Still further important consequences follow from this matter-moulding hypothesis of Riemann. Prior to these views, the principle of sufficient reason appeared to imply that physical space would always turn out to be homogeneous—the same in all places. This does not necessarily mean Euclidean, for, as we know, Riemann’s and Lobatchewski’s geometries also correspond to homogeneous spaces; but all varying degrees of non-Euclideanism from place to place were thought to be excluded a priori. Of the vast realm of possible types of geometries or spaces discovered by Riemann, only the homogeneous types survived; a situation which Weyl finds appropriately expressed by the classical line: “Parturiunt montes, nascetur ridiculus mus.”
But with the new views advocated by Riemann the situation changes entirely; for now the texture, structure or geometry of space is defined by the metrical field, itself produced by the distribution of matter. Any non-homogeneous distribution of matter would then entail a variable structure or geometry for space from place to place.
Now the question arises, What is the nature of this metrical field? Is it merely a name we are giving to the structure of space caused by matter? This view appears impossible, for space of itself, being a mere void, is not amenable to structure. Is the metrical field a direct emanation from matter, a rarefied form of matter? Or again is it a reality of a category differing from matter (call it the ether), which in the absence of matter would be amorphous, and which could only be forced into a structure by the influences due to matter and transmitted from place to place through the ethereal substance? In this case we should again be led to the view that what we commonly call the structure or geometry of real space reduces to the structure of the matter-moulded ether-filling space.
Riemann’s exceedingly speculative ideas on the subject of the metrical field were practically ignored in his day, save by the English mathematician Clifford, who translated Riemann’s works, prefacing them to his own discovery of the non-Euclidean Clifford space. Clifford realised the potential importance of the new ideas and suggested that matter itself might be accounted for in terms of these local variations of the non-Euclideanism of space, thus inverting in a certain sense Riemann’s ideas. But in Clifford’s day this belief was mathematically untenable. Furthermore, the physical exploration of space, even in the interior of liquids, seemed to yield unvarying Euclideanism. And here the remarkable irony of the whole situation must be noted. Although experimenters had utilised the most refined apparatus for detecting a possible non-Euclideanism of space and had failed in their efforts, it was reserved for the theoretical investigator Einstein, by a stupendous effort of rational thought, based on a few flimsy empirical clues, to unravel the mystery and to lead Riemann’s ideas to victory.[16] Nor were Clifford’s hopes disappointed, for the varying non-Euclideanism of the continuum was to reveal the mysterious secret of gravitation, and perhaps also of matter, motion and electricity.
Before solving the problem, however, Einstein had been led to recognise that space of itself was not fundamental. The fundamental continuum whose non-Euclideanism was to be investigated was therefore not one of space but one of Space-Time, a four-dimensional amalgamation of space and time possessing a four-dimensional metrical field governed by the matter distribution. Einstein accordingly applied Riemann’s ideas to space-time instead of to space, and attempted to explore the geometry of space-time by a purely rational co-ordination of known empirical facts. He discovered that the moment we substitute space-time for space (and not otherwise), and assume that free bodies and rays of light follow geodesics no longer in space but in space-time, the long-sought-for local variations in geometry become apparent. They are all around us, in our immediate vicinity; and yet we had never realised it. We had called their effects gravitational effects, ascribing them to forces foreign to the geometry of the extension, and never suspecting that they were the result of those very local variations in the geometry for which our search had ever been vain even though we had extended our observations to the depths of the universe. Indeed, it may be said that the theory of relativity is the theory of the space-time metrical field.
While we are on the subject, we may mention that this mysterious metrical field which moulds both space and time appears to be conditioned entirely by the matter of the universe. Such at least are the conclusions which the existence of Einstein’s cylindrical universe would suggest. The problem is, however, still extremely obscure. It is still possible to believe with Eddington and de Sitter that the metrical field or space-time ether-structure might subsist in the absence of matter, contrary to the views of Riemann.
Finally let us consider how these new views will affect the problem of absolute shape and size. We may say that space-time possesses a definite geometry but that this geometry is subject to local variations both in space and in time, as the masses of the universe modify their positions. Yet, as we have mentioned on several occasions, a geometry, even when fully determined and unchanging, does not imply absolute shape and size. It merely regulates the relationships or defines the laws of the mutual dispositions of bodies, and these laws may remain unmodified even though the absolute shape and size of the bodies vary. Restricting ourselves to the problem of absolute size, we shall find that with the finite universe the metrical field yields a universal standard of length given by the radius of the finite universe. As referred to this natural gauge, an object presents a definite size, but obviously there is nothing absolute about the gauge itself. Not only will its magnitude be governed by the total amount of matter in the universe, an amount determined presumably by accident, but furthermore there would be no means of determining the magnitude of this gauge otherwise than by adopting some other gauge, and so on ad infinitum.
CHAPTER V
AN ALTERNATIVE VIEW OF THE NON-EUCLIDEAN GEOMETRIES
THE type of geometry we obtain is dependent, as we know, on our definition of congruence. If we define as congruent displacements the displacements of those bodies which in ordinary life we consider rigid and undeformable, we obtain Euclidean geometry. If, on the other hand, we define as congruent the displacements of those bodies which in ordinary life we should regard as of changeable shape, we generally obtain some non-Euclidean type of geometry.
In order to simplify what is to follow, we shall first consider the particular case of two-dimensional geometry. Consider, then, a plane surface. If on this plane surface we effect measurements with Euclidean rods, we obtain Euclidean results; if we employ rods which according to the Euclidean point of view squirm in an appropriate way when displaced, we obtain the geometry of Riemann or of Lobatchewski, as the case may be. It is to be noted that the plane is the same in all three cases, and yet the geometry we obtain on its surface may be Euclidean or non-Euclidean. It is not the plane itself but our methods of measurement conducted thereon which have changed.