K\"ahler metrics via Lorentzian Geometry in dimension four

TL;DR

This paper constructs K"ahler metrics from 4-dimensional semi-Riemannian manifolds with special vector fields, analyzing their properties, curvature, and providing numerous examples including classical spacetimes and gravitational waves.

Contribution

It introduces a method to derive K"ahler metrics from Lorentzian 4-manifolds using shear, twist, and Lie bracket properties, with explicit curvature formulas and examples.

Findings

K"ahler metrics can be constructed on open sets of 4-manifolds with specific vector field properties.

The Ricci and scalar curvatures of these K"ahler metrics are explicitly computed.

Examples include de Sitter, Kerr, NUT, and SKR metrics, demonstrating the method's broad applicability.

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.