Einstein Spaces with Symmetry Groups

In this paper, we construct all possible groups of motion (symmetry groups) for empty Einstein spaces admitting a diverging, geodesic, and shear-free ray congruence. (Minkowski space is excluded throughout the discussion.) It is proved that any such Einstein space cannot admit a symmetry group with dimension greater than four. Although the field equations are not solved completely for spaces with groups of dimension one or two, a generalization of the Kerr spinning-mass solution is obtained from the 2-dimensional class. It is shown that all such spaces with 4-dimensional symmetry groups are well known: Schwarzschild, NUT (Newman, Unti, and Tamborino), and a particular hypersurface orthogonal Kerr-Schild metric. The only member of these spaces admitting a 3-dimensional symmetry group is a Petrov Type III hypersurface orthogonal metric.