Revealing the Universe's Secrets: Proofs of Einstein's Law of Gravitation

The reasons for accepting Einstein’s law of gravitation rather than Newton’s are partly empirical, partly logical. We will begin with the former.

Einstein’s law of gravitation gives very nearly the same results as Newton’s, when applied to the calculation of the orbits of the planets and their satellites. If it did not, it could not be true, since the consequences deduced from Newton’s law have been found to be almost exactly verified by observation. When, in 1915, Einstein first published his new law, there was only one empirical fact to which he could point to show that his theory was better than Newton’s. This was what is called the “motion of the perihelion of Mercury.”

The planet Mercury, like the other planets, moves round the sun in an ellipse, with the sun in one of the foci. At some points of its [Pg 132]orbit it is nearer to the sun than at other points. The point where it is nearest to the sun is called its “perihelion.” Now it was found by observation that, from one occasion when Mercury is nearest to the sun until the next, Mercury does not go exactly once round the sun, but a little bit more. The discrepancy is very small; it amounts to an angle of forty-two seconds in a century. That is to say, in each year the planet has to move rather less than half a second of angle after it has finished a complete revolution from the last perihelion before it reaches the next perihelion. This very minute discrepancy from Newtonian theory had puzzled astronomers. There was a calculated effect due to perturbations caused by the other planets, but this small discrepancy was the residue after allowing for these perturbations. Einstein’s theory accounted for this residue, as well as for its absence in the case of the other planets. (In them it exists, but is too small to be observed.) This was, at first, his only empirical advantage over Newton.

His second success was more sensational. According to orthodox opinion, light in a vacuum ought always to travel in straight lines. Not being composed of material particles, it ought to be unaffected [Pg 133]by gravitation. However, it was possible, without any serious breach with old ideas, to admit that, in passing near the sun, light might be deflected out of the straight path as much as if it were composed of material particles. Einstein, however, maintained, as a deduction from his law of gravitation, that light would be deflected twice as much as this. That is to say, if the light of a star passed very near the sun, Einstein maintained that the ray from the star would be turned through an angle of just under one and three-quarters seconds. His opponents were willing to concede half of this amount. Now it is not every day that a star almost in line with the sun can be seen. This is only possible during a total eclipse, and not always then, because there may be no bright stars in the right position. Eddington points out that, from this point of view, the best day of the year is May 29, because then there are a number of bright stars close to the sun. It happened by incredible good fortune that there was a total eclipse of the sun on May 29, 1919—the first year after the armistice. Two British expeditions photographed the stars near the sun during the eclipse, and the results confirmed Einstein’s prediction. Some astronomers [Pg 134]who remained doubtful whether sufficient precautions had been taken to insure accuracy were convinced when their own observations in a subsequent eclipse gave exactly the same result. Einstein’s estimate of the amount of the deflection of light by gravitation is therefore now universally accepted.

The third experimental test is on the whole favorable to Einstein, though the quantities concerned are so small that it is only just possible to measure them, and the result is therefore not decisive. But successive investigations have made it more and more probable that the small effect predicted by Einstein really occurs. Before explaining the effect in question, a few preliminary explanations are necessary. The spectrum of an element consists of certain lines of various shades of light, separated by a prism, and emitted by the element when it glows. They are the same (to a very close approximation) whether the element is in the earth or the sun or a star. Each line is of some definite shade of color, with some definite wave length. Longer wave lengths are towards the red end of the spectrum, shorter ones towards the violet end. When the source of light is moving towards you, the apparent wave [Pg 135]lengths grow shorter, just as waves at sea come quicker when you are traveling against the wind. When the source of light is moving away from you, the apparent wave lengths grow longer, for the same reason. This enables us to know whether the stars are moving towards us or away from us. If they are moving towards us, all the lines in the spectrum of an element are moved a little toward violet; if away from us, toward red. You may notice the analogous effect in sound any day. If you are in a station and an express comes through whistling, the note of the whistle seems much more shrill while the train is approaching you than when it has passed. Probably many people think the note has “really” changed, but in fact the change in what you hear is only due to the fact that the train was first approaching and then receding. To people in the train, there was no change of note. This is not the effect with which Einstein is concerned. The distance of the sun from the earth does not change much; for our present purposes, we may regard it as constant. Einstein deduces from his law of gravitation that any periodic process which takes place in an atom in the sun (whose [Pg 136]gravitation is very intense) must, as measured by our clocks, take place at a slightly slower rate than it would in a similar atom on the earth. The “interval” involved will be the same in the sun and on the earth, but the same interval in different regions does not correspond to exactly the same time; this is due to the “hilly” character of space-time which constitutes gravitation. Consequently any given line in the spectrum ought, when the light comes from the sun, to seem to us a little nearer the red end of the spectrum than if the light came from a source on the earth. The effect to be expected is very small—so small that there is still some slight uncertainty as to whether it exists or not. But it now seems highly probable that it exists.

No other measurable differences between the consequences of Einstein’s law and those of Newton’s have hitherto been discovered. But the above experimental tests are quite sufficient to convince astronomers that, where Newton and Einstein differ as to the motions of the heavenly bodies, it is Einstein’s law that gives the right results. Even if the empirical grounds in favor of Einstein stood alone, they would be conclusive. Whether his law represents the exact truth or not, it is [Pg 137]certainly more nearly exact than Newton’s, though the inaccuracies in Newton’s were all exceedingly minute.

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 [Pg 138]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: [Pg 139]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.[Pg 140]

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, [Pg 141]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 [Pg 142]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. [Pg 143]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.