Lorentzian manifolds with shearfree congruences and Kahler-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/{exptp} is a smooth manifold. If n = 2k, the quotient S = M/{exptp} 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 Kahler manifold of real dimension 2k - 2. Going backwards through this line of ideas, for any quantisable Kahler manifold with associated Sasaki manifold S, we give the local description of all Lorentzian metrics g on the total spaces M of A-bundles pi : M -> S, A = S-1, 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 pas 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 x 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. (C) 2021 Elsevier B.V. All rights reserved.