MASS AND ENERGY

V. A. Fock

According to the terminology adopted in mechanics, the mass of a body is a measure of its inertia (inertial mass). On the other hand, the word “mass” is used in the sense of a body’s ability to create a gravitational field and to experience a force in this field (gravitating or weight mass). Inertia and the ability to create a gravitational field are entirely different manifestations of the properties of matter. The fact that the measures of these different manifestations are denoted by a single word is, however, not accidental, but is due to the fact that both properties always exist jointly and are always proportional to one another; thus, with a proper choice of units, the measure of either property can be expressed by one and the same number. The equality of inertial mass and gravitating mass is an experimental fact confirmed with an enormous degree of accuracy (Eötvös’ experiments). The very definitions of these concepts (based on the corresponding manifestations of the properties of matter) are different. There exists a physical theory—namely, Einstein’s theory of gravitation—in which the fundamental law of equality of inertial mass and gravitating mass is taken into account automatically, in the sense that one and the same constant entering in the solution of the equations figures both as inertial mass and as gravitating mass.

How should one answer the question: are inertial mass and gravitating mass one and the same thing or not? In their manifestations they are different, but their numerical characteristics are proportional to one another. Such a state of affairs is customarily characterized by the word “equivalence.”

An analogous question arises in connection with the concepts of mass and energy (for definiteness, we shall speak of inertial mass). We have just recalled the definition of mass. Energy is usually defined as a measure of the capacity to do work. For the definition of energy, what is essential is, first, the law of conservation of energy and, second, the capacity of different kinds of energy for mutual transformation. These two together are called

by the law of conservation and transformation of energy. The existence of this universal law makes it possible to reduce the measurement of energy of any kind to the measurement of energy of a particular kind, for example mechanical energy, and to express energy of any kind in the same (for example, mechanical) units. Thus, the manifestations of the properties of matter corresponding to mass and energy are unquestionably different. However, the theory of relativity asserts that mass and energy are inseparably connected with one another and, moreover, are proportional to one another. Every change in the energy of a system is accompanied by a change in its inertial mass. This applies not only to changes in the kinetic energy of a body, for which the rest mass remains unchanged, but also to changes in various kinds of internal energy, for which the rest mass changes.

In physics, phenomena are known in which all the energy corresponding to the rest mass of a body can be transformed into radiation energy (which, of course, possesses the same mass). Conversely, the energy of rest mass can arise at the expense of radiation energy. We have in mind the phenomenon of the transformation of an electron–positron pair into a gamma quantum and the inverse phenomenon of the production of such a pair by a gamma quantum.

To every energy \(W\) there should be assigned a mass \(M = W/c^3\), and to every mass \(M\) one can associate the energy \(W = Mc^3\). These two quantities are always proportional to one another, and, expressing them in the same (say, energy) units, they can be measured by one and the same number.

The energy tensor of a field or continuous medium considered in the theory of relativity differs only by the factor \(c^3\) from the mass tensor, and the law of conservation of energy in this theory is at the same time the law of conservation of mass.

Thus, to the question posed above—whether mass and energy are one and the same thing or not—we can give the same answer as in relation to inertial mass and gravitational mass. The manifestations of the properties of matter corresponding to mass and energy are different, but the numerical characteristics of these properties are proportional to one another. In this case one may also speak of equivalence—of the equivalence of mass and energy.

We have just said that in the theory of relativity the law of conservation of energy is at the same time the law of conservation of mass. But here the following question arises. Experience shows us that in the vast majority of known physical processes the mass of a body (determined by weighing) is conserved separately, and its energy (determined by the heat released or by the work performed) is conserved separately. Thus, in fact, two laws of conservation are observed. How is this to be reconciled with the fact that in the theory of relativity only one law is formulated?

To this question one can give the following answer. There is only one strict conservation law: for the total mass of a body \(M\) and for the corresponding total energy of the body \(W\). But the overwhelming part of the energy (and of the corresponding rest mass) usually does not participate in transformations and is conserved separately. Thereby the remaining, active part of the energy participating in transformations is also conserved.

The division of energy into a “passive” part, not participating in transformations (in the given process), and an “active” part, capable of passing into other forms, can be traced in the example of the relativistic equations of motion of a continuous elastic medium. This division is most clearly manifested in the approximate formulas for the components \(T^{00}\) and \(T^{0i}\) \((i=1,2,3)\) of the relativistic mass tensor. Let us recall that the quantity \(T^{00}\) is the total mass density, while the three quantities \(T^{0i}\), multiplied by the speed of light \(c\), represent the mass flux.

The aforementioned approximate formulas can be written in the form

\[ T^{00}=\rho+\frac{1}{c^{2}}S;\qquad cT^{0i}=\rho v_i+\frac{1}{c^{2}}S_i. \tag{1} \]

Here \(\rho\) and \(\rho v_i\) are the density and flux of that part of the mass which does not participate in transformations (\(v\) is the velocity of the medium). These quantities separately satisfy the conservation law

\[ \frac{\partial \rho}{\partial t}+\operatorname{div}(\rho v)=0. \tag{2} \]

The scalar \(S\) and the vector \(S_i\) represent the density and flux of energy introduced by Umov (1874) (the scalar and Umov vector). For an elastic body these quantities have the form

\[ S=\frac{1}{2}\rho v^{2}+\rho \Phi, \tag{3} \]

\[ S_i=v_i\left(\frac{1}{2}\rho v^{2}+\rho\Phi\right)-\sum_{k=1}^{3}p_{ik}v_k, \tag{4} \]

where \(p_{ik}\) is the three-dimensional stress tensor, and \(\Phi\) is the potential energy per unit mass. The quantities \(S\) and \(S_i\) satisfy the conservation law established by Umov*)

\[ \frac{\partial S}{\partial t}+\sum_{i=1}^{3}\frac{\partial S_i}{\partial x_i}=0. \tag{5} \]

) See N. A. Umov, Selected Works*, Gostekhizdat, 1950. The explicit expressions for \(S_i\) given by Umov on p. 175 differ from (4) by the absence of terms with the potential energy \(\Phi\) and satisfy (together with the density of kinetic energy) a conservation law in a form somewhat different from (5) [on the right-hand side of (5) there must stand the work of the elastic forces per unit volume per unit time].

Thus, the equation

\[ \frac{1}{c}\frac{\partial T^{00}}{\partial t}+\sum_{i=1}^{3}\frac{\partial T^{0i}}{\partial x_i}=0, \tag{6} \]

expressing the law of conservation of the total mass (and of the corresponding total energy), is satisfied by virtue of (2) and (5).

As is seen from (1), the total mass density \(T^{00}\) is composed of the density \(\rho\) of the “passive” part of the mass and the density \(S c^2\) of its “active” part.

We have said above that the total mass of a body determined by weighing (which also includes its variable part) is practically conserved, despite the fact that the body emits or absorbs energy. This is explained simply by the insufficient accuracy of weighing, together with the fact that the overwhelming part of the mass of ordinary bodies is its passive part.

Changes in the active part of the mass, however, can be traced with much greater accuracy by measuring the corresponding part of the energy (i.e., by means of calorimetric methods, and not by weighing).

It is natural to raise the question of the deeper reason why, under ordinary conditions, the overwhelming part of the energy is bound so firmly that it is in a completely passive state. Why does even an insignificant part of it not leave this state and disturb the balance of the active part of the energy? Relativity theory by itself cannot answer this question. The answer, in our opinion, should be sought in the realm of quantum regularities, one of whose characteristic features is the existence of stable states with discrete energy levels. For elementary particles, the energy corresponding to the rest mass can either be transformed into an active form of energy (for example, into radiation) as a whole, or else it is not transformed at all. A small leakage of mass is impossible. This has been verified experimentally in the case of the electron and the positron, but it should also be expected for other elementary particles. Since the overwhelming part of the mass of atoms is in the form of the mass of elementary particles, the impossibility of a small leakage of mass must also hold for atoms. In addition, the discreteness of energy levels must be kept in mind.

Thus, in our opinion, the reason for the particular strength of the bond of the passive part of the energy is of a quantum character.

The relative character of the division of energy (with the corresponding mass) into passive and active parts should be emphasized. In ordinary chemical reactions, not only the intranuclear energy, but also the energy of the inner electron shells of atoms behaves passively. At very high temperatures, when complete or nearly complete ionization of atoms becomes possible,

energy of the internal electron shells acquires an active character. Finally, in processes connected with the rearrangement of atomic nuclei, intranuclear energy also becomes active. However, even then the energy corresponding to the rest mass of the heavy elementary particles that make up the nuclei continues to remain in a passive state.

The especially strong connectedness of the predominant part of energy (with its mass) makes it possible to speak of the law of conservation of mass and the law of conservation of energy as two separate laws, although in the theory of relativity these two laws merge into one.

Both of these laws are associated with the name of Lomonosov. The law of conservation of mass in chemical reactions was discovered and experimentally proved by Lomonosov and later confirmed by Lavoisier. As for the law of conservation of energy, although Lomonosov did not give its exact formulation (this was given only in the nineteenth century by R. Mayer), he was convinced of its existence and came very close to its modern form in his famous letter to Euler in 1748.