Lorentzian manifolds with shearfree congruences and Kähler-Sasaki geometry

Abstract

<p>We study Lorentzian manifolds (<i>M, g</i>) of dimension <i>n</i> ≥4, equipped with a maximally twisting shearfree null vector field p, for which the leaf space <i>S</i> = <i>M</i>/{exp<i>t</i>p}is a smooth manifold. If <i>n</i> =2<i>k</i>, the quotient <i>S</i> = <i>M</i>/{exp<i>t</i>p} 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 2<i>k</i> - 2. Going backwards through this line of ideas, for any quantisable Kähler manifold with associated Sasaki manifold <i>S</i>, we give the local description of all Lorentzian metrics <i>g</i> on the total spaces <i>M</i> of <i>A</i>-bundles π : <i>M</i> → <i>S</i>, <i>A</i> = <i>S</i>¹, ℝ, such that the generator of the group action is a maximally twisting shearfree <i>g</i>-null vector field p. We also prove that on any such Lorentzian manifold (<i>M, g</i>)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 <i>M</i> = ℝ × <i>S</i> for any prescribed value of the Einstein constant. If dim <i>M</i> = 4, the Ricci flat metrics obtained in this way are the well-known Taub-NUT metrics.</p>