Energy scales and black hole pseudospectra: the structural role of the scalar product

Abstract

A pseudospectrum analysis has recently provided evidence of a potential generic instability of black hole (BH) quasinormal mode (QNM) overtones under high-frequency perturbations. Such instability analysis depends on the assessment of the size of perturbations. The latter is encoded in the scalar product and its choice is not unique. Here, we address the impact of the scalar product choice, advocating for founding it on the physical energy scales of the problem. 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 asymptotic expansion of the resolvent, 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, complementarily exploiting 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, supporting the conjecture (built on Burnett’s conjecture on the spacetime high-frequency limit) that Nollert-Price branches converge universally to logarithmic Regge branches in the high-frequency limit and c) aiming at a fully geometric description of QNMs, BMS states are hinted as possible asymptotic/boundary degrees of freedom for an inverse scattering problem.

Pseudospectrum of Reissner-Nordström black holes: quasinormal mode instability and universality

“…This picture is qualitatively identical to a typical normal operator (see Fig. 4 in [37] and [80]). The non-perturbed eigenvalues of L are the same as the original eigenvalues of L, because of the very particular form of the adjoint of L. As a consequence, the pseudospectral boundaries of L provide the distances from points in the complex plane to the actual spectrum of L, making contact with the -contour lines E (L) of the error bound function introduced in [113].…”

“…and the horizon at σ = 1), reflecting the fact that the loss of selfadjointness is related to the flux of energy through these boundaries. In fact, as shown in [80], the flux of energy at the boundaries depends linearly on the function γ(σ) at the boundaries.…”

“…10 of Appendix A (see also Fig. 4 in [37] and the discussion in [80]), where nested circular sets of radius ∼ form around non-perturbed QNMs. In that sense, we can differentiate between spectral stability and instability according to the topographic structure of pseudospectra.…”

“…The scalar product fixes the norm that quantifies the notion of "big" or "small" perturbations ||δL||. On physical grounds, and following [37,45], we argue in favor of the system's energy as the most adequate norm to control the pseudospectra and assess spectral instability, because it conveniently encodes the size of the physical perturbation with respect to the nature of the problem (see [80] for a systematic discussion of this point). On the computational side, calculating the pseudospectrum requires the numerical evaluation of the norm of the resolvent, according to the previous definition.…”

“…Within the hyperboloidal formulation of the wave equation given by Eq. ( 15), the energy norm for a ( -mode) vector u reads [37] (see [80] for a full account of its relation with the total energy of the field on a spacetime slice)…”