The Concept of Mass and Energy in Modern Physics

S. E. Frisch

1. Physical concepts, like the concepts of any other science, are established not independently of the prevailing philosophical systems, despite the opinion widespread among many bourgeois scientists that in their scientific investigations they are free from any philosophical tendencies. To these scientists Engels’s words are fully applicable: “Natural scientists imagine that they free themselves from philosophy when they ignore it or abuse it... In the end they nevertheless prove to be in the thrall of philosophy, and, unfortunately, for the most part of the very worst kind...”*).

The physics of the eighteenth and the first half of the nineteenth centuries developed mainly in a materialist spirit. However, the materialism of the majority of scientists of that time was mechanistic in character. To explain a physical phenomenon meant to reduce it to a mechanical model—to the motion of immutable, isolated particles—atoms, or to mechanical changes in hypothetical continuous media. This mechanistic spontaneous materialism, by virtue of its limitations, was not free from metaphysics. Attempts were made to form notions of the physical properties of objects in isolation from their interrelations with other objects. Notions of the concrete forms of matter with which physics deals were confused with the philosophical concept of matter.

At the end of the last century, among bourgeois scientists, a retreat from materialist positions toward idealism began to be observed more and more often. This retreat was caused to a significant degree by the sharpening of the class struggle. The ruling circles of the capitalist countries began to make use of idealist theories as a means of struggle against the emerging progressive ideology

) F. Engels, Dialectics of Nature*, pp. 164–165, Gospolitizdat, 1950.

of the proletariat. The second reason for the deviation toward idealism was the collapse of old theories under the influence of new scientific discoveries and the need to seek new viewpoints. Lenin pointed to this latter circumstance when he wrote: “The deviation toward reactionary philosophy ... is a growing pain, caused above all by the abrupt breakdown of old established concepts”*).

Reactionary Machian philosophy became especially widespread among bourgeois physicists of the late nineteenth and early twentieth centuries. This spread was due to the seeming “scientific character” of Machism, the veiled nature of its idealistic tendencies, and its greater “adaptability” to physics than other idealistic philosophies. A kind of fashion arose for using Machian terminology; moreover, this fashion was often followed by scientists to whom Machism was, in essence, alien. The erroneous idea became widely disseminated that the definitions presupposed in the formulation of the fundamental physical laws are supposedly arbitrary and independent of the laws themselves, and that the laws acquire one or another physical content depending on the adopted “definitions.”

One can avoid both the indicated and other errors in interpreting the fundamental propositions of physics only by firmly adhering to the theory of knowledge of dialectical materialism. It is necessary clearly to distinguish the philosophical, epistemological conception of matter from those concrete conceptions which physics gives of definite particular forms of moving matter. In this respect we have numerous exhaustive statements by the classics of Marxist-Leninist philosophy. Engels writes: “Therefore, matter and motion can be known only by studying individual substances and individual forms of motion; and insofar as we know the latter, to that extent we also know matter and motion as such”**). And further: “Matter as such is a pure creation of thought and an abstraction. We abstract from the qualitative differences of things when we unite them, as corporeally existing, under the concept of matter. Matter as such, in distinction from definite, existing matters, is therefore not something sensuously existing. When natural science sets itself the aim of finding uniform matter as such and reducing qualitative differences to purely quantitative differences formed by combinations of identical smallest particles, it proceeds in the same way as if, instead of cherries, pears, apples, it wished to see fruit as such...”***).

*) V. I. Lenin, Materialism and Empirio-Criticism, p. 287, Gospolitizdat, 1949.
**) F. Engels, Dialectics of Nature, p. 187, Gospolitizdat, 1950.
***) Ibid., p. 203.

The same idea is expressed by Lenin when he says: “But it is absolutely impermissible to confuse, as the Machists do, the doctrine of this or that structure of matter with the epistemological category—to confuse the question of the new properties of new kinds of matter (for example, electrons) with the old question of the theory of knowledge, the question of the sources of our knowledge, of the existence of objective truth, and so on.”*)

In connection with what has been said, it is worth briefly dwelling on the question recently raised in our Soviet literature of whether the classics of Marxism-Leninism introduce two concepts of matter—the philosophical and the physical.**) From the statements of Lenin and Engels cited above it is quite obvious that there can be no question of different concepts of matter, since physics, like any other science, studies particular, concrete forms of matter and of its motion, understood in their philosophical sense. Consequently, the concepts of matter developed by physics (as by any other science) cannot be opposed to the philosophical concept of matter, just as a more particular concept cannot be opposed to a more general one that embraces this particular concept.

Dialectical materialism asserts that matter is not something immutable, but is in a state of continuous change, development—motion, understood in the general, philosophical sense of the word. “Motion is the mode of existence of matter. Nowhere and never has there been, nor can there be, matter without motion” (Engels***)). Moving matter can be known only by considering the particular, concrete forms of matter and of its motion; moreover, these particular forms of matter and of its motion must be considered not in isolation, but in their interconnection. This latter, extremely important circumstance is expressed in the remarkable words of Comrade Stalin: “…no phenomenon in nature can be understood if taken by itself, in isolation from surrounding phenomena, for any phenomenon in any realm of nature may be turned into an absurdity if considered out of connection with the surrounding conditions, in isolation from them; and, conversely, any phenomenon can be understood and substantiated if considered in its inseparable connection with surrounding phenomena, as conditioned by the phenomena surrounding it.”****)

Finally, for a correct understanding of the place and role of physical discoveries in the general system of our knowledge, it is always necessary

*) V. I. Lenin, Materialism and Empirio-Criticism, p. 112, Gospolitizdat, 1949.

**) See, for example: M. V. Pozner, Science and Life, No. 1, 1949; S. G. Suvorov, Uspekhi fizicheskikh nauk, vol. XLIV, issue 4, 1952.

***) F. Engels, Anti-Dühring, pp. 56–57, Gospolitizdat, 1950.

****) I. V. Stalin, On Dialectical and Historical Materialism, p. 5, Gospolitizdat, 1950.

remember that scientific concepts only approximately and relatively correctly, in accordance with the level of development of science and practice at a given historical stage, reflect objective reality. Absolute truth is composed of the sum of relative truths. Since the world is infinitely varied and inexhaustible, what is possible is not the attainment, but only the continuous approximation to absolute truth. Lenin teaches us: “...dialectical materialism insists on the temporary, relative, approximate character of all these milestones in the knowledge of nature by the progressing science of man”*).

2. Physics forms concepts of particular, concrete forms of matter in motion on the basis of observations, experiments, and the theories that generalize them. The concept of any physical quantity characterizing one or another objective property of the concrete form of matter in motion under consideration can be established only through the totality of the regular interrelations of this quantity with other physical quantities.

Proceeding from this proposition, it is necessary to consider the very important question of the role and place of “definitions” in physics. Undoubtedly, one cannot do without definitions, and an attempt to take into account at once a large number of interrelations would make it impossible to single out the main thing and to understand correctly the phenomenon under study. But a “definition” uses one or a few interrelations, and thanks to this it has, in its way, a preliminary character. Whether the concept introduced by means of such a “definition” corresponds to objectively existing relations is revealed only after a sufficiently broad totality of interrelations and of the laws to which they are subject has been examined.

Let us explain what has been said by means of a simple example taken from mechanics. As is known, in mechanics the elementary work \(\Delta A\) is defined as a physical quantity numerically equal to the product of the displacement \(\Delta s\) of the point of application of the force and the projection of the force \(f_s\) onto the direction of this displacement:

\[ \Delta A = f_s \Delta s. \tag{1} \]

However, such a “definition” has meaning only thanks to the existence of a connection between work and the change in the energy of the system to which the forces performing the work are applied. Science makes use—

*) V. I. Lenin, Materialism and Empirio-Criticism, p. 245, Gospolitizdat, 1949.

...by the concept of work, defined by equality (1), not because we wanted or found it “convenient” to introduce such a concept, but because it reflects the properties of a large number of real processes which are at the same time characterized by a number of other physical quantities. Therefore work is not “the product of force by distance,” but is an “independent” physical quantity.

After what has been said, it becomes perfectly clear that a “definition” can never be considered in isolation from those fundamental laws to which the physical quantity under consideration is subject and which establish its interrelation with other physical quantities.

In the course of the development of physics there arises the need to refine and extend existing “definitions,” or even to replace them with new ones. However, such replacement of one definition by another does not mean the possibility of arbitrariness, but leads (leaving aside cases where the new definition is erroneous) to the fact that the new definition more fully and better reflects reality, that it is a step forward in our process of knowing nature. The need for new definitions in physics usually arises when, under the influence of newly discovered facts, the inadequacy or unsuitability of the old definitions is revealed. At the same time the degree of approximation of the old concept is disclosed, as well as its suitability for describing the real features of only a limited group of facts. Hence it is once again confirmed that the old definition, too, was not accidental or arbitrary, but reflected reality with that degree of approximation which was inevitable at the previously existing level of our knowledge of the nature surrounding us.

1. Proceeding from the general propositions indicated in the preceding section, let us analyze the development in physics of two such basic concepts as the concepts of mass and of energy.

The concept of mass, as is well known, was introduced into physics by Newton in his Mathematical Principles of Natural Philosophy (1686). At the same time, although Newton established the fundamental laws that connect mass with other physical quantities, the very definition of the concept of mass in his work still had a metaphysical character. Newton believed that mass is a measure of the quantity of matter in a body. This definition remained in most textbooks almost to our own day. Thus O. D. Khvolson, in the first volume of his Course of Physics (4th ed., 1914), gave the following definition of mass: “the mass of a body is measured by the quantity of matter contained in it.” This definition was repeatedly criticized, including by...

on Mach’s part, who not without reason asked: why do you know that in one atom of sodium there is more matter than in one atom of hydrogen? However, having made this just criticism of the metaphysical Newtonian definition of mass, Mach hastens to formulate a definition of mass in the spirit of his subjective-idealistic philosophy: he asserts that mass is only a coefficient occurring in certain equations of motion. In the same spirit we encounter definitions also in some contemporary physics textbooks.

For example, A. F. Ioffe) writes: “we may define mass as the derivative of the quantity of motion with respect to velocity, and replace a change in the quantity of motion by an equal impulse of force.” S. E. Khaikin in the first edition of his Mechanics*** ) supposed that “depending on how we choose the method of measuring mass, the character of the assertions contained in Newton’s 2nd and 3rd laws turns out to be different.”

A more correct introduction of the concept of mass is given in the Course of General Physics by N. I. Dobronravov and coauthors) and in E. A. Shtrauph’s Molecular Physics**), where mass is considered as a physical quantity of a special kind, the content of which is revealed on the basis of Newton’s 2nd law.

Proceeding from the general proposition formulated by us at the beginning of Section 2, it must be held that the content of the concept of mass is revealed only when considering a sufficiently broad series of facts and laws that cannot be considered in isolation from one another.

Classical mechanics first arrives at the conception of mass by considering the phenomena of inertia. Here, speaking of the inertia of bodies, one assumes that bodies differ in a certain objective property, manifested in the fact that they acquire unequal accelerations under identical external actions. This property, inherent in all bodies, can be characterized by a certain physical quantity, which is mass.

However, in order to give a quantitative characteristic of mass, it is necessary to be able to establish, in some way independent of the indicated fact, the equality of external actions on different bodies—actions leading to the appearance of accelerations in them. This can be done in various ways; for example, the action of one body on another can be transmitted

) A. F. Ioffe, Course of Physics*, vol. I, 3rd ed., Gostekhizdat, Moscow–Leningrad, 1940.

) S. E. Khaikin, Mechanics, Gostekhizdat, Moscow–Leningrad, 1940.

*) N. I. Dobronravov et al., Course of General Physics, vol. I, 3rd ed., Gostekhizdat, 1941.

**) E. A. Shtrauph, Molecular Physics, Gostekhizdat, 1949.

through some definite third body capable of experiencing elastic deformation (a spring, a stretched cord, etc.), and to assume that the actions are identical if the deformations of this third body are identical. One may proceed from Newton’s third law, formulated in the following way: the actions of two bodies upon one another are always equal in magnitude and directed so that the accelerations caused by them are directed in opposite directions. Experiments show that both of these methods lead to the same result. Having made use of one of these methods for establishing the equality of actions, we can define mass as a quantity inversely proportional to those accelerations which bodies acquire under the influence of an identical action. If two bodies, under the influence of an identical external action, acquire different accelerations \(w_1\) and \(w_2\), then the ratio of their masses \(m_1\) and \(m_2\) satisfies the equality:

\[ \frac{m_1}{m_2}=\left|\frac{w_2}{w_1}\right|. \tag{2} \]

Relation (2) makes it possible to compare quantitatively the masses of different bodies; the mass measured in this way may be called inertial mass, since it is measured on the basis of phenomena denoted as phenomena of the inertia of bodies.

Relation (2), taken in isolation, gives only a limited conception of the physical quantity—mass. For a fuller disclosure of the content of the concept of mass, it is necessary to consider it together with other facts. Only a sufficiently broad set of lawfully interconnected facts convinces us that the physical quantity—mass, whose numerical value can be found by means of relation (2), characterizes one of the fundamental properties of matter, and, as will become clear later, of other kinds of matter as well. Let us indicate the most basic of such facts: a) the mass of an isolated system remains constant under all changes occurring in it; b) the quantity of motion of an isolated system, which is the vector sum of the products \(m_i\mathbf{v}_i\), also remains constant (here \(m_i\) are the masses of the bodies forming the system, and \(\mathbf{v}_i\) are the vectors of their velocities).

Let us now turn to another group of phenomena—to the phenomena of attraction (or gravitational phenomena), which also lead to the conception of mass.

As is well known, Newton also established the so-called law of universal gravitation, according to which any two bodies, whose dimensions are small in comparison with the distance between them, attract one another with a force \(f\) directly proportional to the product

proportional to their masses and inversely proportional to the square of the distance between them:

\[ f \sim \frac{m \cdot m'}{r^2}. \tag{3} \]

If relation (3) is considered in isolation from relation (2), then it defines a new physical characteristic of bodies—the “gravitational mass,” which determines the ability of bodies to create a field of gravity and to experience the action of the gravitational field. What relation the “gravitational mass” bears to the “inertial mass” already introduced by us can be clarified only by observations. Newton himself identified the two masses, proceeding from the fact that the accelerations acquired by all bodies near the surface of the Earth under the influence of the force of attraction are the same, and also from the validity of Kepler’s third law. In order to establish with what degree of accuracy all bodies experience the same acceleration near the surface of the Earth, Newton carried out special experiments with swinging pendulums. These experiments were subsequently repeated with a high degree of accuracy, and other experiments were also performed (Eötvös’ experiment), which could have revealed a difference between “gravitational” and “inertial” masses, if such a difference existed. It turned out that the two masses are not only always strictly proportional to one another, but also always exist together, i.e. there is no body possessing only “inertial mass” and not possessing “gravitational mass,” or conversely. Thus, experiments show the complete impossibility of distinguishing “gravitational” and “inertial” mass. Consequently, both in inertial and in gravitational phenomena we are dealing only with different manifestations of one and the same physical quantity—mass.

In introducing the concept of mass, we began with inertial phenomena and used relation (2) for measuring masses. One could have proceeded otherwise and begun with gravitational phenomena, establishing by means of relation (3) a method of measuring masses analogous to the way in which electric charges are measured on the basis of Coulomb’s law, which is formally similar to the law of gravitation. Such a method of reasoning would not have introduced any fundamental changes, since a complete conception of mass is formed only on the basis of the whole set of facts, including both gravitational and inertial phenomena.

At the same time, if the numerical value of mass is determined by both methods—inertial and gravitational—it will turn out that these two methods are not equally “sensitive.” This difference in “sensitivity” is due to the fact that gravitational forces are practically noticeable only when at least one of the two gravitating masses is large; the forces that must be applied ...

apply, in order to impart to a small mass a relatively small acceleration, is quite measurable. For example, it is practically impossible to detect the force of gravitation between two bodies each of whose masses is equal to 1 g. But the force that must be applied to a body with a mass of 1 g in order to impart to it even a relatively small acceleration is easily measurable.

This circumstance is reflected in the relation between the numerical values of mass obtained once on the basis of Newton’s second law, and a second time on the basis of the law of gravitation, with the same units of measurement for force, length, and time. Suppose that at one time we determine the numerical value of mass, for given units of measurement of force, length, and time, with the aid of Newton’s second law:

\[ f=m_{\mathrm{i}}w, \tag{4} \]

where the subscript “i” on the mass \(m\) means that the numerical value of this mass is determined through an inertial phenomenon. In the system of measurement adopted by us, where the units of force, length, and time have been chosen as the fundamental units, the dimension of mass will be

\[ [m_{\mathrm{i}}]=FL^{-1}T^{2}, \]

where the symbol \(F\) denotes the dimension of force.

Let, next, the numerical value of the very same mass be determined with the aid of the law of gravitation, using the same units of measurement of force, length, and time. Then:

\[ f=\frac{m_{\mathrm{g}}\cdot m'_{\mathrm{g}}}{r^{2}}, \tag{5} \]

where the subscript “g” on the mass \(m\) means that its numerical value is determined through a gravitational phenomenon. The dimension of mass in this method of measurement is

\[ [m_{\mathrm{g}}]=F^{1/2}L. \]

In the first system of measurement the law of gravitation will take the form

\[ f=k\frac{m_{\mathrm{i}}\cdot m'_{\mathrm{i}}}{r^{2}}, \tag{6} \]

where \(k\) is a constant (the “gravitational constant”) having, in our system, the dimension \(F^{-1}L^{4}T^{-4}\) and the numerical value \(k=6.685\cdot10^{-8}\), when force, length, and time are measured respectively in dynes, centimeters, and seconds. From comparison

from equalities (5) and (6) it follows that:

\[ m_r=\sqrt{k}\cdot m_i. \tag{7} \]

The presence in formula (6) of the small numerical factor \(k \simeq 6.7\cdot 10^{-8}\) signifies the indicated lesser “sensitivity” of gravitational phenomena to the manifestation of mass than of inertial phenomena. We have dwelt on this circumstance in detail, since we shall need it later.

The identity of “gravitational” and “inertial” masses is fully clarified in Einstein’s theory of gravitation. In the equations of this theory there appears one physical quantity, manifesting itself both as “inertial mass” and as “gravitational mass.” The gravitational constant \(k\) is expressed through the coefficients \(g_{ik}\), which determine the metric of space, which in turn is determined by the available masses and their spatial distribution.

To summarize, we may say: the totality of a large number of diverse facts leads us to the concept of the physical quantity, mass, which characterizes very general properties of matter (and, as will be shown below, of other forms of matter as well), manifesting themselves both in inertial phenomena and in phenomena of gravitation.

1. The content of the concept of energy, like that of the concept of mass, can be disclosed only on the basis of considering a large totality of mutually connected facts. These facts pertain to the domain of the transformation of some forms of motion of matter into others and indicate the existence of such a physical quantity which remains constant in all these transformations. Attempts to find a physical quantity conserved in the interactions of bodies initially concerned only mechanical phenomena. The dispute that took place at the end of the seventeenth century between Leibniz and the Cartesians about the “true measure of forces” is well known; the former took this measure to be the product \(mv^2\), while the latter took it to be the product \(mv\). M. V. Lomonosov posed the question in a considerably more general and profound sense, writing in 1748: “All changes occurring in nature are of such a state that as much as is taken away from one body, so much is added to another. Thus, if somewhere some matter is diminished, it is increased in another place; as many hours as someone devotes to wakefulness, just so much he takes away from sleep. This universal natural law extends also to the very rules of motion: for a body that by its force moves another loses as much of its own as it imparts to another, which receives motion from it”

The further substantial development of the law of conservation of energy was obtained only a hundred years after Lomonosov, in the middle of the nineteenth century, in the works of Mayer and Helmholtz. However, we shall not dwell on these works, since Mayer’s arguments were of too general a character, while Helmholtz, on the contrary, confined himself to considering only mechanical phenomena. We shall pass at once to the statements of Engels, who for the first time fully revealed the fundamental, philosophical content of the law of the transformation and conservation of energy. Engels wrote: “But when we bring these manifold forms of phenomena under the single general name of motion, the point is by no means merely that our reason unites them together. On the contrary, these forms themselves prove by their action that they are forms of one and the same motion, for under certain circumstances they pass into one another... And it follows in this way that to a definite quantity of motion of one form there always corresponds an exactly definite quantity of motion of another form, and, moreover, once again it is immaterial from which form of motion the unit of measure is borrowed by which this quantity of motion is measured...” *).

As is clear from the quotation given, Engels emphasized the essentiality of the mutual transformation of some kinds of motion ) into others**. Consequently, the matter must above all concern the finding of such a physical quantity as would be a common measure for all concrete kinds of motion of matter considered in physics in their mutual transformations. The possibility of finding such a general “measure of motion” follows from the enormous number of physical experiments which have shown that it is possible to establish equivalent relations between any kinds of action upon a system.

Owing to the mutual transformability of all kinds of motion into one another, one and the same change of a system may occur as the result of different external actions. For example, a given body may be heated by a definite number of degrees \(T_2 - T_1\), under specified conditions (for instance, at constant pressure), by means of very different external actions: mechanical, electrical, luminous, etc. This circumstance makes it possible to establish an equivalent relation between different actions. Since among these actions there can always also be a mechanical one, measurements make it possible to establish to what quantity of mechanical work, determined by equality (1): \(\Delta A = f_s \cdot \Delta s\), the given action is equivalent. It is essential to note that the equivalence thus established with mechanical action—

) F. Engels, Dialectics of Nature, p. 52, Gospolitizdat, 1950.
*) The concept of motion here, as everywhere below, is used in its general philosophical sense.

together with this also establishes the equivalence of any kinds of actions among themselves. To illustrate this circumstance, it is appropriate to recall one very ingenious experiment of Joule, which consisted in the following: a closed metallic wire was placed in a small vessel with water. The vessel was located between the poles of an electromagnet and rotated as a whole, together with the water and the wire contained in it. Then an electric current was induced in the wire, which heated the water. Measurements showed that to each unit of mechanical work expended on rotating the vessel there corresponded the same heating of the water as if it had been produced directly as a result of work against frictional forces.

The indicated possibility of estimating any external physical actions leading to a change in the form of motion of matter in equivalent quantities of mechanical work found its expression in Engels’ words: “The measure of motion is work”) and “work is a change in the form of motion, considered from its quantitative side”*).

Having at our disposal a way of quantitatively comparing any external actions, one may pose the problem of characterizing the system itself by means of such a physical quantity \(E\), the change of which, when the system passes from one state to another, would be determined by external actions expressed in quantities of mechanical work equivalent to them. Denoting by \(E_I\) and \(E_{II}\) the values of the physical quantity \(E\) corresponding to two states of the system \(I\) and \(II\), we obtain:

\[ E_{II}-E_I=\sum \Delta A, \tag{8} \]

where \(\sum \Delta A\) is the sum of the mechanical equivalents of all external actions under whose influence the system has passed from state \(I\) to state \(II\). Obviously, such a quantity \(E\) can be introduced into consideration only in the case that it proves to be a single-valued function of the state of the system. Whether this is so or not can be shown only by experiment.

A vast number of experiments show that in any cyclic process the sum of the mechanical equivalents of all external actions is equal to zero. It then follows from relation (8) that \(E\) is indeed a single-valued function of the state of the system, since as a result of the course of a cyclic process (state \(II\) coincides with state \(I\)) one obtains—

) F. Engels, Dialectics of Nature, p. 60, Gospolitizdat, 1950.
*) Ibid., p. 70.

It is assumed that \(E_{II}=E_I\). The physical quantity \(E\) is the energy of the system.

The concept of energy following from relation (8) was precisely clarified only in 1887 by Planck*).

By an isolated system we mean one upon which no external actions are exerted; we obtain that, for an isolated system, \(\sum \Delta A=0\), and consequently, by (8), \(E_{II}=E_I\). Hence it follows that the energy of an isolated system is constant, whatever processes may occur in it. This proposition is the law of conservation of energy.

To summarize, we may say that an enormous number of interconnected facts indicate the possibility of objectively characterizing the concrete forms of moving matter considered in physics by means of a physical quantity—energy—which is a single-valued function of the state of the system, whose change is determined by the sum of the mechanical equivalents of all external actions on the system. The energy of an isolated system remains constant under any changes occurring in such a system.

The latter circumstance indicates that energy is precisely that general characteristic of various concrete forms of moving matter which remains unchanged under their mutual transformations.

The content of the concept of energy is fully revealed only through the entire totality of the indicated facts and laws. Any attempt to “define” energy by some brief phrase will inevitably reduce to the use of a limited group of facts, which can reveal the properties of energy only partially.

The concept of energy, like that of any other fundamental physical quantity, has been subjected to numerous idealistic and mechanistic distortions. Various idealistic distortions consisted in the fact that energy was regarded either as something external in relation to matter, or energy was identified with matter (“energeticism” of Ostwald). The latter point of view was at the same time metaphysical, since it replaced all concrete forms of moving matter by a single one—“energy.” Like every metaphysical point of view, this one too proved fruitless and inconsistent with reality.

*) M. Planck, The Principle of Conservation of Energy. Russian translation, GONTI, 1938.

Mach tried in every way to belittle the significance of the law of conservation of energy, asserting that this law cannot lay claim to any greater significance than any other particular law of physics, for example Boyle–Mariotte’s law. Among Mach’s followers the thought was repeatedly expressed that relation (8) can always be formally satisfied by inventing, alongside the known ones, new kinds of energy, and that therefore it has no special physical meaning. The unsoundness of this point of view is easy to show, although, indeed, as new physical phenomena are discovered, new kinds of energy have to be introduced into consideration. The point is that the possibility of introducing a new kind of energy is always founded on an experimental fact indicating that, in a cyclic process in which the newly discovered kind of transformations also participates, the sum of the mechanical equivalents of all external actions still remains equal to zero. Thus, what is involved is the extension of the old concept of energy in accordance with newly discovered objective phenomena of nature.

The formulation of the law of conservation of energy may change as our knowledge develops. But the fact of the lawful mutual convertibility of the various forms of motion of matter remains unshakable. This is what Engels had in mind when he wrote: “every form of motion has proved capable and compelled to transform itself into any other form of motion. Having reached this form, the law has attained its final expression. Through new discoveries we can supply it with new confirmations, give it a new, richer content. But to the law itself, as it is expressed here, we can add nothing more. In its universality, in which both form and content are equally universal, it is incapable of any further extension: it is an absolute law of nature”*).

Metaphysical and mechanistic points of view on energy were reflected in attempts to present energy as a special kind of “substance,” or, speaking more crudely, as a special kind of “fluid” contained in bodies and flowing from some bodies to others. The concept of energy became established in physics in the second half of the nineteenth century, when mechanistic attempts to explain various physical phenomena with the aid of hypothetical “fluids” (caloric, electrical fluid, etc.) had for the most part already been abandoned. Nevertheless, in the doctrine of energy there became established a terminology connected with these mechanistic tendencies; to this day we speak of energy “contained” in a body, of a “flow” of energy, etc. Although in itself the indicated

*) F. Engels, Dialectics of Nature, pp. 178–179, Gospolitizdat, 1950.

terminology and may be retained, but any attempts to revive on its basis mechanical conceptions of energy are unsound. Such an attempt was made recently in our Soviet literature by T. P. Kravets*), who, proceeding from conceptions of the “flow” of energy and from the fact of the finiteness of the velocity of propagation of any physical actions, tried to ascribe “substantiality” to energy.

Energy cannot be represented in the form of a special “substance,” i.e. a special kind of matter, since energy is a characteristic of the state of concrete kinds of moving matter and is completely determined through the parameters pertaining to the given kind of matter.

1. Let us now consider the question of the connection between energy and mass. The discovery of such a connection is among the most important achievements of twentieth-century physics. The general conclusion that a definite connection exists between the energy and the mass of a system was obtained from the theory of relativity by Einstein in 1905. At the present time, as a result of direct measurements made in nuclear transformations, it may be regarded as empirically established that a change in the energy of a system by an amount \(\Delta E\) leads to a change in its mass by the amount:

\[ \Delta m=\frac{\Delta E}{c^2}, \tag{9} \]

where \(c\) is the speed of light in vacuum, equal to \(\sim 3\cdot 10^{10}\) cm/sec. Energy and mass are mutually connected in the sense that an increase in the energy of a system leads to a simultaneous increase in its mass, and a decrease in energy—to a decrease in mass. Here all manifestations of mass are involved, i.e. when the energy changes, both the inertial and the gravitational properties of the system change.

Relation (9) led to the necessity of revising many propositions of classical physics, including the conditions under which the laws of conservation of mass and energy are fulfilled. Let us examine both these laws in more detail.

The law of conservation of mass asserts that the mass of an isolated system remains constant under any changes occurring inside the system. Similarly, the law of conservation of energy asserts that the energy of an isolated system is constant under all changes of the system. Let us pose the question: what system is called “isolated,” are the criteria—

*) T. P. Kravets, UFN, vol. XXXVI, p. 338, 1948.

criterion of isolation in the two cases? Turning to the history of science, it is easy to see that the criteria of isolation in the establishment of the two laws were different. M. V. Lomonosov, who experimentally proved the existence of the law of conservation of mass, placed various substances in a sealed glass vessel and weighed it before and after the substances had reacted chemically. The reaction was caused by heating the vessel. Both weighings gave one and the same result. Lavoisier and all the other investigators who, after Lomonosov, verified the validity of the law of conservation of mass in chemical reactions proceeded in the same way. From the manner in which these experiments were set up it is clear that the criterion of isolation of the system was the absence of exchange of matter (in the solid, liquid, or gaseous states) between the system and the surrounding bodies. External actions (heating, etc.), however, could be exerted on the system. We shall call such a criterion of isolation the criterion of partial isolation.

The indicated partial isolation is, evidently, insufficient for the constancy of energy. In order that the energy of a system remain unchanged, not only must exchange of matter between the system and the surrounding bodies be absent, but all external actions on the system must also be absent. Consequently, for the fulfillment of the law of conservation of energy a different, more rigorous criterion of isolation was advanced than for the fulfillment of the law of conservation of mass. This more rigorous criterion of isolation we shall call the criterion of complete isolation*).

Relation (9) between energy and mass shows that, when partial isolation of a system is observed, in reality the mass cannot remain constant. Possible changes in the energy of the system, when this criterion is observed, lead to a change in its mass. For example, heating some substance sealed in a glass vessel will lead to an increase in its energy and, consequently, according to relation (9), to an increase in its mass. The circumstance that such a change of mass was not experimentally detected either by Lomonosov or by all subsequent scientists is explained by the fact that a very large exchange of energy between the system and the surrounding bodies is required for the change in the mass of the system to become noticeable. This circumstance finds its expression in the presence of the small multiplier

\[ \frac{1}{c^{2}} = 1.1 \cdot 10^{-21}\ \text{sec}^{2}/\text{cm}^{2} \]

in expression (9). Both methods

*) In reality complete isolation can never be exactly realized. However, this has no fundamental significance for the further arguments, since we can come arbitrarily close to the conditions of complete isolation.

of detecting a change in a system by determining external actions from their mechanical equivalents and by determining mass are not equally “sensitive,” just as gravitational methods and methods based on phenomena of inertia are not equally “sensitive” for determining the mass of a body. Nevertheless, with respect to accuracy the mass of a system remains constant only when the system is completely isolated. Thus we arrive at the conclusion: for a completely isolated system both conservation laws hold: 1) its mass remains unchanged, 2) its energy remains unchanged.

In the physics literature one repeatedly encounters assertions that the fact of the interrelation between mass and energy allegedly implies that the laws of conservation of mass and of energy, taken separately, are not obeyed, and that supposedly only the sum of the mass and energy of an isolated system remains constant. It is also often said that “mass is transformed into energy,” and so on. As is clear from what has been said, such assertions are entirely erroneous: in a completely isolated system each of the physical quantities—mass and energy—remains constant; consequently, no “transition” of one of these quantities into the other occurs*).

Nineteenth-century physics allowed for the existence of such kinds of matter as do not possess mass. For example, it was assumed that radiation (light) manifests neither inertial nor gravitational properties and that, consequently, the concept of mass was inapplicable to it. However, the very existence of light pressure, first established experimentally by P. N. Lebedev, compelled one to ascribe a definite amount of motion to a light flux and thereby indicated the presence of inertial properties in radiation. The existence of an interrelation between energy and mass, expressed by relation (9), leads to the conclusion that any concrete kind of matter considered in physics possesses mass. Indeed, an energy characteristic is applicable to any kind of matter considered in physics, since all forms of motion of matter are mutually transformable. But since any concrete kind of matter possesses energy, then by relation (9) it also possesses mass.

From the existence of an interrelation between energy and mass there follows one more important consequence. So long as it was believed that, for the conservation of the mass of a system, it was necessary only that there be no exchange of substance with surrounding bodies, it turned out that the mass of the system

) On the erroneousness of ideas about the “transformation of mass into energy” and conversely, see also the article by N. F. Ovchinnikov, “Mass and Energy,” Priroda*, No. 11, 1951.

does not depend on its state. By virtue of relation (9), however, mass proves to be dependent on the state of the system, since with a change in the state of the system its energy changes and, consequently, a corresponding change of mass takes place.

Direct measurements show, for example, that the sum of the masses of a certain number of protons and neutrons situated at sufficiently large distances from one another differs from the mass of a system of the same number of protons and neutrons forming an atomic nucleus. When an atomic nucleus is formed, a large amount of energy \(\Delta E\) is liberated, which is transferred to surrounding bodies, as a result of which the mass of the nucleus becomes smaller by the amount \(\Delta m = \dfrac{\Delta E}{c^2}\) than the sum of the masses of the protons and neutrons entering into the composition of the nucleus.

The dependence of mass on the state of the system was first discovered in the example of the dependence of mass on the velocity of motion (experimentally this was established for fast electrons). To bourgeois scholars, who adhered to the metaphysical definition of mass as a quantity characterizing the “quantity” of matter, this fact seemed a refutation of materialist philosophy.

The complete untenability of such a conclusion, as is known, was exposed by Lenin, who pointed out: “Matter disappears”—this means that the limit up to which we have known matter until now disappears; our knowledge goes deeper; such properties of matter disappear which formerly seemed absolute, immutable, primary (impenetrability, inertia, mass, etc.), and which are now found to be relative, inherent only in certain states of matter. For the sole “property” of matter with whose recognition philosophical materialism is bound up is the property of being objective reality, of existing outside our consciousness*).

1. By virtue of the interconnection between energy and mass expressed by relation (9), a change in the state of a system under the influence of external actions can be characterized by a change in either of these two quantities. In this respect energy and mass are “equivalent” to one another.

However, both these quantities, as we have indicated, are not equally “sensitive” for characterizing a change in the state of a system—

) V. I. Lenin, Materialism and Empirio-criticism*, p. 243, Gospolitizdat, 1949.

we can. A change in a system caused, for example, by its heating (within the limits of an ordinary change of temperature) is easily detected by the change in its energy, but is practically impossible to detect by the change in its mass.

The indicated differences in energy and mass do not yet mean that it is impossible to unite them in a more general physical concept. As we have seen, “inertial” and “gravitational” masses, despite their likewise different “sensitivity” for characterizing any one property of a system, can be united in a single concept of mass. Therefore the question naturally arises whether the fact of the interconnection between energy and mass does not imply the possibility of uniting them in one more general physical concept. The theory of relativity gives an affirmative answer to this question. In this theory the so-called “mass tensor”*) \(T^{ik}\) \((i, k = 0, 1, 2, 3)\) is introduced, which is a function of the state of the system. The invariant \(T^{ik}\) has the dimension of mass density; the same invariant, multiplied by \(c^2\), has the dimension of energy density. For a completely isolated system, the expression

\[ \frac{\partial T^{00}}{\partial x^0} + \sum_{k=1}^{3} \frac{\partial T^{0k}}{\partial x^k} = 0 \]

gives the laws of conservation of mass and energy, while the expressions

\[ \frac{\partial T^{i0}}{\partial x^0} + \sum_{k=1}^{3} \frac{\partial T^{ik}}{\partial x^k} = 0 \]

\[ (i = 1,\ 2,\ 3) \]

are the law of conservation of momentum. Thus, one very general physical quantity (the “mass tensor”) simultaneously characterizes the whole totality of phenomena which, in classical physics, were separately characterized by such physical quantities as mass and energy.

From the indicated point of view, the unification of the concepts of energy and mass takes place analogously to the way in which the unification of the concepts of “inertial” and “gravitational” masses was effected in the single concept of mass.

However, it is necessary to determine whether such an analogy is sufficiently complete and whether there are no circumstances indicating—

*) The term “mass tensor” was proposed by V. A. Fock. In the works of bourgeois physicists this tensor is usually called the “tensor of matter,” which leads to confusion between the concepts of the concrete kinds of matter with which physics deals and the general philosophical concept of matter.

indicating the absence of complete “equivalence” between energy and mass.

We have indicated that any change in energy \(\Delta E\) leads to a simultaneous change in the mass of the system by the amount \(\Delta m=\dfrac{\Delta E}{c^2}\). But the question of whether it is possible to achieve any change in the mass of a system by means of external actions (for example, by the performance of work by external forces) still remains unresolved. If the answer to this question is negative, then energy and mass are not fully “equivalent” to one another.

To answer the question posed, let us examine, from the point of view of modern physics, the properties of certain concrete systems. As a first example let us choose a system consisting of \(N\) atoms, each of which has rest mass \(m_{0i}\) (by the rest mass \(m_{0i}\) is meant that value of the mass which an isolated atom, in the normal state, has with respect to the reference system in which it is at rest). Let us assume that no nuclear transformations take place in our system, so that the number of atoms \(N\) is constant. The mass of such a system is equal to:

\[ M=\sum_N m_{0i}+\sum_N \frac{E_{ki}}{c^2}+\sum_N \frac{E_{pi}}{c^2}+\sum_N \frac{E_{\text{exc } i}}{c^2}+\frac{E_{\text{rad}}}{c^2}, \tag{10} \]

where \(E_{ki}\) and \(E_{\text{exc } i}\) are the kinetic energy of the atom and the excitation energy of its electron shell; \(E_{pi}\) is the potential energy, including that part of the potential energy which is caused by the interaction between the atoms forming the system under consideration; \(E_{\text{rad}}\) is the energy of the radiation contained within the system.

The energy of the same system is equal to:

\[ E=c^2\sum_N m_{0i}+\sum_N E_{ki}+\sum_N E_{pi}+\sum_N E_{\text{exc } i}+E_{\text{rad}}. \tag{11} \]

If the system is completely isolated, then in the processes taking place within it only the relative value of the separate kinds of energy changes, while the mass \(M\) of the system, determined by equality (10), and its total energy \(E\), determined by equality (11), remain constant. Thus, in accordance with what was said above, under the condition of complete isolation each of the quantities—the mass \(M\) and the energy \(E\)—is conserved separately.

In ordinary terrestrial conditions the following inequality holds:

\[ \sum_N m_{0i} \gg \sum_N \frac{E_{ki}}{c^2}+\sum_N \frac{E_{pi}}{c^2}+\sum_N \frac{E_{\text{excit }i}}{c^2}+\frac{E_{\text{rad}}}{c^2} \tag{12} \]

and then, with a sufficient degree of approximation, the mass of the system is equal to:

\[ M=\sum_N m_{0i}, \tag{10a} \]

whereas the energy is still given by formula (11). This relation expresses the fact noted by us earlier, that in ordinary terrestrial conditions the practical laws of conservation of mass and energy are fulfilled “separately,” i.e., under different conditions of isolation of the system. For the constancy of the mass of a system \(M\), in accordance with formula (10a), it is practically sufficient that the number of atoms \(N\) forming the system remain unchanged.

However, even if one takes into account the relatively small changes of the mass \(M\) that may occur under conditions of partial isolation of the system, the proposition still remains valid that the greater part of the mass of the system under consideration—namely, that part which is due to the rest mass of the atoms*)—cannot be exhausted solely through external actions (work).

This circumstance led V. A. Fock) to divide mass into “passive” and “active” parts, of which the second can be changed “more easily” under the influence of external actions*).

Let us now include nuclear transformations within the scope of our considerations, but without processes of transformation of electrons and positrons into photons and conversely. Then, instead of atoms, one should consider “elementary” particles, i.e., protons, neutrons, electrons, etc., with their rest masses \(m_{0i}\). In this case, in expressions (10) and

*) By the rest mass of an atom we here mean the mass of the atom relative to a reference system in which it is at rest; moreover, we consider an isolated, unexcited atom. If these latter requirements are not fulfilled, then the rest mass in some definite part also depends on the state of the system, i.e., it may change under the influence of external actions.

) V. A. Fock, Mass and Energy. Published in the present issue.

***) The sum of the rest masses in expression (10) is not always large in comparison with the remaining terms. For example, in very large nebulae the term \(\dfrac{E_{\text{rad}}}{c^2}\) may be of the same order as the sum of the masses of the particles composing the nebula. The greater part of the mass of such a nebula belongs to the “active” form.

From (11) one must separate out the binding energy of the particles in the nuclei, \(E_{\mathrm{bind}\, i}\). Then instead of (10) we obtain:

\[ M=\sum_N m_{0i}+\sum_N \frac{E_{\mathrm{bind}\, i}}{c^2} +\sum_N \frac{E_{ki}}{c^2} + \sum_N \frac{E_{pi}}{c^2} +\sum_N \frac{E_{\mathrm{exc}\, i}}{c^2} +\frac{E_{\mathrm{rad}}}{c^2}, \tag{13} \]

where now \(N\) denotes the number of elementary particles. In nuclear transformations the number of atoms making up the system changes; at the same time the changes in the binding energy \(E_{\mathrm{bind}}\), as is known, can be very large. Nevertheless, when the system is completely isolated*) and nuclear transformations occur in it, its mass \(M\) remains unchanged. Other kinds of energy also change greatly, in correspondence with the change in \(E_{\mathrm{bind}}\). If, for example, nuclear reactions proceed with a decrease in \(E_{\mathrm{bind}}\), then in expression (10) \(E_{ki}\), \(E_{\mathrm{exc}\, i}\), and \(E_{\mathrm{rad}}\) increase greatly, i.e., the system heats up strongly and the radiation density in it increases considerably.

If, however, only the condition of partial isolation is fulfilled, which we now understand in the sense that no exchange can occur of particles possessing a finite rest mass \(m_{0i}\) between the system and the surrounding bodies, then, in the presence of nuclear transformations, the mass of the system \(M\) may change noticeably. Yet even in this case the relative change of mass is small. Indeed, the average binding energy in nuclei, calculated per nucleon (i.e., proton or neutron), is approximately \(8\ \mathrm{MeV}\), whereas the energy \(c^2 m_{0i}\) corresponding to the rest mass of one nucleon is about \(900\ \mathrm{MeV}\). Consequently, even in nuclear transformations the mass of a partially isolated system changes insignificantly—by an amount not exceeding \(1\%\) of the total mass of the system. Thus, the mass of the systems indicated decomposes into two parts—one, independent of the state of the system, and another, dependent on the state, the first of them being considerably larger than the second (except in cases where the term \(E_{\mathrm{rad}}/c^2\) is large; see the footnote on p. 187). Only the second of them

*) It should be borne in mind that for a system in which radioactive transformations occur, we are at present unable to realize the conditions of complete isolation with a sufficient degree of approximation. This is due to the fact that there are no known methods of retaining the neutrino. Only the process of the neutrino’s arising at the expense of other forms of matter is known (for example, in \(\beta\)-transformations of nuclei); the “reverse” process—the transition of a neutrino into other forms of matter—has not yet been discovered. The search for such a process, evidently, is one of the urgent problems of modern nuclear physics.

is an “active” form; it can change as a result of external actions and, consequently, only at its expense can work be obtained.

Until now we have excluded from consideration the process of conversion of electron–positron pairs into photons and the inverse process of the creation of electron–positron pairs by photons. Since photons do not possess rest mass, in the presence of such processes the number of particles in the system having nonzero rest mass changes. As a result, the mass of such a system, under the condition of partial isolation, may, generally speaking, change in any way whatever. For example, if the system initially consisted only of positrons and electrons, then as a result of the conversion of pairs its mass may pass entirely into the “active” form and may be completely exhausted by the performance of work against external forces.

Summarizing, we arrive at the conclusion that only in systems in which a complete conversion of forms of matter possessing rest mass into forms of matter not possessing rest mass is possible are mass and energy fully “equivalent” to one another in the sense that not only does every change of energy lead to a change of mass, but the mass of the system can also be completely exhausted as a result of external actions (the performance of work). Systems in which there is no conversion of matter from forms possessing rest mass into forms not possessing rest mass do not have mass and energy fully “equivalent” to one another in the above-indicated sense of the word.

If processes of conversion of forms of matter possessing rest mass into forms not possessing rest mass are possible in all cases (even through any number of intermediate transformations), then the grounds for considering mass and energy separately generally lose their meaning.

However, it seems to us that at the present time it is still impossible definitively to arrive at such a conclusion. The point is that the conversion of particles with finite rest mass into particles with zero rest mass is apparently connected with the existence of “symmetric” particles, i.e., particles having equal and opposite charges in sign*). Precisely such a pair of particles is the electron and the positron. With respect to the proton, however, no symmetric particle is known to us (a “negative proton”), and therefore the question of whether protons can be converted into particles with zero rest mass remains open. The same is true of the question of the conversion of neutrons into particles with zero rest mass. If such a conversion does not exist, then

*) We exclude from consideration, as little studied, the processes of conversion of mesons, for example the $\pi$-meson into a $\mu$-meson.

a significant part of the mass of ordinary systems—namely, the rest mass of protons and neutrons—must be assigned to the “passive” part of mass, which cannot be generalized in a single concept with energy.

Thus, in our opinion, the present level of knowledge still does not make it possible to decide definitively whether the concepts of mass and energy can be fully generalized in a single, more general concept, analogous to the way this can be done with respect to the concepts of “inertial” and “gravitational” mass. As in all questions of physics, so in this case only experiment can provide the final solution and make it possible to broaden our ideas about the material world surrounding us in accordance with its real properties.