Perturbations of Plane Waves and Quadratic Quasinormal Modes on the Lightring

We study second order gravitational perturbations on plane wave spacetimes from both the metric and curvature perturbation points of view. For the former, we explicitly use the isometries of the background to introduce tensor oscillator harmonics, which render Einstein's equations algebraic around symmetric plane waves. For the latter, we formulate the first and second order Teukolsky equations in a Geroch-Held-Penrose covariant way. Both approaches are useful in their own right, and together with our discussion on gauge freedom, they provide a foundation for the study of higher-order gravitational dynamics around plane wave spacetimes. Taking the perspective that these plane wave spacetimes arise from Penrose limits, we subsequently use these results to explore the nonlinear gravitational dynamics close to black hole lightrings. Specifically, we define and discuss quadratic quasinormal mode ratios, observe that they satisfy emergent selection rules, and make publicly available a code to compute them.