Complex potential equations

The results of a previous paper are applied to a study of the class of Kerr-Schild metrics in general relativity. These metrics g have a form which involves the flat (Minkowski) space-time metric. It is already known that for a certain restricted subclass of these metrics, the vacuum Einstein field equations can be written in the form: (inverted delta γ)2 = γ4; (inverted delta)2γ = 0 where γ is a complex potential. Using the methods developed in a previous paper, such space-times are characterized by means of a special family of complex surfaces in three-dimensional Euclidean space, and the exact solutions for the metric g are consequently recovered. It is also shown that the field equations for a much wider class of Kerr-Schild metrics can be expressed in terms of a potential formalism, not only in the vacuum case, but also for many electrovacuum solutions. (author)