energy concentrated in some body increases its mass; the further question arises, to what extent the entire inertial mass of a body, observed by us when forces act upon it, reduces to such an energetic mass. About this—somewhat later; but let us note also that electromagnetic theory, especially its modification given by Lorentz in his electron theory, makes it possible, in a very general form, to indicate that the quantity of “apparent” mass receives a contribution not only from electromagnetic energy proper, but also from the work expended by electromagnetic forces and, consequently, transformed into other forms of energy; not only electrostatic,

and magnetic, thermal, light, elastic, chemical—in short, any energy imparted to a body increases the mass of the body by the corresponding amount.

Let us dwell on two or three particular examples: 1) We have seen above that the energy of an electrostatic charge (and therefore, in particular, of the elementary charge—the electron) is concentrated in the space around the charge, gradually decreasing from the surface of the charged body to the parts of space remote from it. Hence the mass of the charge is also distributed throughout all space; the mass of the electron (i.e. the electron itself!) is not concentrated in some small region of radius about \(3\cdot 10^{-13}\) cm, but is smeared out through all space. We see that the concept of the “smearing out” of the electron existed much earlier than it was developed—of course, on an entirely different scale—by modern quantum theory with its “uncertainty principle.”

2) I should like to say a few words about the coefficient \(\frac{4}{3}\), which figures in the expression for the “apparent” mass of an electrostatic charge and of radiant energy. In the example of the latter the question is resolved very easily: in order to create a cavity filled with radiant energy, it is necessary: a) to expand the cavity, doing work against the forces of light pressure; the work of pressure is equal to \(v\frac{\varepsilon'}{3}\) (\(\varepsilon'\)—energy density, \(v\)—volume of the cavity); b) to fill the cavity with energy of the same density, for which \(\varepsilon'v\) of radiant energy is required, in all

\[ \frac{4}{3}\,\varepsilon'v=\frac{4}{3} \]

of that energy which is contained inside the cavity. This is where the factor \(\frac{4}{3}\) comes from. The general expression for the mass will be

\[ m=\frac{\varepsilon'}{c^2}. \]

3) The question of the inertial mass of kinetic energy is very difficult. Attention was also drawn to this by V. I. Lenin*). One can save the situation by noting that, for example, the kinetic energy of a moving charge is, properly speaking, as we saw above, the energy of the magnetic field formed around the electron. It is already much easier to associate the idea of inertia with a field (tension along lines of force, etc.).

4) And, finally, one last remark. We know nothing about the “free ether”—about empty space lying between interacting bodies. We begin to sense the varied actions of this “field” when we introduce some body into it: an electrified ball, a mirrored cavity, etc. In the case of the latter we see, in fact, not the inertia of the radiant

* See V. I. Lenin, Materialism and Empirio-criticism, section 3 of Chapter V.

energy, but a change in the inertia of emptiness. In the case of the charged sphere we saw a change in its inertia, and not the inertia of the energy associated with it. Are we not making a logical leap, identifying the result of our investigation of inertia with the energy that appears in our formulas? At present we consider such a mode of action the most natural, and its result—the further substantivation of energy—the most plausible solution of the question of the fundamental substance in nature. But if someday this doctrine should lead us into a dead end, into which the natural-scientific worldview has more than once fallen in search of especially broad generalizations, then in that case it is precisely here that one may seek a way out of the difficulties that have arisen.

8. FURTHER DEVELOPMENT OF THE THEORY

Strictly speaking, it is not quite correct to speak of a further development—the circle of ideas of which we wish to speak further arose in parallel with what has been set forth above. The matter concerns the transmission by radiation not only of energy (for example, solar heat), but also of force, and the question arises in connection with the finite velocity of radiation. Suppose that a certain quantity of radiant energy has left the Sun and set out toward the Earth. After 500 seconds (in round numbers) it will reach the surface of the Earth and will exert pressure upon it. From the point of view of mechanics it is now necessary to pose two questions:

Does the fundamental law of dynamics—the law of equality of action and reaction—remain valid in this case? That is, in other words: does the radiation also press upon the Sun—at the moment when it leaves it? And the second question: how does this law remain valid if the first action is separated from the second by an appreciable interval of time—in the case described, by as much as 500 seconds (more than eight minutes)?

These questions were posed more than fifty years ago—in the nineties of the last century. In the book by J. J. Thomson already mentioned by us[^14] we can find mention of them on p. 9. We shall return to the quantitative side of the question (there it is analyzed) later. From the qualitative side they are solved as follows: from the point of view of Newton’s 2nd law, we call the force with which one body acts on another that amount of motion which the first body gives to the second per unit time.

\[ f=\frac{\Delta(mv)}{\Delta t}, \]

where \(\Delta(mv)\) is the amount of motion imparted, and \(\Delta t\) is the small interval of time during which the transfer takes place. Thus, if we want the laws of mechanics to continue to hold in the case

for radiation, the radiating body must give up a certain amount of motion. It is simplest of all to suppose that this amount of motion is given up together with the radiant energy and is transported together with it, i.e., simply, by it. Then we shall arrive at the conclusion that radiant energy is endowed with a certain amount of motion. From this it is but one step to recognizing that it is endowed with mass. This is a new approach to the idea developed above.

The quantitative side of the question was solved, as has been said, already by Thomson, but not completely. He stopped halfway in a curious manner, noting only that the amount of motion is transported in the same direction as the energy. Now we can write a more meaningful equation:

\[ g=\frac{S}{c^{2}}, \]

where \(g\) is the density of the transported amount of motion, \(S\) is the energy flux (see above), and \(c\) is the speed of light. We see that between the two quantities of interest to us there exists not only the relation that they propagate in the same directions—they are simply proportional to one another.

From the point of view of the vector of the amount of motion, the question of the interaction between the Sun and the Earth is interpreted as follows: in giving up radiant energy, the Sun also gives up an equivalent amount of motion; at the same time it experiences a “recoil,” i.e., a pressure force in the direction opposite to the flux given up. The amount of motion does not disappear—it is transported together with the radiant energy, reaches the surface of the Earth, and is given up to the latter. Thus, the law of conservation of the amount of motion is satisfied.

In this form the question was posed and solved by the recently deceased M. Planck in 1908.[^17] In no less general a form it had been formulated by Lorentz in his first book expounding his electron theory,[^18] and by M. Abraham, who applied the concept of “electromagnetic amount of motion” to the solution of the question of the equations of motion of the electron.[^19]

The existence of this electromagnetic amount of motion, distributed throughout the volume, somewhat complicates the consideration of the question of light pressure. As is known, Maxwell himself, who predicted this pressure, derived it by very unrigorous reasoning, supposing that the electric and magnetic forces acting in a light wave produce in the ether the same stresses as in a static, i.e., unchanging, field. Meanwhile, in light waves they change hundreds of billions of times per second. A new derivation of the forces of light pressure was needed, but it gave a reassuring result: in periodic waves the amount of motion transported by them is also expressed by a periodic function; for such a periodic func-

...whose change over an entire period or over a whole number of periods is equal to zero—the Maxwell conclusion for periodic processes can be drawn as though no electromagnetic momentum existed.

In this connection one cannot fail to recall the participation of one remarkable work by our compatriot, Professor of Yuryev University A. I. Sadovsky^20. At the beginning of the 1890s he wrote a study of the rotational forces which light rays must exert on a crystalline plate (the “Sadovsky effect”) and on a rotating plane of polarization and an absorbing plate. At the beginning of his book A. I. Sadovsky, following the strict custom of earlier years, set down those necessary assumptions which had to be made for the subsequent derivations. Among these assumptions is the following: “Rapidly varying light fields act in the same way as static ones.” The work of A. I. Sadovsky—perhaps precisely because of this “excessive” conscientiousness of his—had no success as a dissertation... Only later was it properly appreciated by P. S. Ehrenfest, who was then living in Russia.

9. THEORY OF RELATIVITY

We have deliberately expounded all the preceding without mentioning the theory of relativity, in order to show that the whole circle of ideas which took shape in physics on the question of energy and its mechanical attributes arose and developed independently of Einstein’s theory. No sufficiently erudite physicist will deny this, if he simply consults the chronology. But all these questions were taken up from its own point of view by the theory of relativity as well—and it solved them with extraordinary simplicity and ease, proceeding solely from its basic principles, making no additional hypotheses—not using, for example, the foundations of Maxwellian and Lorentzian electrodynamics. As is known, the theory of relativity produced a stunning impression by the boldness of its solutions, the unexpectedness and categorical character of its results. But those who did not accept it also did not accept many of those conclusions for which the theory of relativity is responsible to the same extent as are the previous—classical, more customary—theories. This applies completely to the concept of the inertial mass of energy. It follows entirely from the theory of relativity as well. For the first time, in its most general form, the relation

\[ m=\frac{\varepsilon}{c^2} \]

was written precisely in the theory of relativity.

It must be mentioned that, at a later stage of its development—in the so-called general theory of relativity—A. Ein-

stein shows that energy—in particular radiant energy as well—must be endowed with other mechanical attributes, for example, weight, so that a ray, passing by a large mass (near the Sun, for example), must be attracted by it and correspondingly curved. Arising near an attracting center, a ray must “redden”—the length of the emitted wave must shift toward longer waves, etc.

In short, universal gravitation must act between portions of energy just as it acts between portions of matter. All forms of energy, including kinetic energy, obey this rule. Our contemporary V. A. Fok carried out calculations further than others and showed that, if in general relativity one considers the action, say, of the solar system on a point very remote from it, then it will be the same as if the entire mass of the Sun and of all the planets were acting, plus the kinetic energy of all the bodies of the solar system, divided by the square of the speed of light²¹.

From this point of view one may assert that the theory of relativity has introduced further order into our conceptions of energy, has led to the establishment of additional common features between mass and energy: the latter has acquired the properties of mass not only inertial but also gravitating.

But there are in the system of the theory of relativity also deeply destructive elements, manifested, perhaps, most vividly precisely in the question that interests us—the kinship between mass and energy. The theory of relativity definitively completed the destruction of the concept of the ether, begun even before it by physicists concerned with questions of phenomena in moving bodies. They had already shown earlier that, in order to explain all these phenomena, it is impossible to endow the ether with noncontradictory properties: to explain some phenomena the ether must remain immobile in a moving body; to explain others it must be dragged along by the moving body. The theory of relativity, denying a system of absolute coordinates in space, generally throws overboard from physical knowledge the notion of a physical world medium filling the geometrical space of the universe.

This creates great difficulties for us in interpreting the concepts of the field: what are field stress, field energy, field forces, mass in a field, if the material substratum of this field—the substratum bearing all its physical properties—does not exist?

Then Lenin’s words that “motion without matter is impossible” stand forth in all their rigor: what can we oppose to them, if there is no ether, no stresses in the ether, etc.? Only one thing: for the time being we know too little about the physical structure of the field and must await the time when further investigations clarify this question.

On the other hand, the theory of relativity, more than all its predecessors, accustomed us to the idea of the equivalence between mass and energy. Let us cite the following example: Aston, who discovered isotopes of stable atoms, drew attention to deviations in the atomic weights of more complex atoms from integral ratios: in the middle part of Mendeleev’s table the atomic weights are somewhat (slightly!) less than the total weight of the neutrons and protons making up the atom. Where, then, did the mass “disappear” in the construction of the atom?—No one hesitated to admit that the atom is a strongly exothermic compound and, in its formation, released an enormous quantity of heat. It is this that caused the loss of mass.

If, for example, we learned under terrestrial conditions to synthesize hydrogen into helium, we would likewise obtain tremendous quantities of heat in the form of the equivalent of the lost mass, since 4 protons have a mass of 4.030, and helium—4.000; this means that from 4 grams of hydrogen six million large calories would be released! Indeed, the whole question of “atomic” energy, with all its political, military, and cosmic possibilities, is connected with a single equation:

\[ m=\frac{\varepsilon}{c^2}. \]

10. QUANTUM THEORY

Soon (in 1950) we shall be celebrating the semicentennial anniversary of quantum theory. Its creator, the recently deceased German scientist M. Planck, lived only a little short of the anniversary of his fundamental discovery. Let us recall that in his theoretical investigation of the dependence of “black” radiation on wavelength and temperature, Planck \(^{22}\) made the temporary assumption that energy is divided into the tiniest, yet nevertheless indivisible, portions—“quanta,” as he called them. He later wished to abandon this assumption by taking the size of the quantum to be infinitely small. But this could not be done: when the magnitude of the quantum was decreased, the formula obtained by Planck degenerated into another formula, long since refuted by experiment, the Rayleigh–Jeans formula. Planck’s original formula, however, agreed remarkably well with the results of all experiments.

Thus, unexpectedly, the idea of quanta was born, and energy, alongside ponderable matter and electricity, revealed discreteness, or an atomistic character—a new similarity between energy and matter. We shall not speak of the subsequent successes of this idea. Let us note some differences between the “atoms of energy” and the atoms of ponderable matter.

To satisfy the previously found properties of radiation, Planck had to suppose that the quanta corresponding to different rays are different, namely, that the energy contained in one quantum,

“proportional to the number of oscillations characterizing the radiation:

\[ \varepsilon = h \nu, \]

where \(h\) is an exceedingly small “Planck constant.” From this point of view the quanta of infrared light \((\lambda = 500\ \mu)\) must be exceedingly small, whereas the quanta of hard X-rays \((\lambda = 0.01\ \text{Å})\) are comparatively enormous, and the interval of this multiplicity would occupy approximately 50 octaves—a diversity far greater than the diversity of atoms (from 1 to 240).

But whereas the magnitude of atoms changes from one to another always by an integral number of units, the magnitude of a single quantum may differ—and in fact does differ—by any arbitrarily small amount, since we may suppose \(\nu\) to vary by any amount. This is a very fundamental distinction. An atom cannot be broken apart except by taking from it an integral number of neutrons or protons. Quanta we have learned to change by the most varied methods by any arbitrarily small amount.

Quanta in general represent another class of particles, as is evident from the following considerations: simple reasoning leads us to the following relation: if the mass of a particle at rest is equal to \(m_0\), then its mass at velocity \(v\) is given by the formula:

\[ m^2 (c^2 - v^2) = m_0^2 c^2. \]

If \(c \ne v\), then

\[ m = \frac{m_0}{\sqrt{1 - \frac{v^2}{c^2}}}. \]

This is the well-known law according to which the mass of a particle changes with the magnitude of its velocity. Such are atoms, neutrons, protons, electrons, and positrons. But if \(v = c\), we can say nothing about \(m\); however, with respect to \(m_0\) it follows quite convincingly that it is equal to zero, i.e. only those particles can move with the speed of light whose initial, so-called rest mass is equal to zero. Such are quanta, or photons.

Thus all particles in nature fall into two kinds: some—with initial mass, others—without initial mass.

Can particles of one kind pass into the other? It is assumed that an electron and a positron, colliding with one another, cease to exist, producing at the same time two quanta of enormous magnitude. This phenomenon has been called by the entirely unsuitable name of “annihilation,” i.e. the destruction of matter. Of course, no destruction takes place here; there is only a transition of particles with initial mass into particles without initial mass.

CONCLUSION

We are approaching the end of our survey. We have gathered all the necessary material and can proceed to its general interpretation. Energy appears to us as a certain substance, in every respect similar to ponderable matter and endowed with all those properties that compel us to regard ponderable matter as a substance: it is indestructible and cannot be created; it is localized in space; it moves and is transmitted; it possesses inertial mass; it is ponderable; it is divided into atoms. An exact law of equivalence between energy and matter is established. It may be asserted that both are, in equal measure, what we call matter.

CITED LITERATURE

1. Julius Robert Mayer, “Remarks on the Forces of Inanimate Nature” (1842); idem: “Organic Motion in Its Connection with Metabolism” (1845). In the edition Classics of Natural Science, GTTI, Moscow–Leningrad, 1933.

2. Joule’s Scientific Papers, vol. I. On the thermal effects of magneto-electricity and on the mechanical value of heat (1843), as well as a whole series of subsequent works (1844, 1845, 1848, 1850, 1867, 1878).

3. H. von Helmholtz, On the Conservation of Force, “Classics of Natural Science,” GIZ, Moscow, 1922; the same, Moscow–Leningrad, 1934, as well as numerous speeches and reports by the same author in his Vorträge u. Reden.

4. N. A. Umov, “The Theory of Simple Media and Its Application to the Derivation of the Fundamental Laws of Electrostatic and Electrodynamic Interactions.” Memoirs of Novorossiysk University, 10, pp. 1–60 (1871).

5. P. P. Lazarev, N. A. Umov. 1940. (Jubilee edition for the 135th anniversary of the Moscow Society of Naturalists.)

6. J. Clerk Maxwell, A Treatise on Electricity and Magnetism, part I, chapter V, § 111.

7. M. Faraday, Experimental Researches in Electricity, vol. 1, series 11, chap. 5 (pp. 520–539).

8. N. A. Umov, “Equations of the Motion of Energy in Bodies” (1874).

9. A. I. Bachinsky, An Essay on the Life and Works of N. A. Umov. Moscow, 1916.

10. P. P. Lazarev, N. A. Umov (1846–1915), Moscow, 1940 (jubilee edition of the Moscow Society of Naturalists).

11. Lagrange, Mécanique analytique, II, p. 5 (cited from the 1815 edition).

12. Poynting, “On the Transfer of Energy in the Electromagnetic Field,” Phil. Trans., part II, 343 (1884).

13. Faraday. See above, 7, p. 439, § 1077.

1. J. J. Thomson, “On the electric and magnetic effects produced by the motion of electrified bodies,” Phil. Mag., 1881, or in his book Recent Researches in Electricity and Magnetism (1893), pp. 16–23.

2. A. G. Stoletov, Collected Works, II, p. 319 ff.

3. V. I. Lenin, Materialism and Empirio-Criticism, ed. 1931, p. 220 ff.

4. M. Planck, Phys. Zeits., 1908, p. 828.

5. H. A. Lorentz, Versuch einer Theorie etc., p. 28 (ed. 1906).

6. M. Abraham, Theorie d. Elektrizität, p. 24 (ed. 1920).

7. A. I. Sadovsky, “Ponderomotive Actions of Electromagnetic and Light Waves on Crystals,” Scholarly Notes of Yuriev University, 1899.

8. V. A. Fok, ZhETF 9, p. 375 (1939).

9. M. Planck, Ann. d. Phys. 4, p. 553 (1901).