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

TL;DR

This paper demonstrates that nonlinear effects in black hole ringdown quasi-normal modes can be effectively modeled as adiabatic modes using the Penrose limit, leading to simple analytical expressions that match numerical results.

Contribution

It introduces a novel approach to model QNM nonlinearities as adiabatic modes via the Penrose limit, simplifying analysis of black hole ringdowns.

Findings

Derived simple analytical expressions for QNM nonlinearities.

Model successfully reproduces existing numerical results.

Interpreted QNMs as adiabatic modes near large diffeomorphisms.

Abstract

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.