Symplectic 4-manifolds via Lorentzian geometry

TL;DR

This paper demonstrates how Lorentzian geometry in four dimensions can be used to construct symplectic forms, revealing a novel link between null vector fields and symplectic geometry, with applications to Kerr spacetime.

Contribution

It introduces a method to generate symplectic forms from null vector fields in Lorentzian 4-manifolds, including explicit examples on Kerr spacetime.

Findings

Null vector fields can produce exact symplectic forms.

Null surfaces tangent to these fields are Lagrangian.

Application to Kerr spacetime with Liouville vector fields.

Abstract

We observe that, in dimension four, symplectic forms may be obtained via Lorentzian geometry; in particular, null vector fields can give rise to exact symplectic forms. That a null vector field is nowhere vanishing yet orthogonal to itself is essential to this construction. Specifically, we show that on a Lorentzian 4-manifold $(M,g)$, if ${\boldsymbol k}$ is a complete null vector field with geodesic flow along which $\text{Ric}({\boldsymbol k},{\boldsymbol k}) > 0$, and if $f$ is any smooth function on $M$ with ${\boldsymbol k}(f)$ nowhere vanishing, then $dg(e^f{\boldsymbol k},\cdot)$ is a symplectic form and ${\boldsymbol k}/{\boldsymbol k}(f)$ is a Liouville vector field; any null surface to which ${\boldsymbol k}$ is tangent is then a Lagrangian submanifold. Even if the Ricci curvature condition is not satisfied, one can still construct such symplectic forms with additional information from ${\boldsymbol k}$; we give an example of this, with ${\boldsymbol k}$ a complete Liouville vector field, on the maximally extended "rapidly rotating" Kerr spacetime.