All attempts to overcome this duality were unsuccessful so long as we confined our speculations to the space-time of Einstein’s theory. But there always existed the possibility that space-time was of a less simple category than had been assumed by Einstein. If, then, we could conceive of a generalised type of space-time possessing a more complicated metrical field, it might be possible to identify the fundamental
’s of electromagnetics with certain particularities of structure of this generalised space-time and of its metrical field. Weyl’s theory is an attempt in this direction; and though it is still in a highly speculative stage, everything seems to suggest that Weyl is on the right track. Einstein himself has accepted a modified form of Weyl’s theory to the extent of contributing to its advancement.
CHAPTER XXXVII
THE THEORIES OF WEYL AND EDDINGTON
THE general theory of relativity reduces to an application of Riemann’s purely mathematical discoveries, to the real world of space-time in which we live. A further extension of the theory became possible when Weyl had succeeded in carrying the geometrical study of manifolds beyond the point where Riemann and his successors had left it. Weyl describes the situation in the following words:
“Inspired by the weighty inferences of Einstein’s theory to examine the mathematical foundations anew, the present writer made the discovery that Riemann’s geometry goes only halfway towards attaining the ideal of a pure infinitesimal geometry. It still remains to eradicate the last element of geometry ‘at a distance,’ a remnant of its Euclidean past.”[126]
In order to understand Weyl’s theory, which may be regarded as a generalisation of Einstein’s, it will be necessary to enquire into the nature of Weyl’s geometrical contributions. Let us recall that when discussing the geometry of space, we said that the geometry we attributed to conceptual space was defined by the numerical results of measurement which were obtained when we had defined the type of conceptual measuring rods which we intended to employ. The behaviour of these measuring rods was arbitrary to a great extent, and according to the behaviour we attributed to them, the space was Euclidean, Riemannian or Lobatchewskian.
However, certain limitations were imposed by Riemann. Riemann assumed that two unit rods which coincided at a point
would continue to coincide when displaced to another point