On electric phenomena in gravitational fields

Abstract

It is a consequence of general relativity that all electromagnetic and optical phenomena are influenced by a gravitational field. Indeed, the first prediction of relativity-theory, namely, the bending of light-rays when they pass near a massive body such as the sun, was a p articular application of this principle. Evidently, therefore, the classical electromagnetic theory must be rewritten in order to take account of the interaction between electromagnetism and gravitation; but beyond laying down general principles, comparatively little progress has been made hitherto in this task, the mathematical difficulties of solving definite electrical problems in a gravitational field being somewhat formidable. The subject is, however, of some interest to atomic physics; for if we assume that the atom has a massive nucleus with electrons in its immediate neighbourhood, the behaviour of such electrons (especially with regard to radiation) will be affected by the gravitational field of the nucleus. In the present paper two kinds of gravitational field are considered, namely, the field due to a single attracting centre ( i, e ., the field whose metric was discovered by Schwarzschild) and a limiting form of it. Within these gravita­tional fields we suppose electromagnetic fields to exist. Strictly speaking, the electromagnetic field has itself a gravitational effect, i.e. , it changes the metric everywhere; but this effect is in general; small, and we shall treat the ideal case in which it is ignored, so we shall suppose the metric to be simply that of the gravitational field originally postulated. The general equations of the electro­magnetic field are obtained, and particular solutions are found, which are the analogues of well-known particular solutions in the classical electromagnetic theory; notably the fields due to electrons at rest, electrostatic fields in general, and spherical electromagnetic waves. The results of the investigation are for the most part expressible only in terms of Bessel functions and certain new functions which are introduced; but in some interesting cases the electro­magnetic phenomena can be represented in term s of elementary functions, as, for instance, the electric field due to an electron in a quasi-uniform gravitational field (equations (15) and (19) below) and the spherical electromagnetic waves of short wave-length about a gravitating centre (equation (43) below).