Electromagnetism as an aspect of geometry?: Already unified field theory—The null field case

Abstract

The already unified field theory of Rainich-Misner-Wheeler is extended to include null electromagnetic fields. This generalization involves two difficulties: the imposition of new integrability conditions on the complexion in the nonnull regions, and the actual determination of the electromagnetic field in the null regions. These difficulties are surmounted: a prescription is given for determining in every case whether or not a given geometry (Riemannian manifold) arises from a solution of the Einstein-Maxwell equations. Only in a very special set of solutions containing null regions of four dimensions in space-time does the electromagnetic field not follow uniquely (up to a constant duality rotation) from the geometry. All such solutions are explicitly displayed. Hence, except in these latter very special cases, we find that the electromagnetic field—even in the null case—leaves such a distinctive imprint on the geometry in which it lies that the electromagnetic field follows from the geometry. This remarkable fact—the basis of already unified field theory—is discussed.