On Weyl alignment preserving Kaluza-Klein reduction of vacuum

Abstract

We study null alignment properties of Weyl tensors related via Kaluza$\unicode{x2013}$Klein reduction of vacuum spacetimes by one spatial Killing direction. Kaluza$\unicode{x2013}$Klein reduction is a method that relates spacetimes of different dimensionality. Weyl tensor null alignment is used in a recently proposed generalization of the Petrov algebraic classification of spacetimes to higher dimensions. Concentrating on the case where the two considered null directions are parallel in a gauge where they are perpendicular to the Maxwell potential, we express the relations between Riemann tensor null frame components of the original and reduced spacetime; we do the same for the Weyl tensors and also for optical matrices and non-geodeticities. Based on this, we point out basic consequences regarding reduction of Kundt spacetimes, and of spacetimes admitting a geodetic null direction. Finally, we work out the necessary and sufficient conditions for a Kaluza$\unicode{x2013}$Klein lift to preserve Weyl alignment type with respect to the related null directions. In the cases of types III and N with non-vanishing Maxwell field, where both spacetimes turn out to be Kundt, we show explicit solutions for the scalar potential in six dimensions and greater, and discuss some qualitative differences from the four-dimensional case.