K\"ahler metrics via Lorentzian Geometry in dimension four

Amir Babak Aazami; Gideon Maschler

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arXiv · 1711.10011

K\"ahler metrics via Lorentzian Geometry in dimension four

Amir Babak Aazami·Gideon Maschler

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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.

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BibTeX RIS

Amir Babak Aazami, Gideon Maschler. 2017-11-27. K\"ahler metrics via Lorentzian Geometry in dimension four. https://arxiv.org/abs/1711.10011

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