Exact Solutions for Colliding Neutrino Waves and the Absence of Mutual Scattering for a Single Grassmann Mode

We study the head-on collision of two plane neutrino waves in the coupled Einstein-Weyl system, treating the massless Weyl spinor as an odd Grassmann valued field. We show that the mixed components of the Einstein equations force the two counter propagating waves to be built on a single Grassmann mode, whose bilinear $\epsilon =\eta {\eta }^{\ast }$ is nilpotent. With this structure the full coupled system is solved exactly in the interaction region for arbitrary amplitude and phase profiles, and the metric and spinor fields are matched to the single wave regions by the non-impulsive O’Brien-Synge junction conditions. The metric functions contain no terms coupling the two waves; the Weyl scalar ${\mathrm{\Psi }}_2$ vanishes, and ${\mathrm{\Psi }}_0$, ${\mathrm{\Psi }}_4$ are proportional to the neutrino energy fluxes. The waves therefore traverse each other without mutual scattering; we trace this to the nilpotency of the single mode bilinear. We discuss the limits of this result and indicate where cross couplings can arise.