Lorentzian manifolds with shearfree congruences and K\'ahler-Sasaki geometry
| Autor: | Alekseevsky, Dmitri V., Ganji, Masoud, Schmalz, Gerd, Spiro, Andrea |
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| Rok vydání: | 2020 |
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| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: 37 pages; in v4, we corrected a sign in (5.1) and, in cascade, made adjustments in the subsequent formulas; the changes correspond to a change of orientation and have no effect in any result; we also improved the presentation in Sections 2 and 4 and added ackowledgments |
| Databáze: | arXiv |
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