Symplectic 4-manifolds via Lorentzian geometry

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 $\mathit{k}$ is a complete null vector field with geodesic flow along which $$\text{Ric}$({\boldsymbol k},{\boldsymbol k}) > 0$,andif$f$isanysmoothfunctionon$M$with${\boldsymbol k}(f)$nowherevanishing,then$dg(e^f{\boldsymbol k},\cdot)$isasymplecticformand${\boldsymbol k}/{\boldsymbol k}(f)$ is a Liouville vector field; any null surface to which $\mathit{k}$ 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 $\mathit{k}$; we give an example of this, with $\mathit{k}$ a complete Liouville vector field, on the maximally extended “rapidly rotating” Kerr spacetime.