Nonlinear Effects in Black Hole Ringdown Made Simple: Quasi-Normal Modes as Adiabatic Modes

The nonlinear nature of general relativity manifests prominently throughout the merger of two black holes, from the inspiral phase to the final ringdown. Notably, the quasi-normal modes generated during the ringdown phase display significant nonlinearities. We show that these nonlinear effects can be effectively captured by zooming in on the photon ring through the Penrose limit. Specifically, we model the quasi-normal modes as null particles trapped in unstable circular orbits around the black holes and show that they can be interpreted as adiabatic modes, perturbations that are arbitrarily close to large diffeomorphisms. This enables the derivation of a simple analytical expression for the QNM nonlinearities for Schwarzschild and Kerr black holes which reproduces well the existing numerical results.