alone. The difficulty consists in understanding how it is that the electron does not explode under the mutual repulsion of its negatively charged parts. Some kind of counterbalancing pressure, the Poincaré pressure, seems to be necessary, and it appears to be impossible to account for this mysterious pressure in terms of the electromagnetic Maxwellian stresses
alone. Following Poincaré, Mie attacked the problem. His procedure was to introduce additional electromagnetic-field magnitudes which would be effective in the interior of the electron, but whose action in outside space would be negligible. By this means, he succeeded in building up matter out of purely electromagnetic quantities. Einstein comments on Mie’s theory in the following words:
“In spite of the beauty of the formal structure of this theory, as erected by Mie, Hilbert and Weyl, its physical results have hitherto been unsatisfactory. On the one hand, the multiplicity of possibilities is discouraging, and, on the other hand, those additional terms have not as yet allowed themselves to be framed in such a simple form that the solution could be satisfactory.”
Following this attempt, Einstein proved that it was possible to obviate the introduction of Mie’s supplementary electromagnetic terms provided the space of the universe were assumed to possess a slight residual curvature, such as would exist in the cylindrical universe. It would then be this residual universal space-curvature which would act as a negative pressure preventing the electron from exploding. If we note that the universal curvature concerns the metrical field, hence the field of gravitation, we see that the energy of the electron must be composed of an electromagnetic and of a gravitational part. In Einstein’s own words: “Of the energy constituting matter, three-quarters is to be ascribed to the electromagnetic field, and one-quarter to the gravitational field.”
Einstein considers that these facts constitute a powerful argument in favour of the finite universe. At all events, it must be conceded that the hypothesis of the finite universe appears to impose itself in a number of different ways. Not only astronomical considerations bearing on the low velocities of the stars, not only mathematical reasons connected with the boundary conditions, not only philosophical urges concerned with the relativity of rotation and the relativity of inertia, but also atomic or electronic phenomena, drive us towards the same solution.
Let us now pass to a study of the electromagnetic tensor itself. We can see that even if the preceding attempts had proved successful, there would still remain a duality between gravitation and matter or electricity, or, we might say, between the metrical field of space-time and the tensor
. This latter tensor would still appear as an irreducible foreign entity present in space-time, marring the unity of nature. It was the aim of Weyl’s theory to rid science of this duality.