Symplectic 4-manifolds via Lorentzian geometry

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 $\mathbf{k}$ is a complete null vector field with geodesic flow along which $\text{Ric}(\mathbf{k},\mathbf{k})>0$, and if $f$ is any smooth function on $M$ with $\mathbf{k}(f)$ nowhere vanishing, then $dg(e^f\mathbf{k},\cdot )$ is a symplectic form and $\mathbf{k}/\mathbf{k}(f)$ is a Liouville vector field; any null surface to which $\mathbf{k}$ 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 $\mathbf{k}$; we give an example of this, with $\mathbf{k}$ a complete Liouville vector field, on the maximally extended "rapidly rotating" Kerr spacetime.

9 pages; v3: final published version; new Remarks 3, 4, and 6 added; section 2 and former Lemma 1 removed.