27850 2509.03598 10.1007/JHEP12(2025)148 148 148 Perturbations of plane waves and quadratic quasinormal modes on the lightring Regular Article - Theoretical Physics 1 55 2025 12 18 2025 9 29 2025 10 28 2025 12 18 2025 12 18 Villum Fonden (VILLUM Foundation) VIL37766 Danish National Research Foundation (DNRF) DNRF162 Danish National Research Foundation (DNRF) 184 European Commission 101007855 European Commission 101131233 Heising-Simons Foundation 2021-2817 U.S. Department of Energy (DOE) DE-SC0011632 National Science Foundation (NSF) AST-2307146 National Science Foundation (NSF) PHY-2513337 National Science Foundation (NSF) PHY-090003 National Science Foundation (NSF) PHY-20043 National Aeronautics and Space Administration (NASA) 21-ATP21-0010 John Templeton Foundation 62840 Italian Ministry of Foreign Affairs and International Cooperation (MAECI) PGR01167 The Author(s) 2025 Open Access. This article is distributed under the terms of the Creative Commons Attribution License ( CC-BY 4.0 ), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. 13130 2025 2025 12 12 Kwinten Fransen kfransen@caltech.edu David Pereñiguez dpereni1@jhu.edu Jaime Redondo-Yuste jaime.redondo.yuste@nbi.ku.dk https://ror.org/05dxps055 grid.20861.3d 0000 0001 0706 8890 Walker Burke Institute for Theoretical Physics California Institute of Technology Pasadena CA 91125 USA https://ror.org/035b05819 grid.5254.6 0000 0001 0674 042X Center of Gravity Niels Bohr Institute Blegdamsvej 17 2100 Copenhagen Denmark https://ror.org/00za53h95 grid.21107.35 0000 0001 2171 9311 William H. Miller III Department of Physics and Astronomy Johns Hopkins University 3400 North Charles Street Baltimore Maryland 21218 USA Abstract 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. Keywords Black Holes Classical Theories of Gravity ArXiv ePrint: 2509.03598