Parallel spinors, pp-waves, and gravitational perturbations

TL;DR

This paper demonstrates that vacuum gravitational perturbations of 4D pp-wave space-times can be represented using Hertz potentials satisfying the wave equation, linking perturbation theory with spinor and complex geometry.

Contribution

It introduces a method to express gravitational perturbations via Hertz potentials in pp-wave backgrounds, extending the framework to complex and other space-times.

Findings

Perturbations are expressible as real parts of Hertz potentials.

Scalar potentials satisfy the wave equation.

Connections to complex structures and generalized spinors.

Abstract

We prove that any real, vacuum gravitational perturbation of a 4-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.