The A B C of Relativity

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 [Pg 164] 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 [Pg 165] 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 [Pg 166] 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 [Pg 167] 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 [Pg 168] 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 [Pg 169] doubt that the hypothesis of a finite universe is preferable. This, I think, is the extent of what can be said in its favor. Let us now see what the two finite universes are like. The difference between them is that in Einstein’s world it is only space that is queer, whereas in De Sitter’s time is queer too. Consequently Einstein’s world is less puzzling, and we will describe it first. In Einstein’s world, light travels round the whole universe in a time which is supposed to be something like a thousand million years. The odd thing is that all the rays of light which start (say) from the sun will meet again, after their enormous journey, in the place where the sun was when they started. The case is exactly analogous to that of a number of travelers who set out from London to go round the world in great circles, all traveling at the same rate in different aeroplanes. One starts due north, passes the North Pole, then the South Pole, and finally comes home. Another starts due south, reaches the South Pole first and then the North Pole. Another starts westward, but he must not continue to travel due west, because then he would not be traveling on [Pg 170] a great circle. Another starts eastward, and so on. They all meet in the antipodes of London, and then they all meet again in London. Now if instead of aeronauts going round the earth you take rays of light going round the universe, the same sort of thing happens: they all meet first at the antipodes of their starting point, and then meet again at their starting point. That means to say that a person who is near the antipodes of the place where the sun was about five hundred million years ago will see what is apparently a body as bright as the sun then was (except for the small amount of light that has been stopped on the way by opaque bodies), and having the same shape and size. And a person who is near where the sun was a thousand million years ago will see what is apparently a body just like what the sun was a thousand million years ago. And the same applies to the antipodes of the sun fifteen hundred million years ago, and to the place of the sun two thousand million years ago, and so on. This series only ends when it carries us back to a time before the sun existed. But all these suns are only ghosts; that is to say, you could pass through them without experiencing resistance, and they do not exert [Pg 171] gravitation. They are, in fact, like images in a mirror: they exist only for the sense of sight, not for any other sense. It is rather disturbing to reflect that, if this theory is true, any number of the objects we see in the heavens may be merely ghosts. They are like ghosts in their habit of revisiting the scenes of their past life. Suppose a star had exploded at a certain place, as stars sometimes will. Every thousand million years its ghost would return to the scene of the disaster and explode again in the same place. There is, however, considerable doubt whether rays of light could perform the journey with sufficient accuracy to produce a clear image. Some would be stopped by matter on the way, some would be turned out of the straight course by passing near heavy bodies, as in the eclipse observations described in Chapter IX, and for one reason or another their return would not be punctual and exact. There are various reasons for doubting whether Einstein’s universe can be quite right.[12] Some of these are rather complicated. But there is one objection which is easily appreciated: in Einstein’s theory, [Pg 172] absolute space and time re-enter by another door. The ghostly sun is formed in the “place” where it was a thousand million years ago. Both the “place” and the period of time are in a sense absolute. We saw as early as Chapter I that “place” is a vague and popular notion, incapable of scientific precision. It seems hardly worth while to go through such a vast intellectual labor if the errors we set out to correct are to reappear at the end. De Sitter’s world is even odder than Einstein’s, because time goes mad as well as space. I despair of explaining, in non-mathematical language, the particular form of lunacy with which time is afflicted, but some of its manifestations can be described. An observer in this world, if he observes a number of clocks, each of which is perfectly accurate from its own point of