Destabilizing the Fundamental Mode of Black Holes: The Elephant and the Flea

Abstract

Recent work applying the notion of pseudospectrum to gravitational physics showed that the quasinormal mode spectrum of black holes is unstable, with the possible exception of the longest-lived (fundamental) mode. The fundamental mode dominates the expected signal in gravitational wave astronomy, and there is no reason why it should have privileged status. We compute the quasinormal mode spectrum of two model problems where the Schwarzschild potential is perturbed by a small "bump" consisting of either a Pöschl-Teller potential or a Gaussian, and we show that the fundamental mode is destabilized under generic perturbations. We present phase diagrams and study a simple double-barrier toy problem to clarify the conditions under which the spectral instability occurs. Introduction.The advent of gravitational-wave (GW) astronomy [1, 2] and of very long baseline interferometry [3,4] opened exciting new windows to the invisible Universe. Black holes (BHs) play a unique role in the endeavor to test our understanding of general relativity (GR) and in the search for new physics [5][6][7][8][9][10][11].According to the singularity theorems [12,13], classical GR must fail in BH interiors. Quantum mechanics in BH spacetimes also leads to puzzling consequences, such as the information paradox [14][15][16]. It is tempting to conjecture that a theory of quantum gravity will resolve these issues, but the scale and nature of quantum gravity corrections to BH spacetimes is unknown. Uniqueness results in vacuum GR imply that BHs are the simplest macroscopic objects in the Universe [17], and BHs do not "polarize" in binary systems [18][19][20][21][22][23]. The simplicity of BHs (whether isolated or in binaries) implies that they are ideal laboratories to probe the limitations of GR, as long as environmental effects or astrophysical uncertainties can be ignored. In this Letter we ask an important question: is it really possible to ignore environmental effects?One of the tools to test the Kerr geometry is BH spectroscopy [24-26], now a thriving field [27][28][29][30][31][32][33][34]. If a compact binary merger leads to the formation of a rotating BH, as predicted in GR, the spacetime should asymptote to the Kerr metric through a relaxation process during which it can be described as a perturbation of the Kerr metric. The late-time GW signal (the "ringdown") is a superposition of damped exponentials with complex frequencies known as the quasinormal modes (QNMs), which can be computed within perturbation theory as poles of the associated Green's function [35][36][37]. The residues corresponding to these poles in the complex frequency plane dictate the amplitude of the response. To model a ringdown signal using Kerr QNM frequencies in vacuum, we should take into account the surrounding matter (even if it can be considered as a small perturbation). This is the

Stability of the Fundamental Quasinormal Mode in Time-Domain Observations: The Elephant and the Flea Redux

“…Recent work analyzing spectral stability through calculations of the pseudospectrum [17][18][19][20] confirmed early predictions that the QNM spectrum should, in fact, be spectrally unstable under small perturbations [21]. In particular, the fundamental QNM frequency can have corrections of order one when a tiny perturbation is added to the potential [22]. Physical perturbations of the potential can have various origins, including nonlinear effects within vacuum GR [23][24][25][26][27][28][29][30], ordinary matter [31][32][33], dark matter [34], or modifications of Einstein's theory of gravity [35,36].…”

“…The fundamental mode of the Regge-Wheeler equation is unstable if the perturbative bump is sufficiently large or if it is located sufficiently far away [22]. By definition, we will say that "destabilization" occurs when a quantity that characterizes the BH's response (e.g., the QNM frequency or the waveform amplitude) changes by an amount much larger than the magnitude of the perturbation.…”

“…Following Ref. [22], we will assume that the bump is localized around some radius r * = a and that it goes to zero faster than V RW as r * → ∞. We will study for concreteness a Gaussian bump of the form…”

“…The goal of this paper is to investigate whether the instability of the fundamental QNM obtained in [22] translates into an instability of the mode as observed (either theoretically or in actual experiments) in a time-domain analysis. In Sec.…”

Computing the quasinormal modes and eigenfunctions for the Teukolsky equation using horizon penetrating, hyperboloidally compactified coordinates

“…§ For other recent attempts to investigate the stability of the quasinormal mode solutions under perturbations to the underlying Kerr spacetime, see [15,16,17,18].…”