Non-linear Black Hole Ringdowns: an Analytical Approach

TL;DR

This paper develops an analytical framework for understanding non-linear effects in black hole ringdowns, revealing how second-order quasi-normal modes are influenced by the light ring and initial conditions.

Contribution

It introduces a second-order perturbation theory approach that analytically describes non-linear black hole ringdown modes, emphasizing the role of the light ring.

Findings

Second-order modes are peaked around the light ring.

Amplitude ratios of second- to first-order modes depend weakly on initial conditions.

Provides insight into non-linear effects in gravitational wave signals.

Abstract

Due to the nature of gravity, non-linear effects are left imprinted in the quasi-normal modes generated in the ringdown phase of the merger of two black holes. We offer an analytical treatment of the quasi-normal modes at second-order in black hole perturbation theory which takes advantage from the fact that the non-linear sources are peaked around the light ring. As a byproduct, we describe why the amplitude of the second-order mode relative to the square of the first-order amplitude depends only weakly on the initial condition of the problem.