ON THE RELATIONSHIP BETWEEN MASS AND ENERGY

E. V. Shpolsky

1. Recently the question of the relationship between mass and energy has acquired great urgency. Although the fundamental relation

\[ E = mc^2 \]

has received exhaustive experimental confirmation and has acquired outstanding scientific and practical importance, in its interpretation and even in its formulation crude errors have repeatedly been made and are still being made. That is why discussion of questions concerning the mass—energy relationship, and the search for correct formulations for expressing the facts and theoretical propositions belonging to this circle of questions, are highly timely and expedient. To accomplish this task, it seems useful to us to recall the elementary derivations of the relation \(E = mc^2\) and consistently to consider examples of its application to processes that are as simple as possible. The present article is devoted to this.

1. Before proceeding to discuss the subject of interest to us in substance, it is necessary to state clearly what we understand by mass and energy. In mechanics it is generally accepted that the mass of a body is to be understood as a measure of its inertia. We shall adhere to this definition.

By energy we understand, generally speaking, the capacity to do work. This definition looks simplest when mechanical phenomena are under discussion. However, the whole force of the law of conservation and transformation of energy consists precisely in the fact that it indicates a common measure and quantitatively exact relations among different kinds of energy. Indeed, owing to the mutual transformability of different kinds of energy, one and the same change in a system can be effected in different ways, with the aid of different kinds of energy. Among these kinds there is always also mechanical action. Since, by virtue of the law of conservation of energy, there is always a definite numerical relation among the different kinds of energy that bring about one and the same change,

one can reduce the measurement of any kind of energy to the measurement of mechanical energy. We express this by the term “equivalence” and speak, for example, of the equivalence of heat and mechanical work.

This most important philosophical aspect of the law of conservation of energy is very well expressed in the following words of Engels: “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 understanding groups 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 one into another... And this takes place in such a way that a definite quantity of motion of one form always corresponds exactly to a definite quantity of motion of another form, and, again, it is immaterial from which form of motion the unit of measure is borrowed by which this quantity of motion is measured...”*).

Let us note that the most general quantitative definition of the concept “energy,” given by Planck, makes use precisely of this mutual transformability of the various kinds of energy. This definition reads: “Energy is the sum of the mechanical equivalents of all external actions performed by a system when the latter, in any way whatever, passes from the given state into some state taken as fundamental (zero)”**). We shall not dwell here on the development and justification of this definition. We note only that the mention of the fundamental (zero) state indicates that, since this state may be chosen arbitrarily, in the numerical expression of energy there is always an arbitrary constant. The presence of this arbitrary constant is most clearly seen in the example of mechanical energy. The latter, as is known, is the sum of kinetic and potential energy; potential energy, however, is defined only through its derivative, i.e. includes an arbitrary constant.

The discovery of the law of conservation and transformation of energy and the rapid development that then followed of a new science—thermodynamics—was one of the most important successes of natural science in the second half of the nineteenth century. But this same development also gave rise to the idealistic distortion that was Ostwald’s “energetics.” In Materialism and Empirio-criticism, Lenin subjected “energetics” to devastating criticism and showed that it is a source of new idealistic attempts to think motion without matter and that it represents an attempt “...to cover up old epistemological errors with a ‘new’ terminology”***).

) F. Engels, Dialectics of Nature*, p. 52, 1950.

) See, for example, M. Planck, Thermodynamics, GIZ, 1925, p. 47.

*) V. I. Lenin, Works, 4th ed., vol. 14, p. 258.

ON THE CONNECTION BETWEEN MASS AND ENERGY

1. Let us now return to the question of mass. The Newtonian definition of mass as the quantity of matter has repeatedly been criticized from a purely logical point of view. However, an irreparable blow to this definition was dealt by the development of the electron theory, namely by the proof of the existence of electromagnetic mass. To make this clear, let us recall how the necessity of the existence of electromagnetic mass is justified. A change in the velocity of a body, for example from 0 to \(v\), requires the expenditure of a certain amount of work. If this body carries an electric charge, then the work will be greater, since a charged moving body is surrounded not only by an electrostatic but also by a magnetic field, and additional work must be expended to create the latter. Conversely, if a moving charged body is stopped, the magnetic field must then disappear. But the disappearance of the magnetic field, by the law of induction, will cause the appearance of an additional electric field, and it is easy to see that at the point where the center of inertia of the body is located, this electric field will have such a direction that it will accelerate the braking charged body. Everything will take place as if the presence of the electromagnetic field of the charged body caused the appearance of additional inertia, i.e., of an additional mass.

Highly instructive is the way in which the electromagnetic mass is calculated, for example the electromagnetic mass of the electron. Assuming that the electron is a sphere and specifying the distribution of electric charge in this sphere (for example, assuming that the electricity is distributed over the surface), one calculates the total energy of the field of the moving sphere (for the case of motion with small velocities). To this end, one first calculates the energy density of the magnetic field at some point of space lying outside the electron, and then performs the integration over all space, except for that part of it lying inside the electron. In this way, for the total field energy one obtains

\[ \Delta E=\frac{1}{3}\frac{e^2}{r_0 c^2}v^2. \]

If this energy is added to the kinetic energy of the moving electron, then one obtains

\[ E=\frac{1}{2}\left(m+\frac{2}{3}\frac{e^2}{r_0 c^2}\right)v^2. \]

The second term in parentheses is the electromagnetic mass

\[ \Delta m=\frac{2}{3}\frac{e^2}{r_0 c^2}. \tag{1} \]

Thus, from the point of view of the classical electron theory, the total mass must consist of two parts:

\[ m+\Delta m, \]

where the second term

\[ \Delta m=\frac{2}{3}\frac{e^2}{r_0c^2} \tag{2} \]

is the electromagnetic mass. We see that this electromagnetic mass is by no means localized in the electron itself, but is due to its field extending to infinity, i.e., as it were “spread out” over all space. This is field mass. The question is what the relation is between \(m\) and \(\Delta m\), i.e., between the “mechanical,” or, more precisely, non-electromagnetic mass and the electromagnetic mass. As is known, at the beginning of the present century the conviction was widespread that \(m\) is in fact equal to zero, i.e., that all mass is of electromagnetic origin. This opinion was justified by the fact that, in the case of motion with arbitrarily large velocities, the law derived by Lorentz for the variation of electromagnetic mass with velocity,

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

received experimental confirmation.

However, since the theory of relativity showed that such a dependence on velocity must hold for mass of any origin, this argument fell away. Moreover, at the present time one may definitely assert that even in the case of the electron only part of the mass is of electromagnetic origin*).

It is well known that the discovery of electrons and the establishment of their electromagnetic mass at the beginning of the twentieth century gave the idealists a pretext for attacking materialism. We know, however, that Lenin, in Materialism and Empirio-Criticism, showed that the new facts of physics only confirm dialectical materialism.

1. We have thus shown that the presence of an electric charge increases the inertia of a body. We shall now show in a more general way that to every excess of energy in the form of the energy of an electromagnetic field \(\Delta E\) there corresponds an excess of mass \(\Delta m\), equal to

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

For the proof, let us imagine a perfectly homogeneous tube \(AB\) of length \(L\) (Fig. 1); its center of inertia is located exactly at the midpoint, at the point \(O\). Suppose now that the bottom of the cylinder \(A\) has an excess energy \(\Delta E\) in the form of the energy of an electromagnetic field, while the bottom \(B\) “za-

*) On the present state of the question of field mass, see D. Ivanenko and A. Sokolov, Classical Field Theory. Gostekhizdat, 1951, pp. 180 ff., 221 ff. See also V. Weiskopf, “Modern Achievements in the Theory of the Electron,” UFN 41, 165, 1950.

“blackened” so that it completely absorbs the electromagnetic wave incident upon it. Suppose that at some moment \(A\) emits its excess energy in the form of a “flash,” as a result of which a short train of electromagnetic waves \(S\) begins to propagate in the tube from left to right. Let us now take into account the light pressure experimentally proved by Lebedev. Owing to this pressure, at the moment of emission the tube experiences a recoil directed from right to left and equal to

\[ \frac{\Delta E}{c}. \]

Under the influence of this recoil the tube will begin to move and will be displaced to the left until the train of waves \(S\) reaches the wall \(B\), where it is completely absorbed. As a result of this the tube experiences a push to the right, under the influence of which it stops, having moved through a distance \(x\). This distance is easy to calculate.

Fig. 1.

Let us denote the mass of the tube by \(M\), and its velocity by \(v\). Then we have, first of all:

\[ Mv=\frac{\Delta E}{c}; \qquad v=\frac{\Delta E}{Mc}. \]

Next, taking into account that the time during which the tube moved is equal to

\[ t=\frac{L}{c}, \]

we find the distance \(x\)

\[ x=vt=\frac{\Delta E L}{Mc^{2}}. \tag{3} \]

We have thus arrived at the conclusion that the center of inertia has shifted by the distance \(x\). But this contradicts the fundamental laws of mechanics, namely the law of conservation of momentum, by virtue of which the center of inertia of a system cannot be displaced under the influence solely of internal forces. This contradiction can be eliminated in only one way: to the excess energy \(\Delta E\) there must correspond an excess mass \(\Delta m\). In that case, before the flash the left end of the tube will possess this excess mass \(\Delta m\) and will therefore be situated correspondingly closer to the center of mass; and after the absorption of the energy \(\Delta E\) at the right end \(B\) and the corresponding increase of its mass by \(\Delta m\), the tube must move as

by such an amount that the center of inertia remains in place. Taking into account that the mass \(\Delta m\) has moved through the distance \(L\), we have:

\[ x=\frac{\Delta m}{M}L, \]

whence, using (3), we find:

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

which was what had to be proved.

In this derivation we made no use at all of the theory of relativity. Moreover, from the point of view of relativity theory, there are inaccuracies in this derivation, as L. I. Mandelstam pointed out. Einstein gave another derivation, free from these inaccuracies. Let us consider a body moving uniformly with velocity \(v\) and having such a shape that, during radiation, the recoil will be compensated by symmetry. Suppose, for example, that the body has the shape of a plate and emits plane waves in both directions. According to relativity theory, in the coordinate system relative to which the velocity of the body is \(v\), the momentum carried away by the plane waves is equal to

\[ \Delta p=\frac{\Delta E\cdot v}{c^2}. \]

The momentum of the radiating body must decrease by this amount. But since its velocity does not change (because the recoils due to the emission of electromagnetic waves are directed in opposite directions and compensate one another), the mass must change. If the mass of the body before radiation was \(M\), and after radiation \(M'\), then by the law of conservation of momentum

\[ Mv=M'v+\Delta E\,\frac{v}{c^2}, \]

whence

\[ M'=M-\frac{\Delta E}{c^2}, \]

so that

\[ \Delta M=\frac{\Delta E}{c^2}. \]

1. Until now we have been justifying the relation between mass and energy for the case of the electromagnetic field. Let us now consider a number of examples which will show us that this relation has a very general character and is applicable to any kind of energy. To this end we shall first discuss the simplest process—the central collision of two perfectly elastic spheres—and show that applying to this case the law of conservation of mass and the law of conservation of the projection of the quantity of motion at once leads to the relativistic formula for the dependence of mass on velocity when

condition that the velocity-addition theorem of the theory of relativity is used*).

Suppose we have two absolutely identical ideally elastic spheres, the mass of each of which in the coordinate system \(S'\), where they are at rest, is equal to \(m_0\). Let us consider their central collision, assuming that the spheres move along the \(x'\)-axis (Fig. 2). Let the velocity of the first relative to \(S'\) be \(+v'\), and that of the second \(-v'\). At the moment of collision the spheres come to rest, after which they exchange velocities: the first moves with velocity \(-v'\), the second with velocity \(+v'\). Let us introduce another coordinate system \(S\), whose velocity relative to \(S'\)

Fig. 2.

is, say, \(-V\). Then the velocity of \(S'\) relative to \(S\) will be \(V\), and the velocities of the spheres relative to \(S\) will be \(v_1\) and \(v_2\). In this case, according to the velocity-addition theorem of classical mechanics before the collision,

\[ v_1 = v' + V, \qquad v_2 = -v' + V, \tag{4} \]

whereas according to the velocity-addition theorem of the theory of relativity,

\[ v_1 = \frac{v' + V}{1 + \dfrac{v'V}{c^2}}, \qquad v_2 = \frac{-v' + V}{1 - \dfrac{v'V}{c^2}} . \tag{5} \]

In order to distinguish our spheres, let us denote their masses in the coordinate system \(S\), respectively, by \(m_1\) and \(m_2\). We shall now make use of the laws of conservation of mass and of the projections of momentum, which must hold in any coordinate system. Denoting by \(M\) the sum of the masses of the particles at the moment of collision, i.e. at the moment when their velocities are identical and equal to zero in the system \(S'\)

*) The arguments given in this and in the following paragraphs belong to G. N. Lewis and R. Tolman. See, for example, R. Tolman, Phil. Mag. 23, 375 (1912).

and are equal to \(+V\) in the system \(S\), let us write:

\[ \left. \begin{aligned} m_1 + m_2 &= M,\\ m_1 v_1 + m_2 v_2 &= MV. \end{aligned} \right\} \tag{6} \]

These equalities assert that the sum of the masses of the particles and the sum of their \(x\)-components of momentum before the collision and at the moment of collision are equal to one another. A simple calculation immediately shows that if one uses the theorem of addition of velocities of Newtonian mechanics and substitutes, in place of \(v_1\) and \(v_2\), their expressions according to (4), then from (6) one obtains \(m_1=m_2\), i.e., that mass does not depend on velocity. (Let us recall that, by assumption, the balls are identical, and therefore—in the coordinate system in which both of them are at rest—their masses are the same.) If, however, one uses the theorem of addition of velocities of relativistic mechanics and substitutes in (6), in place of \(v_1\) and \(v_2\), their expressions according to (5), then, after some calculations*) which we omit, one obtains

\[ \frac{m_1}{m_2} = \frac{\sqrt{1-\dfrac{v_2^2}{c^2}}}{\sqrt{1-\dfrac{v_1^2}{c^2}}}. \]

This shows that the masses of both balls, equal when the balls are at rest and equal in that case, for example, to \(m_0\), are inversely proportional to \(\sqrt{1-\dfrac{v^2}{c^2}}\) if they move with velocity \(v\). Thus,

\[ m=\frac{m_0}{\sqrt{1-\dfrac{v^2}{c^2}}}, \tag{7} \]

which is what we wished to show.

1. It is not difficult to convince oneself that the formula (7), obtained by the indicated general method, for the dependence of mass on velocity already contains, in implicit form, the relation between mass and energy

\[ \Delta E = c^2 \Delta m. \tag{8} \]

Indeed, we shall now show that formula (8) can be obtained from formula (7), or, conversely, that formula (7) can be obtained from formula (8), without any additional hypotheses or postulates whatsoever. Let us consider a body of mass \(m\), moving with velocity \(v\) (for simplicity we shall assume that the body moves along the \(X\)-axis).

) See, for example, E. V. Shpol’skii, Atomic Physics*, vol. I, p. 515, Gostekhizdat, 1951.

The fundamental law of mechanics in its relativistic formulation states

\[ d(mv)=Fdt. \tag{9} \]

Further, by the definition of work,

\[ dE=Fds=Fvdt. \tag{10} \]

Carrying out the differentiation on the left-hand side of (9), we write:

\[ mdv+vdm=Fdt. \tag{11} \]

Differentiating (7), we obtain:

\[ dm=\frac{m_0 vdv}{c^2\sqrt{\left(1-\frac{v^2}{c^2}\right)^3}}. \tag{12} \]

Substitute (12) into (11); after simple transformations we find:

\[ \frac{m_0\,dv}{\sqrt{\left(1-\frac{v^2}{c^2}\right)^3}}=Fdt. \tag{13} \]

Finally, combining (10) and (13), we obtain:

\[ dE=\frac{m_0 vdv}{\sqrt{\left(1-\frac{v^2}{c^2}\right)^3}} \]

or, taking (12) into account,

\[ dE=c^2dm, \]

which was to be shown.

By analogous reasoning, taking formula (8) as the basis and using (9) and (10), one can obtain formula (17) for the dependence of mass on velocity. Thus it is proved that any arguments based on the dependence of mass on velocity are implicitly based on the connection between mass and energy.

Let us emphasize that in the reasoning of this paragraph the discussion concerned kinetic energy, since uniform motion of a body was considered. Thus it has been shown that formula (8), previously derived by us for the case of the energy of an electromagnetic field, also holds for the kinetic energy of a moving body. This connection between the change in kinetic energy and the change in mass becomes especially clear in the transition to the nonrelativistic case.

Assuming that \(\frac{v}{c}\ll 1\), we expand in (7) the factor \(\left(1-\frac{v^2}{c^2}\right)^{-\frac12}\) in a power series and use the first two terms of the expansion. Then we obtain

\[ m=m_0+\frac{1}{c^2}\frac{1}{2}m_0v^2=m_0+\frac{E_{\mathrm{kin}}}{c^2}, \tag{14} \]

as was to be expected.

It may seem strange that the presence of kinetic energy corresponds to an excess of mass

\[ \Delta m=\frac{E_{\mathrm{kin}}}{c^2}. \]

The following considerations, which we borrow from lectures on the theory of relativity*) by H. A. Lorentz, show that if mass is defined as a measure of inertia, then there is nothing surprising in this connection.

“Let us consider a closed vessel in which there is gas. How do we infer, from the laws of motion of this system, that there is gas in the vessel? Let us imagine that the vessel has the form of a rectangular parallelepiped and that we impart to it a constant acceleration—say, from left to right. We shall assume that the walls of the vessel are rough on the inside and that therefore, after striking a wall, the molecules have a motion composed of the translational velocity of the wall and thermal motion. For simplicity let us imagine that the gas is so rarefied that we may neglect collisions of the molecules with one another. If the system had a constant velocity, the molecules would have it as well, and impacts on the right and on the left wall would be equally intense, just as if the vessel were at rest. But if the motion is accelerated, then at the moment of impact the molecule does not yet have the full velocity of the translational motion of the vessel; it has precisely the velocity of translational motion that the vessel had earlier, at the preceding collision with the wall. It is easy to see that, as a consequence of this, impacts on the left wall will be more intense, and impacts on the right wall less intense, than in uniform motion. Thus it follows that the vessel experiences from the gas a resultant force directed to the left—and that therefore, for acceleration, a greater force is required than in the case when the vessel is empty. We express this by saying that the mass of the system has become greater. As for the increase of mass which, according to Einstein’s law, is due to an increase of internal energy, i.e. to an intensification of the molecular motion of the gas, there can be no doubt that it can be explained if one takes into account the intensity of the impacts of the molecules on the opposite walls. However, these collisions must be calculated according to the laws of the theory of relativity.”

Let us note, incidentally, that this reasoning is completely analogous to that by which, proceeding from the existence of light pressure, the necessity of ascribing mass to radiation was justified. It was specifically pointed out that if a vessel filled with equilibrium radiation is given an acceleration to the right, then the pressure

) H. A. Lorentz, Das Relativitätsprinzip. Drei Vorlesungen gehalten in Teylers Stiftung zu Haarlem.* — B. Teubner. Leipzig u. Berlin, 1914, p. 29.

radiation on the left wall will be greater than on the right, from which it follows that the vessel filled with radiation must have a greater mass.

1. Let us now consider a simple example that will allow us to take one more step toward generalizing the result obtained.

Suppose that in some coordinate system there is at rest a ball \(K\) (Fig. 3), composed of two exactly identical hemispheres \(K'\) and \(K''\). In this coordinate system let the mass of the ball be \(M_0\), and consequently the mass of each hemisphere will be

\[ m_0=\frac{1}{2}M_0. \]

Let us now divide our ball into its constituent hemispheres, move them apart by some distance, and then give them equal but oppositely directed velocities \(v\). At the moment when the hemispheres \(K'\) and \(K''\) touch, we again obtain a ball at rest in the same coordinate system. However, we shall now denote it not by \(K\), but by \(K^*\), for the reason that, although \(K^*\) and \(K\), with respect to the substance composing them, are completely identical, and although they both are at rest in one and the same coordinate system, their masses are not equal; namely, the mass of \(K^*\) is greater than the mass of \(K\). Indeed, before the collision the masses of \(K'\) and \(K''\) are equal to

\[ \frac{m_0}{\sqrt{1-\dfrac{v^2}{c^2}}} = \frac{\dfrac{1}{2}M_0}{\sqrt{1-\dfrac{v^2}{c^2}}}, \]

and consequently the sum of their masses, i.e. the mass of \(K^*\), will be

\[ M_0^* = \frac{2m_0}{\sqrt{1-\dfrac{v^2}{c^2}}} = \frac{M_0}{\sqrt{1-\dfrac{v^2}{c^2}}}. \]

Fig. 3.

But it is obvious that

\[ M_0^*>M_0, \tag{15} \]

namely,

\[ \Delta m=M_0^*-M_0 = M_0\left(\frac{1}{\sqrt{1-\dfrac{v^2}{c^2}}}-1\right). \tag{16} \]

This result at first glance seems paradoxical. How

this be, that two bodies, quite identical with respect to the substance composing them, both at rest in one and the same coordinate system, possess different masses in this same coordinate system? However, there is in fact no paradox here, since the bodies \(K\) and \(K^*\), identical in all other respects, differ in their internal state. Let us consider two limiting cases:

a) The body \(K\) (and hence also \(K'\) and \(K''\)) is ideally elastic. In this case an elastic deformation arises in the sphere \(K^*\), under whose influence the hemispheres \(K'\) and \(K''\) again fly apart with the same velocities \(v\) (but, of course, oppositely directed). Thus \(K^*\) differs from \(K\) by the presence of the energy of elastic deformation \(\Delta E\), to which there also corresponds an excess of mass \(\Delta m\).

b) The body \(K\) is absolutely inelastic. In this case, after the collision of \(K'\) and \(K''\), the sphere \(K^*\) that has formed is preserved, but it heats up. The sphere \(K^*\) in this case too differs from \(K\) by an excess of internal energy \(\Delta E\), and consequently by an excess of mass \(\Delta m\).

Thus, in both cases the excess of mass is connected with the excess of energy \(\Delta E\)—either in the form of the energy of elastic deformation, or in the form of heating.

Let us, however, examine this question from a somewhat different point of view. At first sight it might seem that in the operations described the law of conservation of mass is violated. In reality, of course, this is not so: the law of conservation of mass is fulfilled quantitatively exactly, and is expressed by equation (15).

The difference of the mass \(M_0^*\) from the mass \(M_0\), which we have called the rest mass, is connected with the fact that the sphere \(K^*\) possesses an excess of energy in comparison with the sphere \(K\). This does not mean, of course, that energy has “turned into” mass; it means only that mass and energy are inseparably connected with one another and that therefore to every excess of energy there corresponds an excess of mass and, conversely, to an excess of mass—an excess of energy.

In the case of an ideally elastic body, the kinetic energy of the hemispheres \(K'\) and \(K''\) is transformed into the energy of elastic deformation of the sphere \(K^*\), with conservation of the excess mass \(\Delta m\). In the case of an absolutely inelastic body, the kinetic energy of the ordered motion of the hemispheres \(K'\) and \(K''\) in the collision is transformed into the kinetic energy of the chaotic motion of the molecules of \(K^*\), i.e., into heat, again with conservation of the excess mass \(\Delta m\). Since the energy of a body is a single-valued function of state, the same increase of mass must occur upon heating by the same number of degrees, by whatever method this heating is brought about.

Let us note that the excess mass \(\Delta m\), expressed by formula (16), in the nonrelativistic approximation (for \(\dfrac{v}{c} \ll 1\)) can be represented—

set out in the form (cf. formula (14))

\[ \Delta m=\frac{1}{c^2}\frac{1}{2}M_0v^2=2\frac{E_{\mathrm{kin}}}{c^2}, \]

where by \(E_{\mathrm{kin}}\) is meant the kinetic energy of each half-sphere (with rest mass \(\frac{1}{2}M_0\)) moving with velocity \(v\), which once again emphasizes that mass is inseparably connected with any kind of energy, not excluding the kinetic energy of the translational motion of a body.

1. On the basis of what has been said, in any exchange of energy—whether this exchange is due to collision or heating, or to the absorption of light—there also occurs an exchange of mass. Namely, to every change of energy \(\Delta E\) there corresponds a change of mass \(\Delta m\), equal to

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

It goes without saying that, owing to the enormous magnitude \(c^2=9\cdot 10^{20}\sim 10^{21}\), the changes of mass in ordinary processes of heating or collision are so small that they have no practical significance whatever. Only in nuclear reactions, where \(\Delta E\) is very large, can \(\Delta m\) acquire—and usually does acquire—a measurable value. It is well known that it is precisely in this way that, in recent years, the relation between mass and energy has been subjected to experimental verification and has been confirmed in the most irreproachable manner. Moreover, at present the most accurate method of determining the masses of isotopes consists precisely in determining the energies of nuclear reactions. The reason for this lies in the immeasurably greater “sensitivity” of the energy characteristic of a process as compared with its mass characteristic. Indeed, the latter is \(c^2\) times smaller than the former. I shall note, finally, that in nuclear physics, where one has to deal with enormous energies, the latter are often measured in units of mass, with 1 mass unit of energy being equivalent to 931 MeV.

1. Up to now we have used as examples the simplest phenomena—such as the impact of perfectly elastic or perfectly inelastic spheres—and have deliberately avoided turning to nuclear reactions, which are usually used when considering the question of the relation between mass and energy. Let us now consider how matters stand in the case of nuclear processes.

In considering collision, we became convinced that the mass of a body depends on its state. We saw, for example, that the masses

ideal elastic spheres \(K\) and \(K^*\), which differ from one another only in that the sphere \(K^*\) possesses an internal energy of elastic deformation, are not the same, although in all other respects the spheres do not differ from one another in any way. The situation is analogous in the case of nuclear processes. Consider a nucleus composed of \(Z\) protons and \(N\) neutrons. Let us denote by \(M'\) the sum of the rest masses of these elementary particles

\[ M'=\sum_{i=1}^{Z} m_p^{(i)}+\sum_{K=1}^{N} m_n^{(k)} . \tag{17} \]

It is not difficult to see that the mass of the nucleus, which contains these \(Z+N\) particles, is not equal to \(M'\). Indeed, in calculating the mass of a stable nucleus it is necessary to take into account not only the rest masses of its constituent elementary particles, but also the mass corresponding to their binding energy \(\Delta E\). Since, however, we are dealing with a stable nucleus, its binding energy is negative. Therefore the total mass of the nucleus \(M\) is equal to

\[ M=\left(\sum m_p+\sum m_n\right)-\frac{\Delta E}{c^2} = M' - \frac{\Delta E}{c^2}. \tag{18} \]

Thus, the mass of the nucleus is less than the sum of the masses of the neutrons and protons by the amount \(\dfrac{\Delta E}{c^2}\), i.e. by the amount of mass corresponding to the binding energy.

Let us now consider the process of formation of the nucleus from a somewhat different point of view. Suppose that our \(Z\) protons and \(N\) neutrons are initially located at a great distance from one another. The total mass of the whole system will simply be equal to the sum of the rest masses

\[ \sum m_p+\sum m_n, \]

since, so long as the elementary particles are at a great distance from one another, their interaction energy is equal to zero. Now bring these particles together to a distance of the order of the radius of action of the nuclear forces. In doing so, a stable nucleus with mass \(M\) will arise and some amount of energy \(\Delta E\) will be liberated (equal to the binding energy with the opposite sign). The total mass will therefore be

\[ M+\frac{\Delta E}{c^2}. \]

The law of conservation of mass now gives

\[ \sum m_p+\sum m_n = M+\frac{\Delta E}{c^2} \]

or

\[ M=\left(\sum m_p+\sum m_n\right)-\frac{\Delta E}{c^2} = M' - \frac{\Delta E}{c^2}, \]

i.e. again equality (18).

If the space in which the process of formation of a stable nucleus takes place is strictly isolated from the surrounding medium, so that not only exchange of matter with this external medium is impossible, but also exchange of energy, in whatever form this energy might be released (including hard \(\gamma\)-rays), then the mass of the system before and after the reaction will be one and the same: no transformation of part of the mass into energy occurs. If, however, the system is not isolated, or if the isolation is incomplete in the sense that leakage of matter (i.e., particles with finite rest mass) is prevented, but leakage of energy remains possible, then a part of the total mass of the system, equal to \(\dfrac{\Delta E}{c^2}\), will be “dispersed” among the surrounding bodies, and this part will be lost for the nucleus that has formed.

This fact, in countless popular books and articles up to recent times, has usually been described as a “transformation of mass into energy”). Such a formulation is completely erroneous: mass is not transformed into anything and cannot be transformed, since there is no mass without energy and no energy without mass; both are properties of matter, inseparably connected with one another*).

10. Let us now consider the following important question: do the laws of conservation of mass and energy hold “separately,” or in fact is there only one general law of conservation of mass—energy, while these laws taken separately are invalid? On this

) This erroneous formulation is given, unfortunately, also in my little book Atomic Energy, published in 1946. I note, however, that in my larger book Atomic Physics* neither in its first edition (1944) nor in subsequent editions (1948 and later years) does this error occur.

) In the general press one sometimes encounters the assertion that the occurrence of a mass defect \(\Delta M\) is necessarily connected with the formation of photons. Thus, for example, in the article by S. Melokhin “Mass and Energy” (Uchitel’skaya gazeta of 9 July 1952) we read: “The so-called ‘mass defect’ (i.e., loss of mass) is the transformation of part of matter into photons, which also possess a specific (??) mass,” or “... in nuclear reactions a certain part of matter is transformed into radiation, into a field.” Both these assertions are valid only in certain particular cases, namely, in the case of reactions \((n,\gamma)\) or \((p,\gamma)\). As general propositions these assertions are incorrect: there is an enormous number of nuclear reactions in which no photons arise. These include, for example, the often cited reaction \(\mathrm{Li}^7(p,\alpha)\mathrm{He}^4\). In this reaction the mass defect corresponds exactly to the kinetic energy of the particles flying apart in opposite directions. Since, however, the relation \(\Delta E = c^2 \Delta m\) is valid for any kind of energy, there is no need to resort to an assumption contradicting experiment in order to explain the mass defect.

there is great confusion on this point in the popular literature. In some articles you encounter the assertion that there is only one law. But then it remains unclear how two laws could have been established at different times and quite independently of one another: the law of conservation of mass and the law of conservation of energy. In other articles it is asserted—especially often recently—that the laws of conservation of mass and energy must necessarily hold “separately.” This assertion is even given philosophical significance: the opposite assertion is declared idealistic. But then an oppressive contradiction arises with what is said in textbooks of theoretical physics, where it is proved mathematically that the strict conservation law, valid for any energies, is one.

In fact, however, this collision is imaginary: with an exact formulation of the conditions both assertions prove to be correct. As the most general assertion, the theorem proved in textbooks of theoretical physics is, of course, correct, namely that the strict conservation law is one. It is not difficult to see, however, that in the relativistic case (i.e. when \(v\) has any values up to \(v \cong c\)) fulfillment of the law of conservation of mass automatically entails fulfillment of the law of conservation of energy. Indeed, from the equality

\[ \sum_i m_i=\mathrm{const} \tag{19} \]

there automatically follows

\[ c^2\sum_i m_i=\sum_i m_i c^2=\mathrm{const}. \tag{20} \]

The first of these equalities expresses the law of conservation of mass, the second—the law of conservation of total relativistic energy (let us note that \(m_i\) is everywhere not the rest mass, but the mass corresponding to the given velocity). The converse assertion is, of course, also true: if equality (20) is valid, then by dividing both sides by \(c^2\) one obtains (19). Since both equalities are consequences of one another, they give not two equations, but only one.

Let us now turn to the so-called nonrelativistic case, i.e. to the case when \(\dfrac{v}{c} \ll 1\). Under this condition, as we have already seen more than once, the mass \(m_i\) corresponding to the velocity \(v\) can be represented with sufficient accuracy as the sum of the rest mass and the mass corresponding to the kinetic energy

\[ m_i=m_i^0+\frac{1}{c^2}\,\frac{1}{2}m_i^0 v^2. \]

Therefore the left-hand side of the law of conservation of mass can also be

represent in the form of the sum

\[ \sum_i m_i^0+\sum_i \frac{E_i}{c^2}=\text{const}, \tag{21} \]

where by \(E_i\) is meant the total kinetic energy of the system of \(i\) masses. But it is obvious that in the nonrelativistic case (more precisely, in the case when the energy is less than \(m_0c^2\)) the rest masses of the particles cannot change, as a result of which the first term of the left-hand side is constant and we obtain, instead of equation (19), two equations

\[ \sum_i m_i^0=\text{const}_1;\qquad \sum_i E_i=\text{const}_2 \tag{22} \]

—the first expresses the law of conservation of mass, the second the law of conservation of energy.

Finally, one last remark in connection with the conservation laws. If processes accompanied by the release of energy occur in a system (for example, chemical reactions), then the law of conservation of mass must take the form (21), i.e., with allowance for the mass corresponding to the energy released. Therefore, strictly speaking, the mass in a given system will be conserved only in the case when the strictest measures have been taken to isolate the system from surrounding bodies. However, in all works that proved the law of conservation of mass, namely, in the experiments of Lomonosov, Lavoisier, and also in the later experiments carried out by Landolt already at the beginning of the twentieth century, the absence of exchange of matter with surrounding bodies was ensured, but no measures were taken to ensure the absence of exchange of energy. Nevertheless the law of conservation of mass in all cases proved to be fulfilled—understood to be within the limits of experimental error. Here, of course, there is likewise no contradiction. Indeed, the energies released in ordinary chemical reactions correspond to such negligible masses (we recall that the factor \(\frac{1}{c^2}\), which converts ergs into grams, is equal to \(\sim 10^{-21}\)!) that the error due to neglecting the exchange of energy was more than covered by other unavoidable experimental errors.

Nuclear reactions are a different matter. In them enormous energies are released or absorbed, and isolation must therefore be stricter: measures must be taken to prevent not only leakage of matter, but also leakage of energy—including in the form of the energy of \(\gamma\)-rays.

If, however, the task is to use the energy released in nuclear reactions, then the process should be carried out in such a way as to ensure the fullest and easiest possible removal of the liberated energy. In this case the mass of the reaction products will, of course, be less than the sum

masses of the initial substances. The missing mass does not “disappear,” does not “turn into energy,” but undergoes “scattering,” i.e., is distributed among an enormous number of bodies external to the given system. It is not uninteresting to cite some figures. In so-called nuclear reactors or “piles,” where a nuclear chain reaction takes place, from each kilogram of \(U^{235}\) there are obtained about 989 g of fission products, about 10 g of neutrons, and about 1 g is scattered, being associated with the kinetic energy and excitation energy of the fission products. Of this last one gram, about 700 mg goes into kinetic energy, and 100 mg in the subsequent transformation of the fission products goes into \(\gamma\)-radiation*). The circumstances are still more interesting in the case of the reaction \(\mathrm{Li}^7(p,\alpha)\mathrm{He}^4\), which in essence is also a fission reaction. Here the fission products are obtained in the unexcited state. At \(2\cdot 10^7\) degrees K this reaction can proceed spontaneously with an average transformation time of 1 min. Calculation shows that from 1 kg of a mixture of \(\mathrm{Li}^7+\mathrm{H}^1\), 997.5 g of helium—the product of the reaction—should be obtained, and 2.5 g goes into kinetic energy, which is scattered in the form of heat.

1. Up to now all our considerations have referred to differences of energy \(\Delta E\) and correspondingly to differences of mass \(\Delta m\): the relation between mass and energy we deliberately wrote in the form

\[ \Delta E = c^2 \Delta m . \]

A natural question arises: can we also extend this relation to the total energy, i.e., write the relation in the form

\[ E = mc^2 . \tag{23} \]

That the answer to this question is not obvious follows at least from the fact that an arbitrary constant enters into the numerical value of the energy. Let us note that in relation (23) \(m\), generally speaking, is not the rest mass, but the mass corresponding to the velocity of the body \(v\), i.e.

\[ E=\frac{m_0c^2}{\sqrt{1-\dfrac{v^2}{c^2}}}. \]

In the nonrelativistic approximation \(\left(\dfrac{v}{c}\ll 1\right)\) we have:

\[ E=m_0c^2+\frac{1}{2}m_0v^2+\frac{3}{8}\frac{m_0v^4}{c^2}+\ldots \]

and correspondingly for the mass

\[ m=m_0+\frac{1}{c^2}\frac{1}{2}m_0v^2+\ldots . \]

) The figures are given from the book Scientific and Technical Foundations of Nuclear Power Engineering*, edited by Goodman, vol. I, Foreign Literature Publishing House, 1948.

In these formulas \(m_0c^2\) may be regarded as an energy constant normalized in a definite way, while \(m_0\) is the invariant mass of the particle, its rest mass. To make our reasoning as clear as possible, let us suppose that our object is an elementary particle—an electron, a proton, etc. Our question can now be formulated as follows: does the constant \(m_0c^2\) have the meaning of the energy that corresponds to the rest mass of an elementary particle? In other words: can we assert that with the rest mass of the electron

\[ m_0 = 9.106 \cdot 10^{-28}\ \text{g} \]

there is inseparably associated the energy

\[ m_0c^2 = 9.106 \cdot 10^{-28} \times 8.99 \cdot 10^{20}\ \text{ergs} = 0.5108\ \text{MeV}? \]

The same question may also be posed for other elementary particles with nonzero rest mass. The answer to this question must above all be provided by experiment. It is well known that, in the case of the electron, such an experimental answer exists. Experiment shows that under certain conditions there occurs the so-called annihilation of an electron and a positron, i.e. the disappearance of this pair of particles and the appearance in their place of two photons with an energy of \(0.51\ \text{MeV}\) each. Considering this process from the point of view of the law of conservation of mass, we must say that the mass is fully conserved, since the mass of the two photons that arise is equal to the mass of the disappearing electron–positron pair. Likewise, in the inverse process of “birth” of a pair at the expense of a photon with energy \(1.02\ \text{MeV}\), both mass and energy are conserved. In this latter case, a pair of particles with finite rest mass arises at the expense of a photon—a particle having no rest mass, but having a mass of motion that corresponds exactly to the energy \(2m_0c^2\), equal to the “rest energy” of the electron–positron pair.

At present there is still no experimental proof of analogous transformations of protons.

However, for the “birth” of a proton–antiproton pair, an energy is required that is at least 1840 times greater than for the birth of an electron–positron pair, i.e. an energy on the order of a billion electron-volts.

On the other hand, it is known that in collisions of high-energy particles, \(\pi\)-mesons arise at the expense of the corresponding excess energy (with a mass of \(270\ \text{MeV}\)). Recently Fermi subjected to theoretical investigation the processes possible in collisions of nucleons of ultra-high energies. He showed that under these conditions one should expect, with the greatest probability, multiple production of \(\pi\)-mesons; the processes of production

pairs nucleon–antinucleon are also possible, but comparatively unlikely*).

Let us note another interesting case of the spontaneous transformation of a particle with nonzero rest mass into a pair of photons. This concerns the recently discovered neutral meson with rest mass about \(300\,m_e\). This particle decays spontaneously into two photons with a very short half-life \(\tau \simeq 10^{-14}\) sec.

In all the cases considered, the mutual transformations of elementary particles occur with the participation of photons.

This, however, is not necessary: there exist phenomena in which elementary particles of greater rest mass are transformed “in flight” into other elementary particles of smaller rest mass, but possessing, correspondingly, greater kinetic energy. The law of conservation of mass is observed in this case as well, provided, however, that the mass corresponding to the kinetic energy of the particle produced is taken into account. Phenomena of this kind are observed in the transformations of \(\pi^{\pm}\)-mesons. These elementary particles possess rest mass \(270\,m_e\) (\(m_e\) is the mass of the electron); having been brought to rest in matter, they transform into other elementary particles—\(\mu\)-mesons (and a neutrino) with rest mass \(210\,m_e\) and with kinetic energy varying continuously from 0 to a maximum value equal to the difference of the masses \(\pi-\mu\), multiplied by \(c^2\). In turn, the \(\mu\)-meson decays spontaneously into an electron and two neutrinos, with a corresponding increase in kinetic energy**).

1. It remains for us to dwell briefly on the last question: what significance do the facts described above and the theoretical considerations have for the most important physical and philosophical problem—the problem of matter. First of all, it is clear that an irreparable blow has been dealt not only to Newton’s definition of mass, but also to the Democritean conception of atoms as small immutable bodies occupying a definite volume inaccessible to other atoms. We know that this in no way shakes materialism, since we remember Lenin’s words: “The recognition of any immutable elements, an ‘immutable essence of things,’ etc., is not materialism, but metaphysical, i.e. anti-dialectical, materialism”***).

It is not superfluous to note that this collapse of the Democritean conception of atoms occurred not only in the sphere of questions relating to the connection between mass and energy, but in the most varied

) See E. Fermi, UFN, vol. XLI, p. 71, 1952.
) See the detailed review by Powell, “Mesons,” UFN, vol. XLV, p. 15, 1951.
) V. I. Lenin, Works, 4th ed., vol. 14, p. 248.

specific problems of physics. When a modern physicist discusses collision problems—collisions of electrons with atoms, of atoms with one another, collisions of neutrons or protons with nuclei—he never regards these processes as the geometrical contact of round particles. On the contrary, he always studies the interaction of the fields with which these particles are associated; he introduces the notion of “effective” cross sections, which have only the most indirect relation to the “sizes” of atoms or nuclei, since in some cases they turn out to be considerably smaller than the “geometrical” sizes of the particles, while in others they exceed them by tens of thousands of times. It is needless to say that the entire experimental foundation of quantum mechanics—the wave properties of free particles, “tunneling” through potential barriers—is in irreconcilable contradiction with Democritean atom-corpuscles.

It seems beyond doubt that these facts, and also those especially interesting to us today and described above, may find a natural interpretation within the framework of the field theory of matter. According to this theory, which is still more a program than a completed theory, it is not particles that create the field, but they themselves are generated by the field; they are peculiar “quanta” of the field, sometimes possessing a finite rest mass, and sometimes—as in the case of photons—not possessing it. This theory still has to answer the question why the field has a “granular” structure, producing discrete elementary particles and atoms. This idea is not new. Attempts were made long ago, for example, to modify Maxwell’s equations in such a way that the existence of electrons would follow from them as a consequence, rather than being introduced into the theory as an experimentally substantiated postulate. Later these views were thoroughly forgotten, and only in the very most recent years have they again been formulated and developed by Soviet physicists—D. I. Blokhintsev and Ya. I. Frenkel*). At present this theory is still far from completion. Apparently, however, the future development of the theory of matter will proceed precisely in this direction.

) See the articles by D. I. Blokhintsev, UFN, Vol. XLII, p. 76, 1950; UFN, Vol. XLIV, p. 101, 1951, and also Ya. I. Frenkel, UFN, Vol. XLII, p. 60, 1950; UFN*, Vol. XLIV, p. 112, 1951.