Let us first consider the two-dimensional analogy. If, on the two-dimensional surface of a sphere, we perform measurements with tiny rigid Euclidean rods, we obtain the numerical results, not of Euclid’s but of Riemann’s two-dimensional geometry. Accordingly, we say that the geometry of a spherical surface is Riemannian, and that a Riemann space is curved and finite. But the word “curved” immediately evokes the idea of a third dimension along which the surface or space is curved. In the present two-dimensional case we have no difficulty in visualising this third dimension, because the space of our actual experience happens to be three-dimensional. And so we note that the spherical surface possesses an outside and an inside, and that if we consider solely the points on the surface, we are excluding from our investigation an indefinite number of points of space both inside and outside the sphere. This procedure is unfortunate; for when we generalise our previous two-dimensional illustration to a three-dimensional spherical space, we are inclined to reason by analogy and to conceive, or endeavour to conceive, of a four-dimensional Euclidean space in which our three-dimensional Euclidean universe must be embedded. The result is that we think of points of space outside our universe, as though our universe did not embody the whole of space.
In the present case this attitude would be entirely erroneous, but the fault lies with the peculiar type of two-dimensional analogy we chose to present. In order to eliminate this error, let us proceed as explained in a previous chapter, and let us suppose that the sphere rests at its south pole on an infinite Euclidean plane, while a point of light is situated at its north pole. If the sphere is translucent, our tiny rods displaced on the spherical surface will cast their shadows on the plane; and, of course, if we effect measurements on the plane by means of these shadows of our rods, we shall obtain exactly the same metrical relationships on the plane as we obtained formerly on the spherical surface. In particular, if a hundred tiny rods lying along a meridian span the distance from the south to the north pole on our sphere, the hundred shadows of these tiny rods will extend across the plane to infinity. Likewise, just as a finite number of postage stamps would cover the sphere’s surface, so now a finite number of shadows, the shadows of these stamps, would cover the entire infinite extent of the plane. Thus we see that when measured with our shadows the plane will be finite and its geometry Riemannian, whereas it would have been infinite and Euclidean had we effected our measurements by displacing ordinary foot-rules over its surface.
The advantage of presenting the problem in this alternative manner is that in the present case the introduction of a third dimension is no longer necessary in order to explain curvature, or non-Euclideanism. We realise that regardless of whether the space of the plane is to be considered as finite and Riemannian or as infinite and Euclidean, exactly the same points of space are included in either case. This two-dimensional illustration can, of course, be extended to three dimensions. By following the same line of reasoning, we will say that the universe of three-dimensional space is infinite, if by placing side by side, in rows, columns and superimposed layers, an indefinite number of congruent cubes, we find it possible to go on piling them up for ever and ever. On the other hand, we will say that the universe of space is finite if this process cannot be continued indefinitely, that is, if after a finite number of cubes have been superimposed, further progress is impossible. This last case is, of course, similar to the one discussed previously when we found it impossible to place an indefinite number of shadows of our postage stamps over the plane.
From these examples it is apparent that the finiteness or infiniteness of our space is contingent on the geometrical behaviour of the cubes which we regard as congruent or equal. From a purely mathematical standpoint, therefore, the problem of the finiteness of the universe can possess no absolute significance, since congruence, or equality of size, is incapable of being defined in any absolute way. Yet from the standpoint of the physicist the problem assumes a very definite significance, for whereas one unique rational definition of congruence is impossible, an empirical definition is easily obtained. By congruent cubes the physicist means those which visually and tactually will appear the same in whatever region of space they may be situated, provided the observer accompanies the cubes in their displacements. He means cubes carved out of the substance of those bodies which men have agreed, from time immemorial, to regard as invariable solids, and which are illustrated by the bodies around us, when maintained under conditions of constant temperature and pressure.
It is as referred to such bodies, therefore, or, again, to the courses of light rays, that the universe, according to Einstein, would manifest itself as finite. This is but another way of saying that our bodies would behave like Riemannian solids (over vast areas at least) instead of like Euclidean ones, as was formerly thought to be the case.
Summarising, we see that this same so-called finite universe would manifest itself as infinite were our rods to behave Euclideanly, whence we may conclude that the universe, whether finite or infinite, may contain exactly the same regions of space, just as our plane contained the same points whether measured with yardsticks or shadows. To assume, therefore, that certain regions of space must necessarily be excluded owing to the finiteness of the universe, would be to take a stand in no wise demanded by the actual conditions of the problem.
CHAPTER XXXV
THE IMPORTANCE OF SPACE-TIME, AND THE PRINCIPLE OF ACTION
IN the course of this book we have seen how a number of difficulties were overcome as soon as we recognised the existence of four-dimensional space-time in lieu of the world of separate space and time of classical science. There are still other aspects of the problem, however, and we shall now mention them briefly.
In the first place, all vectors such as velocities and forces, and all tensors such as Maxwell’s stresses of the electromagnetic field, erstwhile expressed in three-dimensional space, must now be extended and supplemented with additional components so as to yield vectors and tensors in four-dimensional space-time. These new components do not define new physical magnitudes unknown to classical science; in the general case they give us well-known magnitudes, such as work, power, and so on. As a case in point, let us consider electric and magnetic intensities. In classical science these were entirely distinct magnitudes. But when we represent these intensities in terms of space-time, we find that they are given by the various components of one same space-time tensor
