Lorentzian manifolds with shearfree congruences and Kähler-Sasaki geometry
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 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 $\pi: M \to S$, $A = S^1, \mathbb R$, 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.
- Publication:
-
arXiv e-prints
- Pub Date:
- September 2020
- DOI:
- 10.48550/arXiv.2009.07179
- arXiv:
- arXiv:2009.07179
- Bibcode:
- 2020arXiv200907179A
- Keywords:
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- Mathematics - Differential Geometry;
- General Relativity and Quantum Cosmology;
- High Energy Physics - Theory;
- 83C20;
- 83C50;
- 53C25;
- 32V05;
- 32V30;
- 53C17
- E-Print:
- 37 pages