Physical scales in black hole scattering pseudospectra : the role of the scalar product

Edgar Gasperin

A pseudospectrum analysis has recently provided evidence of a potential generic instability of black hole (BH) quasinormal modes (QNM) under high-frequency perturbations. Such instability analysis depends on the assessment of the size of perturbations. This is encoded in the scalar product and its choice is not unique. Here, we address the impact of the scalar product choice, exploiting the physical scales of the problem ; the energy. The article is organized in three parts : basics, applications and heuristic proposals. In the first part, we revisit the energy scalar product used in the hyperboloidal approach to QNMs, extending previous effective analyses and placing them on solid spacetime basis. The second part focuses on systematic applications of the scalar product in the QNM problem : i) we demonstrate that the QNM instability is not an artifact of previous spectral numerical schemes, by implementing a finite elements calculation from a weak formulation ; ii) using Keldysh’s theorem, we provide QNM resonant expansions for the gravitational waveform, with explicit expressions of the expansion coefficients ; iii) we propose the notion of epsilon-dual QNM expansions to exploit BH QNM instability in BH spectroscopy, employing both non-perturbed and perturbed QNMs, the former informing on large scales and the latter probing small scales. The third part enlarges the conceptual scope of BH QNM instability proposing a) spiked perturbations are more efficient in triggering BH QNM instabilities than smooth ones, b) a general picture of the BH QNM instability problem is given, leading to the conjecture (built on Burnett’s conjecture) that Nollert-Price branches converge universally to logarithmic Regge branches in the high-frequency limit and c) aiming for a fully geometric description of QNMs, BMS states are hinted as possible asymptotic/boundary degrees of freedom for inverse scattering data.