Hamilton-Jacobi theory and two-point null cone symmetry on plane waves

We discuss a Hamilton-Jacobi approach to two-point null-cone functions for plane wave spacetimes. In the exact Rosen model, null separation is given by an explicit two-point function that cuts out the null relation, solves the null Hamilton–Jacobi equation, and is exactly symmetric under interchange of the two endpoints on each non-caustic branch. This provides a concrete two-ended model for null geometry. By contrast, the corresponding ordered constructions in Brinkmann-type or Penrose-adapted settings are not manifestly symmetric, even though they encode the same underlying null relation. The talk explains this contrast and uses the Hamilton-Jacobi viewpoint to clarify how the reciprocal Rosen description and the ordered characteristic description fit together. More broadly, I will highlight how this viewpoint makes the two-point doubling of a spacetime into a natural arena for questions of geometric optics and null propagation. Time permitting, I will also discuss connections with Liouville-type equations and related scalar curvature problems on endpoint space.