On finding hypersurface orthogonal Killing fields

Abstract

The conditions for a space-time to admit a hypersurface-orthogonal Killing vector field are studied. Tests are derived first for space-times that admit two commuting Killing vectors, both for the timelike-spacelike (TS) case and the spacelike-spacelike case (SS); these give the condition for linear combinations of the two known Killing vectors to be hypersurface orthogonal. Illustrations are given in some stationary axisymmetric (TS) metrics and cylindrically symmetric (SS) metrics. In the general case, conditions for static symmetry are already known. We give the similar conditions for a spacelike hypersurface-orthogonal Killing field.