Symmetries of the stationary Einstein–Maxwell field equations. I

Abstract

The Einstein equations for stationary axially symmetric gravitational fields are written in several extremely simple forms. Using a tensor generalization of the Ernst potential, we give forms that are manifestly covariant under (i) the external group G of coordinate transformations, (ii) the internal group H of Ehlers transformations and gage transformations, and (iii) the infinite parameter group K of Geroch which combines both. We then show how the same thing can be done to the Einstein–Maxwell equations. The enlarged internal group H′ now includes the Harrison transformations, and is isomorphic to SU(2,1). The enlarged group K′ contains even more parameters, and generates even more potentials and conservation laws.