But the considerations which originally led Einstein to his law were not of this detailed kind. Even the consequence about the perihelion of Mercury, which could be verified at once from previous observations, could only be deduced after the theory was complete, and could not form any part of the original grounds for inventing such a theory. These grounds were of a more abstract logical character. I do not mean that they were not based upon observed facts, and I do not mean that they were à priori fantasies such as philosophers indulged in formerly. What I mean is that they were derived from certain general characteristics of physical experience, which showed that Newton must be wrong and that something like Einstein’s law must be substituted.
The arguments in favor of the relativity of motion are, as we saw in earlier chapters, quite conclusive. In daily life, when we say that something moves, we mean that it moves relatively to the earth. In dealing with the motions of the planets, we consider them as moving relatively to the sun, or to the center of mass of the solar system. When we say that the solar system itself is moving, we mean that it is moving relatively to the stars. There is no physical occurrence which can be called “absolute motion.” Consequently the laws of physics must be concerned with relative motions, since these are the only kind that occur.
We now take the relativity of motion in conjunction with the experimental fact that the velocity of light is the same relatively to one body as relatively to another, however the two may be moving. This leads us to the relativity of distances and times. This in turn shows that there is no objective physical fact which can be called “the distance between two bodies at a given time,” since the time and the distance will both depend on the observer. Therefore Newton’s law of gravitation is logically untenable, since it makes use of “distance at a given time.”
This shows that we cannot rest content with Newton, but it does not show what we are to put in his place. Here several considerations enter in. We have in the first place what is called “the equality of gravitational and inertial mass.” What this means is as follows: When you apply a given force[6] to a heavy body, you do not give it as much acceleration as you would to a light body. What is called the “inertial” mass of a body is measured by the amount of force required to produce a given acceleration. At a given point of the earth’s surface, the “mass” is proportional to the “weight.” What is measured by scales is rather the mass than the weight: the weight is defined as the force with which the earth attracts the body. Now this force is greater at the poles than at the equator, because at the equator the rotation of the earth produces a “centrifugal force” which partially counteracts gravitation. The force of the earth’s attraction is also greater on the surface of the earth than it is at a great height or at the bottom of a very deep mine. None of these variations are shown by scales, because they affect the weights used just as much as the body weighed; but they are shown if we use a spring balance. The mass does not vary in the course of these changes of weight.
The “gravitational” mass is differently defined. It is capable of two meanings. We may mean (1), the way a body responds in a situation where gravitation has a known intensity, for example, on the surface of the earth, or on the surface of the sun; or (2), the intensity of the gravitational force produced by the body, as, for example, the sun produces stronger gravitational forces than the earth does. Newton says that the force of gravitation between two bodies is proportional to the product of their masses. Now let us consider the attraction of different bodies to one and the same body, say the sun. Then different bodies are attracted by forces which are proportional to their masses, and which, therefore, produce exactly the same acceleration in all of them. Thus if we mean “gravitational mass” in sense (1), that is to say, the way a body responds to gravitation, we find that “the equality of inertial and gravitational mass,” which sounds formidable, reduces to this: that in a given gravitational situation, all bodies behave exactly alike. As regards the surface of the earth, this was one of the first discoveries of Galileo. Aristotle thought that heavy bodies fall faster than light ones; Galileo showed that this is not the case, when the resistance of the air is eliminated. In a vacuum, a feather falls as fast as a lump of lead. As regards the planets, it was Newton who established the corresponding facts. At a given distance from the sun, a comet, which has a very small mass, experiences exactly the same acceleration towards the sun as a planet experiences at the same distance. Thus the way in which gravitation affects a body depends only upon where the body is, and in no degree upon the nature of the body. This suggests that the gravitational effect is a characteristic of the locality, which is what Einstein makes it.
As for the gravitational mass in sense (2), i.e., the intensity of the force produced by a body, this is no longer exactly proportional to its inertial mass. The question involves some rather complicated mathematics, and I shall not go into it.[7]
We have another indication as to what sort of thing the law of gravitation must be, if it is to be a characteristic of a neighborhood, as we have seen reason to suppose that it is. It must be expressed in some law which is unchanged when we adopt a different kind of co-ordinates. We saw that we must not, to begin with, regard our co-ordinates as having any physical significance: they are merely systematic ways of naming different parts of space-time. Being conventional, they cannot enter into physical laws. That means to say that, if we have expressed a law correctly in terms of one set of co-ordinates, it must be expressed by the same formula in terms of another set of co-ordinates. Or, more exactly, it must be possible to find a formula which expresses the law, and which is unchanged however we change the co-ordinates. It is the business of the theory of tensors to deal with such formulæ. And the theory of tensors shows that there is one formula which obviously suggests itself as being possibly the law of gravitation. When this possibility is examined, it is found to give the right results; it is here that the empirical confirmations come in. But if Einstein’s law had not been found to agree with experience, we could not have gone back to Newton’s law. We should have been compelled by logic to seek some law expressed in terms of “tensors,” and therefore independent of our choice of co-ordinates. It is impossible without mathematics to explain the theory of tensors; the non-mathematician must be content to know that it is the technical method by which we eliminate the conventional element from our measurements and laws, and thus arrive at physical laws which are independent of the observer’s point of view. Of this method, Einstein’s law of gravitation is the most splendid example.
CHAPTER X:
MASS, MOMENTUM, ENERGY
AND ACTION
The pursuit of quantitative precision is as arduous as it is important. Physical measurements are made with extraordinary exactitude; if they were made less carefully, such minute discrepancies as form the experimental data for the theory of relativity could never be revealed. Mathematical physics, before the coming of relativity, used a set of conceptions which were supposed to be as precise as physical measurements, but it has turned out that they were logically defective, and that this defectiveness showed itself in very small deviations from expectations based upon calculation. In this chapter I want to show how the fundamental ideas of pre-relativity physics are affected, and what modifications they have had to undergo.
We have already had occasion to speak of mass. For purposes of daily life, mass is much the same as weight; the usual measures of weight—ounces, grams, etc.—are really measures of mass. But as soon as we begin to make accurate measurements, we are compelled to distinguish between mass and weight. Two different methods of weighing are in common use, one, that of scales, the other that of the spring balance. When you go a journey and your luggage is weighed, it is not put on scales, but on a spring; the weight depresses the spring a certain amount, and the result is indicated by a needle on a dial. The same principle is used in automatic machines for finding your weight. The spring balance shows weight, but scales show mass. So long as you stay in one part of the world, the difference does not matter; but if you test two weighing machines of different kinds in a number of different places, you will find, if they are accurate, that their results do not always agree. Scales will give the same result anywhere, but a spring balance will not. That is to say, if you have a lump of lead weighing ten pounds by the scales, it will also weigh ten pounds by scales in any other part of the world. But if it weighs ten pounds by a spring balance in London, it will weigh more at the North Pole, less at the equator, less high up in an aeroplane, and less at the bottom of a coal mine, if it is weighed in all those places on the same spring balance. The fact is that the two instruments measure quite different quantities. The scales measure what may be called (apart from refinements which will concern us presently) “quantity of matter.” There is the same “quantity of matter” in a pound of feathers as in a pound of lead. Standard “weights,” which are really standard “masses,” will measure the amount of mass in any substance put into the opposite scales. But “weight” is a properly due to the earth’s gravitation: It is the amount of the force by which the earth attracts a body. This force varies from place to place. In the first place, anywhere outside the earth the attraction varies inversely as the square of the distance from the center of the earth; it is therefore less at great heights. In the second place, when you go down a coal mine, part of the earth is above you, and attracts matter upwards instead of downwards, so that the net attraction downwards is less than on the surface of the earth. In the third place, owing to the rotation of the earth, there is what is called a “centrifugal force,” which acts against gravitation. This is greatest at the equator, because there the rotation of the earth involves the fastest motion; at the poles it does not exist, because they are on the axis of rotation. For all these reasons, the force with which a given body is attracted to the earth is measureably different at different places. It is this force that is measured by a spring balance; that is why a spring balance gives different results in different places. In the case of scales, the standard “weights” are altered just as much as the body to be weighed, so that the result is the same everywhere; but the result is the “mass,” not the “weight.” A standard “weight” has the same mass everywhere, but not the same “weight”; it is in fact a unit of mass, not of weight. For theoretical purposes, mass, which is almost invariable for a given body, is much more important than weight, which varies according to circumstances. Mass may be regarded, to begin with, as “quantity of matter”; we shall see that this view is not strictly correct, but it will serve as a starting point for subsequent refinements.