An analogy will serve to make the point clear. Suppose that on a dark night a number of men with lanterns were walking in various directions across a huge plain, and suppose that in one part of the plain there was a hill with a flaring beacon on the top. Our hill is to be such as we have described, growing steeper as it goes up, and ending in a precipice. I shall suppose that there are villages dotted about the plain, and the men with lanterns are walking to and from these various villages. Paths have been made showing the easiest way from any one village to any other. These paths will all be more or less curved, to avoid going too far up the hill; they will be more sharply curved when they pass near the top of the hill than when they keep some way off from it. Now suppose that you are observing all this, as best you can, from a place high up in a balloon, so that you cannot see the ground, but only the lanterns and the beacon. You will not know that there is a hill, or that the beacon is at the top of it. You will see that people turn out of the straight course when they approach the beacon, and that the nearer they come the more they turn aside. You will naturally attribute this to an effect of the beacon; you may think that it is very hot and people are afraid of getting burnt. But if you wait for daylight you will see the hill, and you will find that the beacon merely marks the top of the hill and does not influence the people with lanterns in any way.
Now in this analogy the beacon corresponds to the sun, the people with lanterns correspond to the planets and comets, the paths correspond to their orbits, and the coming of daylight corresponds to the coming of Einstein. Einstein says that the sun is at the top of a hill, only the hill is in space-time, not in space. (I advise the reader not to try to picture this, because it is impossible.) Each body, at each moment, adopts the easiest course open to it, but owing to the hill the easiest course is not a straight line. Each little bit of matter is at the top of its own little hill, like the cock on his own dung-heap. What we call a big bit of matter is a bit which is at the top of a big hill. The hill is what we know about; the bit of matter at the top is assumed for convenience. Perhaps there is really no need to assume it, and we could do with the hill alone, for we can never get to the top of any one else’s hill, any more than the pugnacious cock can fight the peculiarly irritating bird that he sees in the looking glass.
I have given only a qualitative description of Einstein’s law of gravitation; to give its exact quantitative formulation is impossible without more mathematics than I am permitting myself. The most interesting point about it is that it makes the law no longer the result of action at a distance: the sun exerts no force on the planets whatever. Just as geometry has become physics, so, in a sense, physics has become geometry. The law of gravitation has become the geometrical law that every body pursues the easiest course from place to place, but this course is affected by the hills and valleys that are encountered on the road.
CHAPTER IX:
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 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 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 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 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 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 certainly more nearly exact than Newton’s, though the inaccuracies in Newton’s were all exceedingly minute.