For theoretical purposes, a mass is defined as being determined by the amount of force required to produce a given acceleration: The more massive a body is, the greater will be the force required to alter its velocity by a given amount in a given time. It takes a more powerful engine to make a long train attain a speed of ten miles an hour at the end of the first half-minute, than it does to make a short train do so. Or we may have circumstances where the force is the same for a number of different bodies; in that case, if we can measure the accelerations produced in them, we can tell the ratios of their masses: the greater the mass, the smaller the acceleration. We may take, in illustration of this method, an example which is important in connection with relativity. Radio-active bodies emit beta-particles (electrons) with enormous velocities. We can observe their path by making them travel through water vapor and form a cloud as they go. We can at the same time subject them to known electric and magnetic forces, and observe how much they are bent out of a straight line by these forces. This makes it possible to compare their masses. It is found that the faster they travel, the greater is their mass, as measured by the stationary observer; the increase is greatest as applied to their mass as measured by the effect of a force in the line of motion. In regard to forces at right angles to the line of motion, there is a change of mass with velocity in the same proportion as the changes of length and time. It is known otherwise that, apart from the effect of motion, all electrons have the same mass.
All this was known before the theory of relativity was invented, but it showed that the traditional conception of mass had not quite the definiteness that had been ascribed to it. Mass used to be regarded as “quantity of matter,” and supposed to be quite invariable. Now mass was found to be relative to the observer, like length and time, and to be altered by motion in exactly the same proportion. However, this could be remedied. We could take the “proper mass,” the mass as measured by an observer who shares the motion of the body. This was easily inferred from the measured mass, by taking the same proportion as in the case of lengths and times.
But there is a more curious fact, and that is, that after we have made this correction we still have not obtained a quantity which is at all times exactly the same for the same body. When a body absorbs energy—for example, by growing hotter—its “proper mass” increases slightly. The increase is very slight, since it is measured by dividing the increase of energy by the square of the velocity of light. On the other hand, when a body parts with energy it loses mass. The most notable case of this is that four hydrogen atoms can come together to make one helium atom, but a helium atom has rather less than four times the mass of one hydrogen atom.
We have thus two kinds of mass, neither of which quite fulfils the old ideal. The mass as measured by an observer who is in motion relative to the body in question is a relative quantity, and has no physical significance as a property of the body. The “proper mass” is a genuine property of the body, not dependent upon the observer; but it, also, is not strictly constant. As we shall see shortly, the notion of mass becomes absorbed into the notion of energy; it represents, so to speak, the energy which the body expends internally, as opposed to that which it displays to the outer world.
Conservation of mass, conservation of momentum, and conservation of energy were the great principles of classical mechanics. Let us next consider conservation of momentum.
The momentum of a body in a given direction is its velocity in that direction multiplied by its mass. Thus a heavy body moving slowly may have the same momentum as a light body moving fast. When a number of bodies interact in any way, for instance by collisions, or by mutual gravitation, so long as no outside influences come in, the total momentum of all the bodies in any direction remains unchanged. This law remains true in the theory of relativity. For different observers, the mass will be different, but so will the velocity; these two differences neutralize each other, and it turns out that the principle still remains true.
The momentum of a body is different in different directions. The ordinary way of measuring it is to take the velocity in a given direction (as measured by the observer) and multiply it by the mass (as measured by the observer). Now the velocity in a given direction is the distance traveled in that direction in unit time. Suppose we take instead the distance traveled in that direction while the body moves through unit “interval.” (In ordinary cases, this is only a very slight change, because, for velocities considerably less than that of light, interval is very nearly equal to lapse of time.) And suppose that instead of the mass as measured by the observer we take the proper mass. These two changes increase the velocity and diminish the mass, both in, the same proportion. Thus the momentum remains the same, but the quantities that vary according to the observer have been replaced by quantities which are fixed independently of the observer—with the exception of the distance traveled by the body in the given direction.
When we substitute space-time for time, we find that the measured mass (as opposed to the proper mass) is a quantity of the same kind as the momentum in a given direction; it might be called the momentum in the time direction. The measured mass is obtained by multiplying the invariant mass by the time traversed in traveling through unit interval; the momentum is obtained by multiplying the same invariant mass by the distance traversed (in the given direction) in traveling through unit interval. From a space-time point of view, these naturally belong together.
Although the measured mass of a body depends upon the way the observer is moving relatively to the body, it is none the less a very important quantity. For any given observer, the measured mass of the whole physical universe is constant.[8] The proper mass of all the bodies in the world is not necessarily the same at one time as at another, so that in this respect the measured mass has an advantage. The conservation of measured mass is the same thing as the conservation of energy. This may seem surprising, since at first sight mass and energy are very different things. But it has turned out that energy is the same thing as measured mass. To explain how this comes about is not easy; nevertheless we will make the attempt.
In popular talk, “mass” and “energy” do not mean at all the same thing. We associate “mass” with the idea of a fat man in a chair, very slow to move, while “energy” suggests a thin 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 to the meanings of the terms in popular talk. For the present, we are concerned with the technical meaning of “energy.”