Kähler 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 Kähler 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. In some cases the Kähler metrics are complete. The Ricci and scalar curvatures of gK 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 gK is an SKR metric. For the latter an inverse ansatz is described, constructing g from the SKR metric.

Keywords

• kahler metric
• lorentzian metric
• shear
• twist
• integrable complex structures
• warped products
• de sitter spacetime
• kerr metric
• gravitational waves
• pp waves
• petrov type
• 53b30
• 53b55
• 53c55