Lorentzian manifolds with shearfree congruences and K\"ahler-Sasaki geometry

TL;DR

This paper explores Lorentzian manifolds with shearfree null vector fields, linking their geometry to Sasaki and Kähler structures, and constructs Einstein and Ricci-flat metrics, generalizing classical results to higher dimensions.

Contribution

It provides a local description of Lorentzian metrics with shearfree null fields related to Sasaki and Kähler geometries, and constructs new Einstein and Ricci-flat metrics in higher dimensions.

Findings

Existence of non-trivial electromagnetic plane waves with shearfree null directions.

Construction of Einstein metrics on bundles over Sasaki manifolds.

Generalization of Robinson's theorem to higher dimensions.

Abstract

We study Lorentzian manifolds $(M, g)$ of dimension $n\geq 4$, equipped with a maximally twisting shearfree null vector field $p_o$, for which the leaf space $S = M/\{\exp t p_o\}$ is a smooth manifold. If $n = 2k$, the quotient $S = M/\{\exp t p_o\}$ 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 K\"ahler manifold of real dimension $2k -2$. Going backwards through this line of ideas, for any quantisable K\"ahler manifold with associated Sasaki manifold $S$, we give the local description of all Lorentzian metrics $g$ on the total spaces $M$ of $A$-bundles $\pi: M \to S$, $A = S^1, \mathbb R$, such that the generator of the group action is a maximally twisting shearfree $g$-null vector field $p_o$. We also prove that on any such Lorentzian manifold $(M, g)$ there exists a non-trivial generalized electromagnetic plane wave having $p_o$ as propagating direction field, a result that can be considered as a generalization of the classical $4$-dimensional Robinson Theorem. We finally construct a 2-parametric family of Einstein metrics on a trivial bundle $M = \mathbb R \times S$ 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.