Weyl Neutrinos In Plane Symmetric Spacetimes

We investigate complex quaternion-valued exterior differential forms over 4-dimensional Lorentzian spacetimes and explore Weyl spinor fields as minimal left ideals within the complex quaternion algebra. The variational derivation of the coupled Einstein-Weyl equations from an action is presented, and the resulting field equations for both first and second order variations are derived and simplified. Exact plane symmetric solutions of the Einstein-Weyl equations are discussed, and two families of exact solutions describing left-moving and right-moving neutrino plane waves are provided. The study highlights the significance of adjusting a quartic self-coupling of the Weyl spinor in the action to ensure the equivalence of the field equations.