The gravitational field of a radiating electromagnetic dipole

Abstract

The principle purpose of this paper is to analyze in the context of general relativity the effects of electromagnetic radiation on a gravitational field. The basic outline is that we start from flat-space with a time-dependent electric and magnetic dipole solution of Maxwell's equations. This field, treated as first order, then acts as the source for a gravitational perturbation. First, from the flat-space version of the Bianchi identities, with the Maxwell stress tensor as the (second-order) source, the Weyl tensor is found as a field on the Minkowski background. From this Weyl tensor, we go on and find the spin coefficients and the full metric in this second-order approximation. The physical interpretation and meaning of many of the derived relations are discussed. In particular, we can identify the Bondi energy–momentum—given in terms of the Maxwell field—and their evolution. Of particular attractiveness is the observation of an angular momentum conservation law that comes directly from the asymptotic terms in the Bianchi identities. This law contains a flux term that is a confirmation of a classical E&M result but obtained directly from GR.