A generalisation of Weyl's theory of the electromagnetic and gravitational fields

Abstract

1. In the non-Euclidean geometry of Riemann, the metric is defined by certain quantities, g μν , which are identified by Einstein with the potentials of the gravitational field. H. Weyl has shown that, on removing a rather artificial restriction in Riemann’s geometry, the expression for the metric includes also terms which are identified with the four potentials of the electromagnetic field. I believe that Weyl’s geometry, far-reaching though it is, yet suffers from an unnecessary and harmful restriction; and it is the object of this paper to develop a still more general theory. In passing beyond Euclidean geometry, gravitation makes its appearance; in passing beyond Riemannian geometry, electromagnetic force appears; what remains to be gained by further generalisation? Clearly, the non-Maxwellian binding-forces which hold together an electron. But the problem of the electron must be difficult, and I cannot say whether the present generalisation succeeds in providing the materials for its solution. The present paper does not seek these unknown laws, but aims at consolidating the known laws. I hope to show that, in freeing Weyl’s geometry from its limitation, the whole scheme becomes simplified, and new light is thrown on the origin of the fundamental laws of physics.