Einstein and the universe: A popular exposition of the famous theory

by Charles Nordmann

There are many other reasons why the universe, in consequence of gravitation, does not conform to the laws of Euclid’s geometry.

For instance, in the Euclidean geometry the extent of the circumference has a well-known proportion to its diameter, and this is indicated by the Greek letter π. This proportion, expressing how many times the diameter is contained in the circumference, is equal to 3·14159265 ... etc., but I pass over the rest, as π has an infinite number of decimals. We then ask: In practice is the proportion of circumferences to their diameters really equal to the classic value of π? For instance, is this precisely the proportion of the earth’s circumference to its diameter?[11] Einstein says that it is not, and he gives us the following proof. Imagine two very clever and quick and wizard-like surveyors setting out to measure the circumference and diameter of the earth at the Equator. They both use the same scales of measurement. They begin measuring at the same moment, and they start from the same point on the Equator. But one goes westward and the other eastward, and their speeds are equal, and such that the one who goes westward keeps up with the earth’s rotation, and thus sees the sun all day long stationary at the same height above the horizon. In music-halls, for instance, one sometimes sees an acrobat walking on a rolling ball and keeping to the top of the ball, because the pace of his steps is exactly equal and contrary to the displacement of the spherical surface.

A stationary observer in space—on the sun, let us say—would thus see our surveyor who is going westward, stationary right opposite to him. On the other hand, the surveyor who goes eastward will seem to him to go round the earth, and twice as quickly as if he had remained at the starting-point.

When each of our surveyors, both going at the same speed, has finished his task of measuring the round of the earth, will they both have the same result? Evidently not. As the super-observer in the sun will see, the yard of the surveyor who travels eastward is shortened by velocity in virtue of the Fitzgerald-Lorentz contraction. On the other hand, the yard of the surveyor who travels westward does not experience this contraction, as the super-observer on the sun, in reference to whom he remains stationary, would see.

Consequently the two surveyors reach different figures for the earth’s circumference, the one who travels westward finding a result a few yards less than that of the other. Yet it is obvious that when they proceed to measure the earth’s diameter, travelling at the same speed, the two observers will reach the same figure for it.

Hence the π which expresses the proportion of the earth’s circumference to its diameter on the ground of actual measurement differs according as the measurer travels in the direction of the earth’s rotation or in the opposite direction. Therefore, as the real values of π are different, they cannot be the unique and quite definite figure of classical geometry. Therefore the real universe does not conform to this geometry.

These differences, in the illustration we have given, are due to the earth’s rotation. From the standpoint of gravitation the earth’s rotation has centrifugal effects which modify the centripetal influence of weight. We have seen, moreover, that for the surveyor whose speed equals that of the earth’s rotation the value of π is smaller than for the observer whose speed seems to be double that of the rotation. Thus the effects of weight being the reverse of those of rotation, or of centrifugal force, it follows (it would be just as easy to prove this as the preceding) that the effect of weight is to give π something less than its classical value.

In a word, in the universe real circumferences traced upon gravitating masses, such as stars, are, in proportion to their diameters, less than they are in the Euclidean geometry.

The difference is generally very slight, it is true. But there is a difference. If we put a mass of a thousand kilogrammes in the centre of a circle that is ten metres in diameter, the figure π will differ in reality from its Euclidean value by less than one-thousand-million-billionth.

In the neighbourhood of such formidable masses of matter as the stars are, the difference may be far greater, as we shall see. This is the origin of the divergences between Newton’s law of gravitation and that of Einstein: divergences which observation has settled in favour of the latter. But we will not anticipate.