Lorentzian manifolds with shearfree congruences and Kähler-Sasaki geometry

Abstract

We study Lorentzian manifolds (M,g) of dimension n≥4, equipped with a maximally twisting shearfree null vector field p, for which the leaf space S=M/{exp⁡tp} is a smooth manifold. If n=2k, the quotient S=M/{exp⁡tp} is naturally equipped with a subconformal structure of contact type and, in the most interesting cases, it is a regular Sasaki manifold projecting onto a quantisable Kähler manifold of real dimension 2k−2. Going backwards through this line of ideas, for any quantisable Kähler manifold with associated Sasaki manifold S, we give the local description of all Lorentzian metrics g on the total spaces M of A-bundles π:M→S, A=S1,R, such that the generator of the group action is a maximally twisting shearfree g-null vector field p. We also prove that on any such Lorentzian manifold (M,g) there exists a non-trivial generalised electromagnetic plane wave having p as propagating direction field, a result that can be considered as a generalisation of the classical 4-dimensional Robinson Theorem. We finally construct a 2-parametric family of Einstein metrics on a trivial bundle M=R×S for any prescribed value of the Einstein constant. If dim⁡M=4, the Ricci flat metrics obtained in this way are the well-known Taub-NUT metrics.