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 $\boldsymbol {k}$ 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.