Parallel spinors, pp-waves, and gravitational perturbations

We prove that any real, vacuum gravitational perturbation of a four-dimensional vacuum pp-wave space-time can be locally expressed, modulo gauge transformations, as the real part of a Hertz/Debye potential, where the scalar potential satisfies the wave equation. We discuss relations with complex perturbations, complex space-times, non-linear structures, and real spaces with split (ultra-hyperbolic/Kleinian) signature. Motivated by generalized notions of parallel spinors, we also discuss generalizations of the result to other space-times.