’s, we are led to Weyl’s manifold; by imposing still others, we are led to the classical manifolds, of which Einstein’s space-time was a particular instance. As a matter of fact, even in Eddington’s manifold, certain restrictions of symmetry defining what is known as the affine condition are still imposed upon the
’s; were it not for these restrictions, there would have been 64 of these
’s. So far as is known to the writer, the highly generalised manifolds which would result have not been studied; but it is interesting to note that they can be conceived of, and no one can foresee whether their introduction may not prove necessary eventually. Since Eddington’s geometry is a generalisation of Weyl’s, we may anticipate certain novel features in it. Thus, in Weyl’s geometry, we have seen that the length of a rod would vary were it to be displaced from one point to another. With Eddington’s generalisation, we must assume that a change of length would also arise if the rod were rotated round one of its points, hence if its orientation were modified, even though the rod were not displaced as a whole.
And here a few words may be said with respect to the purpose of these successive generalisations. The ultimate goal is one of unification. We wish to succeed in representing the entire physical universe in terms of the relationships of structure of the fundamental space-time continuum. Einstein’s space-time enabled us to account for gravitation in this way, but the electromagnetic field appeared as a foreign invasion. None of the known elements of structure seemed capable of accounting for it. It seemed as though the world of space-time, gravitating masses and energy might just as well have existed in the absence of this electrical intrusion. Now, we have every reason to believe that such cannot be the case, for since matter is built up of protons and electrons, the annihilation of electricity would entail that of matter, gravitation and energy, and possibly also of space and time. In order to overcome this difficulty, it appeared necessary to generalise our conception of space-time, obtaining thereby a continuum presenting additional elements of structure which might be identified with electric and magnetic forces. And so we were led to Weyl’s generalisation, which permits the electromagnetic field to enter into the general synthesis, no longer as a foreign adjunct, but as a constituent element of space-time structure. But, besides the field, there is the electron, and the field alone does not seem to afford us the possibility of building up the electron; we cannot account for atomicity or for the quanta of action. Therein resides the problem of matter, and it remains to-day an outstanding challenge to science. Neither can we understand the significance of the two different kinds of electricity, the positive and negative. So long as all these mysteries remain unsolved, it is only natural that we should seek new elements of structure by proceeding with further generalisations. Eddington’s theory was conceived of with this object in view.
There exist certain interesting features about Eddington’s theory which we may mention briefly. Inasmuch as the intrinsic magnitudes of the universe must be in-tensors and in-invariants, instead of tensors and invariants, the
’s lose their position of pre-eminence; for the