Non-linear Quasi-Normal Modes of the Schwarzschild Black Hole from the Penrose Limit

The Penrose limit connects a plane wave geometry to the photon ring of a black hole, where the quasi-normal modes are located in the eikonal limit. Utilizing this simplification, we analytically extract the quadratic-level non-linearities in the quasi-normal modes of a Schwarzschild black hole for the $(\mathrm{\ell }\times \mathrm{\ell })\to 2\mathrm{\ell }$ channel. We demonstrate that this result is independent of $\mathrm{\ell }$ and further confirm it through symmetry arguments.