The A B C of Relativity

person full of hustle and “pep.” Popular talk associates “mass” and “inertia,” but its view of inertia is one-sided: it includes slowness in beginning to move, but not slowness in stopping, which is equally involved. All these terms have technical meanings in physics, which are only more or less analogous [Pg 154] to the meanings of the terms in popular talk. For the present, we are concerned with the technical meaning of “energy.” Throughout the latter half of the nineteenth century, a great deal was made of the “conservation of energy,” or the “persistence of force,” as Herbert Spencer preferred to call it. This principle was not easy to state in a simple way, because of the different forms of energy; but the essential point was that energy is never created or destroyed, though it can be transformed from one kind into another. The principle acquired its position through Joule’s discovery of “the mechanical equivalent of heat,” which showed that there was a constant proportion between the work required to produce a given amount of heat and the work required to raise a given weight through a given height: in fact, the same sort of work could be utilized for either purpose according to the mechanism. When heat was found to consist in motion of molecules, it was seen to be natural that it should be analogous to other forms of energy. Broadly speaking, by the help of a certain amount of theory, all forms of energy were reduced to two, which were called respectively “kinetic” and “potential.” These were defined as follows: [Pg 155] The kinetic energy of a particle is half the mass multiplied by the square of the velocity. The kinetic energy of a number of particles is the sum of the kinetic energies of the separate particles. The potential energy is more difficult to define. It represents any state of strain, which can only be preserved by the application of force. To take the easiest case: If a weight is lifted to a height and kept suspended, it has potential energy, because, if left to itself, it will fall. Its potential energy is equal to the kinetic energy which it would acquire in falling through the same distance through which it was lifted. Similarly when a comet goes round the sun in a very eccentric orbit, it moves much faster when it is near the sun than when it is far from it, so that its kinetic energy is much greater when it is near the sun. On the other hand, its potential energy is greatest when it is farthest from the sun, because it is then like the stone which has been lifted to a height. The sum of the kinetic and potential energies of the comet is constant, unless it suffers collisions or loses matter by forming a tail. We can determine accurately the change of [Pg 156] potential energy in passing from one position to another, but the total amount of it is to a certain extent arbitrary, since we can fix the zero level where we like. For example, the potential energy of our stone may be taken to be the kinetic energy it would acquire in falling to the surface of the earth, or what it would acquire in falling down a well to the center of the earth, or any assigned lesser distance. It does not matter which we take, so long as we stick to our decision. We are concerned with a profit-and-loss account, which is unaffected by the amount of the assets with which we start. Both the kinetic and the potential energies of a given set of bodies will be different for different observers. In classical dynamics, the kinetic energy differed according to the state of motion of the observer, but only by a constant amount; the potential energy did not differ at all. Consequently, for each observer, the total energy was constant—assuming always that the observers concerned were moving in straight lines with uniform velocities, or, if not, were able to refer their motions to bodies which were so moving. But in relativity dynamics the matter becomes more complicated. We cannot profitably [Pg 157] adapt the idea of potential energy to the theory of relativity, and therefore the conservation of energy, in a strict sense, cannot be maintained. But we obtain a property, closely analogous to conservation, which applies to kinetic energy alone. As Eddington puts it: the kinetic energy is not always strictly conserved, and the classical theory therefore introduces a supplementary quantity, the potential energy, so that the sum of the two is strictly conserved. The relativity treatment, on the other hand, discovers another formula, analogous to the one expressing conservation, which holds always for the kinetic energy. “The relativity treatment adheres to the physical quantity and modifies the law; the classical treatment adheres to the law and modifies the physical quantity.” The new formula, he continues, may be spoken of “as the law of conservation of energy and momentum, because, though it is not formally a law of conservation, it expresses exactly the phenomena which classical mechanics attributes to conservation.”[9] It is only in this modified and less rigorous sense that the conservation of energy remains true. What is meant by “conservation” in practice is not exactly what it [Pg 158] means in theory. In theory we say that a quantity is conserved when the amount of it in the world is the same at any one time as at any other. But in practice we cannot survey the whole world, so we have to mean something more manageable. We mean that, taking any given region, if the amount of the quantity in the region has changed, it is because some of the quantity has passed across the boundary of the region. If there were no births and deaths, population would be conserved; in that case the population of a country could only change by emigration or immigration, that is to say, by passing across the boundaries. We might be unable to take an accurate census of China or Central Africa, and, therefore, we might not be able to ascertain the total population of the world. But we should be justified in assuming it to be constant if, wherever statistics were possible, the population never changed except through people crossing the frontiers. In fact, of course, population is not conserved. A physiologist of my acquaintance once put four mice into a thermos. Some hours later, when he went to take them out, there were eleven of them. But mass is not subject to these fluctuations: [Pg 159] the mass of the eleven mice at the end of the time was no greater than the mass of the four at the beginning. This brings us back to the problem for the sake of which we have been discussing energy. We stated that, in relativity theory, measured mass and energy are regarded as the same thing, and we undertook to explain why. It is now time to embark upon this explanation. But here, as at the end of Chapter VI, the totally unmathematical reader will do well to skip, and begin again at the following paragraph. Let us take the velocity of light as the unit of velocity; this is always convenient in relativity theory. Let m be the proper mass of a particle, v its velocity relative to the observer. Then its measured mass will be m ——— √(1 - v²) while its kinetic energy, according to the usual formula, will be ½ mv² As we saw before, energy only occurs in a profit-and-loss account, so[Pg 160] that we can add any constant quantity to it that we like. We may therefore take the energy to be m + ½ mv² Now if v is a small fraction of the velocity of light, m + ½ mv² is almost exactly equal to m ——— √(1 - v²) Consequently, for velocities such as large bodies have, the energy and the measured mass turn out to be indistinguishable within the limits of accuracy attainable. In fact, it is better to alter our definition of energy, and take it to be m ——— √(1 - v²) because this is the quantity for which the law analogous to conservation holds. And when the velocity is very great, it gives a better measure of energy than the traditional formula. The traditional formula must therefore be regarded as an approximation, of which the new formula gives the exact version. In this way, energy and measured mass become identified. I come now to the notion of “action,” which is less familiar to the general public than energy, but has become more important in [Pg 161] relativity physics, as well as in the theory of quanta.[10] (The quantum is a small amount of action.) The word “action” is used to denote energy multiplied by time. That is to say, if there is one unit of energy in a system, it will exert one unit of action in a second, 100 units of action in 100 seconds, and so on; a system which has 100 units of energy will exert 100 units of action in a second, and 10,000 in 100 seconds, and so on. “Action” is thus, in a loose sense, a measure of how much has been accomplished: it is increased both by displaying more energy and by working for a longer time. Since energy is the same thing as measured mass, we may also take action to be measured mass multiplied by time. In classical mechanics, the “density” of matter in any region is the mass divided by the volume; that is to say, if you know the density in a small region, you discover the total amount of matter by multiplying the density by the volume of the small region. In relativity mechanics, we always want to substitute space-time for space; therefore a “region” must no longer be taken to [Pg 162] be merely a volume, but a volume lasting for a time; a small region will be a small volume lasting for a small time. It follows that, given the density, a small region in the new sense contains, not a small mass merely, but a small mass multiplied by a small time, that is to say, a small amount of “action.” This explains why it is to be expected that “action” will prove of fundamental importance in relativity mechanics. And so in fact it is. All the laws of dynamics have been put together into one principle, called “The Principle of Least Action.” This states that, in passing from one state to another, a body chooses a route involving less action than any slightly different route—again a law of cosmic laziness. The principle is subject to certain limitations, which have been pointed out by Eddington,[11] but it remains one of the most comprehensive ways of stating the purely formal part of mechanics. The fact that the quantum is a unit of action seems to show that action is also fundamental in the empirical structure of the world. But at present there is no bridge connecting the quantum with the theory of relativity.