On Vacuum Space-Times Admitting a Null Killing Bivector

Abstract

Starting with a vacuum space-time (Rab = 0) which admits a Killing vector field K, a study is made of the subclass where the Killing bivector (KBV) Ka; b is null. Reference is made to an earlier paper [J. Math. Phys. 12, 1088 (1971)] by the author which established some of the general approach and formalism used here. All space-times with the property above turn out to be in the class of expansion-free radiation fields, which are necessarily algebraically special. Of these only Petrov types II, D, and N are allowed; furthermore, those of type N are the pp waves. A result obtained from applying this approach is that expansion-free radiation fields are the only vacuum space-times which admit a geodesic Killing vector field; that field is necessarily lightlike. Finally, since the spaces with symmetry studied by R. P. Kerr and the author [J. Math Phys. 11, 2807 (1970)] had nonzero expansion, the associated bivector to each of those symmetries must necessarily be nonnull.