We now come to a crucial further step made by Einstein, which greatly extended the revolutionary effect of his relativistic point of view, i.e., the demonstration of the equivalence of mass and energy through the by-now very well known formula E=mc2.
To develop this notion, we begin with Eq. (18–37) for the mass of a moving object, . For small v/c we can expand m as a series of powers of v/c, keeping only the terms up to v2/c2. The result is
(19–1)
If we multiply this by c2, we obtain
(19–2)
But m0v2/2 is just the nonrelativistic expression for the kinetic energy T of a body moving with speed v, and m
0c 2
in just a constant. We then have, for such a body
(19–3)
The conservation of total mass of a system, obtained in the previous chapter, then becomes equivalent to the law of conservation of total energy of a collection of bodies, at least in the nonrelativistic limit. But the principle of relativity requires that if such a law holds in any one frame, it will hold in all frames. It follows then that Eq. (19–3) must represent the kinetic energy of a particle in any frame, even when the expansion in terms of v2/c2 is no longer a good approximation.
The significance of this result can be seen more clearly if one transfers m
0c 2
to the other side of the equation. We then write for the energy of the body
The Equivalence of Mass and Energy 71
We can always do this, because in nonrelativistic theory the energy is in any case undefined to within an arbitrary constant. Mathematically speaking, Einstein’s procedure here is equivalent to defining this arbitrary constant, so that the energy of a particle at rest is taken to be
(19–5)
Physically, this corresponds to assuming that even a particle that is not in motion has the rest energy given by (19–5).
What is the meaning of this rest energy? We can perhaps bring this out by noting that a typical object which is visibly at rest is constituted of parts (i.e., molecules, atoms, nuclei, etc.) which are actually in a state of violent movement, such that on the average the effects of the movement cancel out when observed on a macroscopic scale. Nevertheless, according to the arguments given in the previous chapter, all these movements are contributing to the masses of the constituent particles, according to the formula
(19–6)
The total mass of the system is then
(19–7)
where T is the total kinetic energy of the various particles. On multiplying by c2 we have
Now, is there any way of checking experimentally whether the internal state of movement contributes to the mass? The answer is that there are several possible ways of doing this. The most obvious idea would be to raise the temperature of a body and to see if the weight increased by the amount Q/c2, where Q is the heat energy absorbed by the body. The difficulty is that with temperature changes that are available (a few thousand degrees centigrade, at the most), is too small a quantity to be detected by methods that are now available. (This is basically because c2 is such a large number.) Similarly, if we allow two systems to combine chemically and to give off the energy , the sum of the masses should be less than its original value by . But, once again, this is too small to be detected experimentally.
72 The Special Theory of Relativity
Some time after Einstein demonstrated the equivalence of mass and energy theoretically, experimental studies of nuclear transformations were carried out, in which great enough quantities of energy were given off, so that the difference between the sum of the masses of the products and that of the initial reactants was actually measurable with the equipment that was then available. Many such measurements were made; these all confirmed Einstein’s prediction that the change in mass of the whole system is equal to Q/c2.
The experiments cited above show that at least a part of the rest mass of an object can be ascribed to internal movements, in such a way that when these movements alter and give off an energy Q, the mass of the system decreases by Q/c2. But can we verify Einstein’s statement that all of the rest mass of an object can be related in a similar way to an energy?
Some years after the first nuclear transformations were investigated, new particles, called positrons, were discovered, having the same mass as an electron, but opposite charge. It was found that when an electron meets a positron, the two particles can annihilate each other, leaving no particles at all, but giving off gamma rays with total energy (which is ultimately transformed into heat as a result of collisions of the gamma rays with electrons and atoms). In this way it was shown that all of the rest energy of an electron is potentially transformable into other forms of energy, such as heat. Since the discovery of the positron, particles called “antiprotons” (with negative charge and the same mass as that of the proton) have been found, which can similarly annihilate protons. Indeed, it is now known that to each kind of fundamental particle there exists an antiparticle, of the same mass and definitely related properties (such as charge and spin), which combines with the particle to give nothing but energy, in one form or another. Vice versa, it has been shown that a gamma ray colliding with a nucleus can be absorbed, and its energy transformed into the rest energy of, for example, an electron-positron pair, which is created in this process, under conditions in which no such particles existed before. So there has been conclusive experimental proof that either a part or the whole of the “rest energy” of a body can be transformed into other forms of energy, and that the inverse process of transforming other forms of energy into rest energy is also possible.
The only reason that the equivalence of mass and energy was not observed earlier is, as we have already suggested, that before the discovery of nuclear processes, the mass Q/c2, associated with the energy Q, was too small to be detected. Conversely, this means of course that the enormous reserves are “locked” in the rest energy of matter. These reserves are what are being liberated, in part, by nuclear fission in atomic piles, as well as by “fusion” processes that go on spontaneously in the Sun and in the stars.
Einstein gave a simple physical way of seeing why mass and energy are related by the formula E=mc2. To do this, he considered a box of mass M
B, at rest in the laboratory.
Suppose that this box contained a distribution of radiant electromagnetic energy in thermodynamic equilibrium with the walls. Let the energy of this radiation be denoted by ER.
Now, it is well known that electromagnetic energy exerts a radiation pressure in the walls of the box, similar to that produced by a gas. When the box is at rest or in uniform motion, the total force exerted on any one wall is cancelled by that exerted on the
The Equivalence of Mass and Energy 73
opposite wall. But if the box is given an acceleration a, then while the acceleration is taking place the radiation which reflects off the rear wall will gain more momentum than the radiation which reflects off the front wall will lose.
When one carries out a detailed calculation of the resulting changes of pressure on the moving walls, one discovers that the radiation exerts a net force on the box of , which opposes the acceleration. The equation of motion of the system will then be
(19–8)
where F is the applied force. This reduces to
(19–9)
So the radiant energy ER adds an “effective mass” ER/c2 in the sense that it contributes in the same way as such a mass would to the inertia, or resistance to acceleration, which is one of the characteristic manifestations of that physical property called by the name of “mass.”
It can be seen that the case considered by Einstein is very similar to that discussed in the previous chapter, where we studied the effects on the total mass of the internal movements of its various particles. Einstein refers instead to the effects of the internal movements of electromagnetic radiation, thus helping to bring out the point that the contribution to the mass is independent of the nature of the energy.