Asymptotics of linear differential systems and application to quasi-normal modes of nonrotating black holes

TL;DR

This paper introduces a new method for analyzing black hole perturbations directly from first-order systems, facilitating the study of quasi-normal modes in both General Relativity and modified gravity theories.

Contribution

It presents a novel approach that analyzes first-order differential systems directly, avoiding second-order reformulations for studying black hole perturbations.

Findings

Successfully applied to Schwarzschild black holes

Numerical computation of quasi-normal modes using spectral methods

Applicable to modified gravity theories for perturbation analysis

Abstract

The traditional approach to perturbations of nonrotating black holes in General Relativity uses the reformulation of the equations of motion into a radial second-order Schr\"odinger-like equation, whose asymptotic solutions are elementary. Imposing specific boundary conditions at spatial infinity and near the horizon defines, in particular, the quasi-normal modes of black holes. For more complicated equations of motion, as encountered for instance in modified gravity models with different background solutions and/or additional degrees of freedom, we present a new approach that analyses directly the first-order differential system in its original form and extracts the asymptotic behaviour of perturbations, without resorting to a second-order reformulation. As a pedagogical illustration, we apply this treatment to the perturbations of Schwarzschild black holes and then show that the standard quasi-normal modes can be obtained numerically by solving this first-order system with a spectral method. This new approach paves the way for a generic treatment of the asymptotic behaviour of black hole perturbations and the identification of quasi-normal modes in theories of modified gravity.