SU(2,1) generation of electrovacs from Minkowski space

Abstract

For every nonnull Killing vector K of any given electrovac, there exists a group of tranformations ℋK of the gravitational and electromagnetic potentials of Ernst. This is the group which is a nonlinear representation of SU(2,1) and was developed by Kinnersley on the basis of work by Ehlers, Harrison, and Geroch. For every K of Minkowski space (MS), we compute the set ℋK(MS) of all electrovacs derived from MS by noniterative application of HK; the results include appropriate null tetrads, the connection forms, the conform tensors, and (in the discussion) the group of all motions of every member of every ℋK(MS). Each conform tensor is type Npp (plane gravitational wave) or type D, and the principal null vector(s) are also eigenvectors of the Maxwell field. Except for those K which represent infinitesimal rotations about a timelike 2-surface of MS followed by null translations in that 2-surface, each K has a corresponding MS Killing vector L such that the G2 generated by K and L has nonnull surfaces of transitivity and is invertible. The discussion covers properties of the principal null rays and the Maxwell fields, Killing tensors of the results (one of the Npp families admits an irreducible Killing tensor of Segre characteristic [(11)(11)]), and the precise conditions under which a Killing vector of an electrovac is also an MS Killing vector. Also, some deductions are made concerning the Petrov class and principal null ray properties of the second generation electrovacs which would result from further applications of SU(2,1). Those points of MS which are possible singularities of electrovacs in ℋK(MS) are classified. The conditions under which an electrovac in ℋK(MS) has all of R4 (except for curvature singularities) as its domain are found; in particular, such an extension to R4 exists whenever the one-parameter group generated by K has no fixed points or whenever one restricts ℋK to the Ehlers or Harrison transformations.