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 Pöschl-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.
- Received 14 April 2020
- Accepted 3 May 2021
DOI:https://doi.org/10.1103/PhysRevX.11.031003
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.
Published by the American Physical Society
Physics Subject Headings (PhySH)
Viewpoint
ViewpointInstability in Black Hole Vibrational Spectra
Published 6 July 2021
A new analysis of black hole vibrational spectra identifies which frequencies are stable to perturbations—information pertinent for gravitational-wave analysis and quantum gravity modeling.
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Popular Summary
Perturbed black holes resonate as bells do, with characteristic frequencies and decay times. The corresponding complex frequencies, or quasinormal modes, encode geometric and physical information about the black hole, the physics of which sits at the nexus of gravitation, quantum mechanics, and thermodynamics. Interferometric gravitational-wave detectors can measure these quasinormal modes, defining the new field of black hole spectroscopy. But, in contrast with bells, black hole resonating frequencies are potentially unstable under small-scale perturbations. Here, we explore the implications of this instability.
Our demonstration of black hole quasinormal mode instability relies on two key mathematical frameworks: a “hyperboloidal approach” to field dynamics and the notion of “pseudospectrum.” The former characterizes geometrically the gravitational radiation far from a source, and the latter unveils the underlying structures responsible for this instability.
In particular, we introduce pseudospectra to gravitational physics. With these tools, we demonstrate that high-frequency perturbations leave the slowest decaying quasinormal mode unchanged while migrating more damped resonances (overtones) to new quasinormal branches in the complex plane that we dub “Nollert-Price,” in honor of the first researchers to observe this phenomenon.
This high-frequency phenomenon provides a probe into the small scales of black hole physics. Perturbed quasinormal modes in the gravitational-wave signal may yield key astrophysical insights into the debris-populated environments of black holes or into fundamental gravity physics at the quantum scale, enlarging the scope of the black hole spectroscopy program.
