The field of an electron on Einstein’s theory of gravitation

Abstract

Einstein’s Generalised Theory of Relativity has accomplished notable results in the region of astronomical mechanics, but for the moment it seems difficult to go further in this direction until some further progress is made with the theory of the solution of the field equations. Practically all the consequences of the theory which have been established so far are obtained from the solution of the equations corresponding to a single isolated singularity. The exact solutions corresponding to two isolated singularities are urgently required for the further exploration of the theory. The approximate solutions which have been put forward by De Sitter, Dröste and Einstein, though valuable in default of exact solutions, are apt to be misleading and perhaps to exclude effects of far-reaching theoretical importance such as radiation. Meanwhile it is well to examine the consequences of the theory at the other extreme of the realm of physical science, in its relation to atomic phenomena. Here it seems no longer justifiable to consider gravitation and electricity separately. In these microscopic phenomena, where we are free from the effects of averaging, mass and charge seem to be inextricably connected. They exist in certain definite combinations, possibly in only two combinations as electrons and hydrogen nuclei. The gravitational field corresponding to such a charged particle has been investigated by Nordström by the application of the calculus of variations to the Hamiltonian function of the combined fields. Nordström’s work was not brought to our notice until after the present paper was written and we had obtained the same result by direct solution of the field equations. As the methods employed may perhaps be more familiar to English students of the subject it has been thought well to give a brief outline of this proof as an alternative to Nordström’s in the first section of this paper.