Almost Kähler metrics and pp-wave spacetimes

Abstract

We establish a one-to-one correspondence between a class of strictly almost Kähler metrics on the one hand and Lorentzian pp-wave spacetimes on the other; the latter metrics are well known in general relativity, where they model radiation propagating at the speed of light. Specifically, we construct families of complete almost Kähler metrics by deforming pp-waves via their propagation wave vector. The almost Kähler metrics we obtain exist in all dimensions \(2n \ge 4\), and are defined on both \(\mathbb {R}^{2n}\) and \({\mathbb {S}}^1\times {\mathbb {S}}^1 \times M\), where M is any closed almost Kähler manifold; they are not warped products, they include noncompact examples with constant negative scalar curvature, and all of them have the property that their fundamental 2-forms are also co-closed with respect to the Lorentzian pp-wave metric. Finally, we further deepen this relationship between almost Kähler and Lorentzian geometry by utilizing Penrose’s “plane wave limit,” by which every spacetime has, locally, a pp-wave metric as a limit: using Penrose’s construction, we show that in all dimensions \(2n \ge 4\), every Lorentzian metric admits, locally, an almost Kähler metric of this form as a limit.