Isometry Groups with Surface-Orthogonal Trajectories

Abstract

It is shown that the trajectories of an isometry group admit orthogonal surfaces if the sub-group of stability leaves no vector in the tangent space of the trajectories fixed. A necessary and sufficient condition is given that the trajectories of an Abelian group admit orthogonal surfaces. In spacetimes which admit an Abelian G2 of isometries, the trajectories admit orthogonal 2-surfaces if a timelike congruence exists with the following properties: the curves lie in the trajectories and are invariant under G2; ωα and üα are linearly independent and orthogonal to the trajectories.