On Symplectic and Lorentzian geometry in dimension four

Abstract

We observe that, in dimension four, there is a correspondence between symplectic and Lorentzian geometry. The guiding observation is that on a Lorentzian 4-manifold \((M,g)\), 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 if \({\boldsymbol k}\) is a complete null vector field on \(M\) with geodesic flow along which \(\text{Ric}({\boldsymbol k},{\boldsymbol k}) > 0\), and if \(f\) is any 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. We also discuss applications to Weinstein structures: if \((M,g)\) contains a compact Cauchy hypersurface, then the symplectic manifold above yields a trivial Weinstein cobordism.