Kähler metrics via Lorentzian Geometry in dimension four

Amir Babak Aazami, Gideon Maschler

Published 2017-11-27Version 1

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 $\gK$ is constructed, defined on an open set in $M$, which coincides with $M$ in many typical examples. Under certain conditions $g$ and $\gK$ share various properties, such as a Killing vector field or a vector field with a geodesic flow. The Ricci and scalar curvatures of $\gK$ are computed in some cases 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 $\gK$ is an SKR metric. For the latter an inverse ansatz is described, constructing $g$ from the SKR metric.