All the laws of dynamics have been put together into one principle, called “The Principle of Least Action.” This states that, in passing from one state to another, a body chooses a route involving less action than any slightly different route—again a law of cosmic laziness. The principle is subject to certain limitations, which have been pointed out by Eddington,[11] but it remains one of the most comprehensive ways of stating the purely formal part of mechanics. The fact that the quantum is a unit of action seems to show that action is also fundamental in the empirical structure of the world. But at present there is no bridge connecting the quantum with the theory of relativity.
CHAPTER XI:
IS THE UNIVERSE FINITE?
We have been dealing hitherto with matters that must be regarded as acquired scientific results—not that they will never be found to need improvement, but that further progress must be built upon them, as Einstein is built upon Newton. Science does not aim at establishing immutable truths and eternal dogmas: its aim is to approach truth by successive approximations, without claiming that at any stage final and complete accuracy has been achieved. There is a difference, however, between results which are pretty certainly in the line of advance, and speculations which may or may not prove to be well founded. Some very interesting speculations are connected with the theory of relativity, and we shall consider certain of them. But it must not be supposed that we are dealing with theories having the same solidity as those with which we have been concerned hitherto.
One of the most fascinating of the speculations to which I have been alluding is the suggestion that the universe may be of finite extent. Two somewhat different finite universes have been constructed, one by Einstein, the other by De Sitter. Before considering their differences, we will discuss what they have in common.
There are, to begin with, certain reasons for thinking that the total amount of matter in the universe is limited. If this were not the case, the gravitational effects of enormously distant matter would make the kind of world in which we live impossible. We must therefore suppose that there is some definite number of electrons and protons in the world: theoretically, a complete census would be possible. These are all contained within a certain finite region; whatever space lies outside that region is, so to speak, waste, like unfurnished rooms in a house too large for its inhabitants. This seems futile, but in former days no one knew of any alternative possibility. It was obviously impossible to conceive of an edge to space, and therefore, it was thought, space must be infinite.
Non-Euclidean geometry, however, showed other possibilities. The surface of a sphere has no boundary, yet it is not infinite. In traveling round the earth, we never reach “the edge of the world,” and yet the earth is not infinite. The surface of the earth is contained in three-dimensional space, but there is no reason in logic why three-dimensional space should not be constructed on an analogous plan. What we imagine to be straight lines going on for ever will then be like great circles on a sphere: they will ultimately return to their starting point. There will not be in the universe anything straighter than these great circles; the Euclidean straight line may remain as a beautiful dream, but not as a possibility in the actual world. In particular, light rays in empty space will travel in what are really great circles. If we could make measurements with sufficient accuracy, we should be able to infer this state of affairs even from a small part of space, because the sum of the angles of a triangle would always be greater than two right angles, and the excess would be proportional to the size of the triangle. The suggestion we have to consider is the suggestion that our universe may be spherical in this sense.
The reader must not confuse this suggestion with the non-Euclidean character of space upon which the new law of gravitation depends. The latter is concerned with small regions such as the solar system. The departures from flatness which it notices are like hills and valleys on the surface of the earth, local irregularities, not characteristics of the whole. We are now concerned with the possible curvature of the universe as a whole, not with the occasional ups and downs due to the sun and the stars. It is suggested that on the average, and in regions remote from matter, the universe is not quite flat, but has a slight curvature, analogous, in three dimensions, to the curvature of a sphere in two dimensions.
It is important to realize, in the first place, that there is not the slightest reason à priori why this should not be the case. People unaccustomed to non-Euclidean geometry may feel that, even if such a thing be logically possible, the world simply cannot be so odd as all that. We all have a tendency to think that the world must conform to our prejudices. The opposite view involves some effort of thought, and most people would die sooner than think—in fact, they do so. But the fact that a spherical universe seems odd to people who have been brought up on Euclidean prejudices is no evidence that it is impossible. There is no law of nature to the effect that what is taught at school must be true. We cannot therefore dismiss the hypothesis of a spherical universe as in any degree less worthy of examination than any other. We have to ask ourselves the same two questions as we should in any other case, namely: (1) Are the facts consistent with this hypothesis? (2) Is this hypothesis the only one with which the facts are consistent?
With regard to the first question, the answer is undoubtedly in the affirmative. All the known facts are perfectly consistent with the hypothesis of a spherical universe. A very slight modification of the law of gravitation—a modification suggested by Einstein himself—leads to a spherical space, without producing any measurable differences in a small region such as the solar system. The known stars are all within a certain distance from us. There is nothing whatever in the stellar universe as we know it to show that space must be infinite. There can therefore be no doubt whatever that, so far as our present knowledge goes, the hypothesis of a finite universe may be true.
But when we ask whether the hypothesis of a finite universe must be true, the answer is different. It is obvious, on general grounds, that we cannot, from what we know, draw conclusive inferences as to the totality of things. A very slight change in the Newtonian formula for gravitation would prevent masses beyond the limits of the visible universe from having appreciable effects if they existed, and would therefore destroy our reason for supposing that they do not exist. All arguments as to regions which are too distant to be observed depend upon extending to them the laws which hold in our part of the world, and upon assuming that there is not, in these laws, some inaccuracy which is inappreciable for observable distances, but fatal to inferences in which very much greater distances are involved. We cannot, therefore, say that the universe must be finite. We can say that it may be, and we can even say a little more than this. We can say that a finite universe fits in better with the laws that hold in the part we know, and that awkward adjustments of the laws have to be made in order to allow the universe to be infinite. From the point of view of choosing the best framework into which to fit what we know—best, I mean, from a logico-æsthetic point of view—there is no doubt that the hypothesis of a finite universe is preferable. This, I think, is the extent of what can be said in its favor.