Abstract
We study the stability of quasinormal modes (QNM) in asymptotically flat black hole spacetimes by means of a pseudospectrum analysis. The construction of the Schwarzschild QNM pseudospectrum reveals the following: (i) the stability of the slowest-decaying QNM under perturbations respecting the asymptotic structure, reassessing the instability of the fundamental QNM discussed by Nollert [H. P. Nollert, About the Significance of Quasinormal Modes of Black Holes, Phys. Rev. D 53, 4397 (1996)] as an "infrared" effect; (ii) the instability of all overtones under small-scale ("ultraviolet") perturbations of sufficiently high frequency, which migrate towards universal QNM branches along pseudospectra boundaries, shedding light on Nollert's pioneer work and Nollert and Price's analysis [H. P. Nollert and R. H. Price, Quantifying Excitations of Quasinormal Mode Systems, J. Math. Phys. (N.Y.) 40, 980 (1999)]. Methodologically, a compactified hyperboloidal approach to QNMs is adopted to cast QNMs in terms of the spectral problem of a non-self-adjoint operator. In this setting, spectral (in)stability is naturally addressed through the pseudospectrum notion that we construct numerically via Chebyshev spectral methods and foster in gravitational physics. After illustrating the approach with the P\"oschl-Teller potential, we address the Schwarzschild black hole case, where QNM (in)stabilities are physically relevant in the context of black hole spectroscopy in gravitational-wave physics and, conceivably, as probes into fundamental high-frequency spacetime fluctuations at the Planck scale.
Highlights
We have demonstrated the following: (i) Fundamental black hole (BH) quasinormal modes (QNM) are stable under high-frequency perturbations while unstable under modifications of the asymptotics, the latter being consistent with Ref. [44]; (ii) BH QNM overtones are unstable under high-frequency perturbations, instabilities being quantifiable in terms of the energy content of the perturbation, extending results in Refs. [44,45] to show isospectrality loss; and (iii) pseudospectrum contour lines provide the rationale underlying the structurally stable pattern of perturbed Nollert-Price BH QNM branches
The soundness of the results relies on the use of a compactified hyperboloidal approach to QNMs, with the key identification of the relevant scalar product in the problem as associated with the physical energy, combined with accurate spectral numerical methods
We are confident in the soundness of our conclusions: As discussed in detail, the same qualitative behavior is found systematically by other studies not relying on the hyperboloidal approach, in particular, Nollert and Price’s pioneer work
Summary
A. Black hole QNM stability problem and the pseudospectrum. In the context of spectral problems pervading physics, often related to wave phenomena in both classical and quantum theories, this concerns, in particular, the basic question about the stability of the physical spectrum of the system. Thereupon, one needs to assess the following questions: How does the spectrum react to small changes of the underlying system? Is the spectrum stable; i.e., do small perturbations lead to tiny deviations? With small changes in the system leading to drastic modifications of the spectrum? The problem we address here is the spectral robustness of BH QNMs, namely, the stability of the resonant frequencies of BHs under perturbations. Our spectral (in)stability analysis is built upon the notion of the so-called pseudospectrum
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