Now this attitude of regarding the behaviour of rigid rods and light rays as symptoms, and not as causes, must not obscure the fact that rods, light rays and physical phenomena generally are an essential requirement for the discovery of the pre-existing structure of space. There is no sense, therefore, in attempting to determine this structure a priori on the plea that physical measurements, however perfect they may be, can never transcend a certain approximation, and are therefore of no value in permitting us to discover the precise structure of space. We must never forget that we are dealing here with a problem of physics, not with one of pure mathematics. For instance, in Euclidean geometry the mathematician does not have to appeal to measurements with rigid rods in order to determine the value of
, since in virtue of our basic Euclidean postulates and the rules of logical reasoning,
can be determined as accurately as we wish by purely mathematical means. But in the case of the geometry of physical space we are completely at sea. A priori, this geometry might be one of a variety of different types. All we might do would be to assert that the principle of sufficient reason compelled us to regard physical space as homogeneous and isotropic, hence as reducing to a constant-curvature geometry, whether Euclidean or not. Even this belief would be open to question until we knew more about the origin of this structure. At any rate, on a priori grounds the precise geometry of space can never be ascertained. Whether we like it or not, we are compelled to resort to rods, light rays and physical methods of exploration; and whatever geometry we obtain in this way will ipso facto become that of real physical space, to the order of precision of our measurements.
Up to this point we have limited ourselves to discussing the first step in our study of physical space. The structure is viewed as pre-existing; and our rods and light rays, by adjusting themselves to this structure, permit us to objectivise it, as it were. In much the same way a coloured liquid poured into an ideally transparent glass (so transparent as to be invisible) permits us to ascertain the inner shape of the glass. But we now come to the second step. What is the cause of this structure? Is it posited by the Creator, as an intrinsic property of space, or is it the mere manifestation of something more fundamental?
Riemann refused to believe that a structure of the void could exist of itself. He recognised, as before, that our rods would adjust themselves and would thus reveal the pre-existing structure; but this structure in turn would have to be created and conditioned by something else foreign to the void. For this “something else” Riemann appealed to the forces generated by the enormous mass of the totality of the star distribution. It would follow that were there no stars and no matter in the universe aside from our exploration rods, no geometry would be revealed. The rods would not know how to act; light rays would not know where to go; physical space would be unthinkable. With this bold idea of Riemann’s, the homogeneity and isotropy of space (whether Euclidean or non-Euclidean), formerly thought to be imposed by the principle of sufficient reason, was no longer inevitable; for star-matter, the generating cause of this structure, was not distributed equally throughout the universe. Riemann, therefore, considered that a varying non-Euclideanism might be present in space. Rods and light rays would, of course, adjust themselves to this heterogeneous structure, yielding different numerical results from place to place throughout space.
All these theories antedate Einstein’s discoveries by many years, but with the advent of the theory of relativity the entire problem presented itself afresh, with this difference, however—that the fundamental continuum to be explored was four-dimensional space-time and no longer three-dimensional space. The introduction of space-time necessitated an appeal to chronometers or vibrating atoms as well as to material rods, although the progress of light rays through space could also serve to measure duration (owing to the postulate of the invariant velocity of light according to which a ray of light would describe equal spaces in equal times). At all events, the time determinations would have to yield the same results whether computed by light rays or by the vibrations of atoms, since otherwise we should have to assume that the structure of space-time itself could not be the governing and moulding influence regulating the behaviour of material bodies and of all processes of change. In this eventuality a belief in the structure of space or of space-time would be difficult to accept. This structure of space-time has received various names. Weyl calls it the affine-relationship, the guiding field or the metrical field. Einstein refers to it as the ether of space-time or the gravitational ether. In order to avoid confusion with the stagnant ether of classical science, from which it differs in so many respects, we shall refer to it as the “metrical field.” Mathematically, as we know, it is represented by the
-distribution.