COSMOLOGY According to the simple views of space and time which prevailed before the advent of relativity, time had no properties except duration, and space had no properties except extension. Every instant of time was, in its essence, precisely similar to every other instant of time, so that time must go on forever unless it was stopped by something different from time ; it could not end of itself, for this would require the last instant of time to have different properties from all other instants of time. Similarly all elements of space were supposed to be intrinsically of the same nature and thus required that space should extend to infinity in all directions unless it were abruptly ended by something that was not space.
Einstein's theory of relativity changed all this. According to this theory the space-time in the neighbourhood of a gravitating mass is essentially different from space-time which is far away from all masses; the former is curved, while the latter, at least if all gravitating masses are infinitely far away, is not. The cur vature of the space-time in the neighbourhood of the sun causes the planets to describe ellipses, whereas if all masses were in finitely removed, they would describe straight lines. When it is once admitted that all elements of space-time are not intrinsically similar, no reason remains either why space should be of infinite extent or why time should be of infinite duration.
On Einstein's theory, the curvature of space-time at any point of the continuum is proportional to the value of the gravitational potential at the point. To calculate this we must take the different masses of the universe in turn, divide each by its distance from the point in question, and add together the quotients so obtained.
When this is done it is found that, unless there happens to be a large mass very near to the point, distant stars and masses contrib ute far more to the gravitational potential than near ones. The distant masses have to be divided by a greater distance, but are so much more numerous than the near masses that their greater numbers more than compensate for their distance. Accordingly, except in the near neighbourhood of massive bodies, the curvature of the continuum is determined by the distant masses of the uni verse, and there will be a regular and continuous field of cur vature, disturbed only in the neighbourhood of massive bodies, by the additional curvature impressed by these bodies. An analogy
in two dimensions is provided by the spherical surface of the earth, which we may imagine to be deformed here and there by mole-hills or other small undulations; these correspond to the local curvatures which are impressed on the general curvature of space by the proximity of massive bodies.
When our ancestors found that the earth's surface was curved it was natural for them to suppose that its curvature bent it round into a closed surface, so that a traveller continually follow ing a straight path could come back to his starting point. The earth might have been shaped like a saucer or a basin, but it was much simpler and more natural to suppose it to be spherical. In the same way when the space-time continuum is found to be curved with a fairly uniform curvature, it is natural to con jecture that this curvature of itself closes the continuum and so reduces it to finite dimensions. There still remain innumerable different ways in which this can be effected.
The general curvature of the continuum at any point is de termined by the distant masses of the universe, so that if space is closed, the volume of this space and the distribution of masses inside it must determine its curvature at every point. The curva ture in turn fixes the volume and this will not be the volume from which we started unless the masses are of exactly the right amount.
Thus we see that a universe of given size requires masses of a certain amount to maintain it at this size, and, conversely, given masses require a universe of a certain definite size in which to exist. In the simplest case in which the matter is supposed to be spread uniformly throughout the universe, the total mass of the universe is proportional to its radius, or, what is the same thing, both the mass and the radius of the universe vary inversely as the square root of the density of the matter. Thus a great amount of matter requires a large universe in which to exist, and exists in this universe in a highly attenuated state, whereas a small amount of matter must have a small universe in which it exists in a state of high density.