Invariant Approach to a Space-Time Symmetry

A study is made of Killing vector fields in vacuum Einstein spaces with a restriction primarily to those fields whose associated bivector is nonnull. However, a well‐known theorem of Robinson is modified slightly to show that if such a space admits a null bivector associated with a Killing vector, the space must be algebraically special. Consequently, all algebraically general spaces admit a nonnull Killing bivector (KBV) if they admit a symmetry at all. Furthermore, it is shown that if a vacuum Einstein space admits a spacelike or timelike Killing vector field whose associated KBV is nonnull, the Killing trajectories are not geodesics. Computation of invariants from the curvature tensor and the KBV allows an approach which gives a general classification to such spaces which admit at least one hypersurface orthogonal Killing vector field. A few geometrical properties involving the principal null directions of the KBV are also derived for the hypersurface orthogonal cases. In addition, a topological result follows immediately from the behavior of the invariant $\mathfrak{J}.$