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The traditional approach to perturbations of nonrotating black holes inGeneral Relativity uses the reformulation of the equations of motion into aradial second-order Schr\"odinger-like equation, whose asymptotic solutions areelementary. Imposing specific boundary conditions at spatial infinity and nearthe horizon defines, in particular, the quasi-normal modes of black holes. Formore complicated equations of motion, as encountered for instance in modifiedgravity models with different background solutions and/or additional degrees offreedom, we present a new approach that analyses directly the first-orderdifferential system in its original form and extracts the asymptotic behaviourof perturbations, without resorting to a second-order reformulation. As apedagogical illustration, we apply this treatment to the perturbations ofSchwarzschild black holes and then show that the standard quasi-normal modescan be obtained numerically by solving this first-order system with a spectralmethod. This new approach paves the way for a generic treatment of theasymptotic behaviour of black hole perturbations and the identification ofquasi-normal modes in theories of modified gravity.

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