K\"ahler metrics via Lorentzian Geometry in dimension four

Abstract

Given a semi-Riemannian $4$-manifold $(M,g)$ with two distinguished vector fields satisfying properties determined by their shear, twist and various Lie bracket relations, a family of K\"ahler metrics $g_K$ is constructed, defined on an open set in $M$, which coincides with $M$ in many typical examples. Under certain conditions $g$ and $g_K$ share various properties, such as a Killing vector field or a vector field with a geodesic flow. In some cases the K\"ahler metrics are complete. The Ricci and scalar curvatures of $g_K$ are computed under certain assumptions in terms of data associated to $g$. Many examples are described, including classical spacetimes in warped products, for instance de Sitter spacetime, as well as gravitational plane waves, metrics of Petrov type $D$ such as Kerr and NUT metrics, and metrics for which $g_K$ is an SKR metric. For the latter an inverse ansatz is described, constructing $g$ from the SKR metric.