Asymptotics of linear differential systems and application to quasinormal modes of nonrotating black holes
David Langlois, Karim Noui, Hugo Roussille
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 quasinormal 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 behavior 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 quasinormal 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 behavior of black hole perturbations and the identification of quasinormal modes in theories of modified gravity.