Quasi-normal mode expansions of black hole perturbations: a hyperboloidal Keldysh’s approach

Abstract

We study asymptotic quasi-normal mode expansions of linear fields propagating on a black hole background. In order to achieve this we exploit the compactified hyperboloidal approach to cast quasi-normal modes as eigenfunctions of a non-selfadjoint spectral problem and we adopt a Keldysh scheme for the spectral construction of the resonant expansions. The role of the scalar product structure in the Keldysh construction is clarified: although it is key to provide a notion of scale, it is proven to be non-necessary to construct a unique quasinormal mode time-series at null infinity. We present (numerical) comparison with the time-domain signal for test-bed initial data and we demonstrate the efficiency and accuracy of the Keldysh spectral approach. Surprisingly, power-law tails for Schwarzschild following the Price law are also recovered in a Keldysh-like scheme. Finally, the importance of the contribution of highly-damped modes to the early time behaviour is illustrated and presented as an introduction to the question of the convergence of the asymptotic series.