Vishveshwara's waveform revisited: insights from a Keldysh quasinormal mode expansion

We revisit Vishveshwara’s linear scattering on a Schwarzschild black hole to study the evolution of quasinormal mode activation along the full time-domain waveform. Specifically, by adopting a hyperboloidal approach we cast the scattering problem in a non-selfadjoint dynamics setting where a (Keldysh) resonant expansion in a bi-orthogonal system of quasinormal modes can be readily performed. The calculation reveals the neat correlation of the second peak in Vishveshwara’s waveform to the fundamental mode and first overtones, closely following the ringdown waveform pattern of non-linear evolutions. The first peak is completely controlled, for even (Zerilli) perturbations, by the algebraically special mode and the nearby branch cut. This phenomenon is absent for odd (Regge-Wheeler) perturbations. These qualitative features are robust under change of initial data, extending to Reissner-Nordstr"om and Kerr black holes but being absent when departing from asymptotic flatness to spacetime asymptotics without a branch cut. Underlying linear mechanisms might shed light on the role of algebraic speciality in the transition to the ringdown in non-linear evolutions.