Spectral stability of near-extremal spacetimes Huan Yang1 2and Jun Zhang3 4y 1Perimeter Institute for Theoretical Physics Ontario N2L 2Y5 Canada

2025-05-03 0 0 766.69KB 9 页 10玖币

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Spectral stability of near-extremal spacetimes

Huan Yang1, 2, ∗and Jun Zhang3, 4, †

1Perimeter Institute for Theoretical Physics, Ontario, N2L 2Y5, Canada

2University of Guelph, Guelph, Ontario N1G 2W1, Canada

3International Centre for Theoretical Physics Asia-Paciﬁc,

University of Chinese Academy of Sciences, 100190 Beijing, China

4Taiji Laboratory for Gravitational Wave Universe (Beijing/Hangzhou),

University of Chinese Academy of Sciences, 100049 Beijing, China

It has been suggested that the spectrum of quasinormal modes of rotating black holes is unstable against

additional potential terms in the perturbation equation, as the operator associated with the equation is non-self-

adjoint. We point out that a bi-linear form has been constructed previously to allow a perturbation analysis

of the spectrum, which was applied to study the quasinormal modes of weakly charged Kerr-Newman black

holes [1]. The proposed spectral instability should be restated as instability against potential terms that have

inﬁnitesimal “energy” norm that is speciﬁcally deﬁned by the type of inner products introduced by Jaramillo

et al. and preserving the physical meaning of energy. We argue that it is necessary to address the stability of

all previous mode analysis results to reveal their susceptibility to energetically inﬁnitesimal perturbations. In

particular, for near extremal Kerr spacetime, we show that the spectrum of zero-damping modes, which have

slow decay rates, is unstable (with order unity fractional change in decay rates) with ﬁne-tuned modiﬁcation

of the potential. The decay rates are, however, always positive with energetically inﬁnitesimal perturbations.

If ﬁnite potential modiﬁcations are allowed near the black hole, it is possible to ﬁnd superradiantly unstable

modes, i.e., a “black hole bomb” without an explicit outer shell. For the zero-damping modes in near-extremal

Reissner-Nordstr¨

om-de Sitter black holes, which are relevant for the breakdown of strong cosmic censorship,

we ﬁnd that the corresponding spectrum is stable under energetically inﬁnitesimal perturbations.

I. INTRODUCTION

Modal analysis of black hole spacetimes plays an impor-

tant role in gravitational-wave astronomy, as the quasinormal

mode (QNM) excitation and ringdown contribution is a vi-

tal part of generic black hole perturbations, especially for the

postmerger black holes. Black hole spectroscopy, i.e., mea-

suring the frequencies and damping rates of QNMs, can be

used to infer the black hole mass and spin and to test general

relativity. So far, the fundamental mode (`=2,m=2) has

been convincingly detected in some of the LIGO events [2–4],

including GW150914 [5]. There have been claims of detect-

ing high-overtone modes as well [6–8], albeit there are con-

cerns from independent analyses [9]. The detection of QNMs

with higher `generally requires a higher event signal-to-noise

ratio, which may beneﬁt from coherently stacking multiple

events [10].

It has been recently (re)claimed that distant and/or short-

wavelength perturbation of the wave potential of QNMs may

signiﬁcantly change the mode frequencies, despite the per-

turbation amplitudes being inﬁnitesimal. This observation

was initially pointed out in Refs. [11–14] and was recently

revisited using a pseudospectrum analysis [15,16] for the

Schwarzschild spacetime. We emphasize that the spectrum of

a black hole is stable under inﬁnitesimal perturbations, if the

perturbation analysis is performed with the previously con-

structed bi-linear form in Ref. [17]. Instead, the observation

in Ref. [15] should be reinterpreted as instability [18] under

energetically inﬁnitesimal perturbations, the norm of which is

∗hyang@perimeterinstitute.ca

†zhangjun@ucas.ac.cn

deﬁned by inner products preserving the physical meaning of

energy.

It is crucial to examine the spectral stability under ener-

getically inﬁnitesimal perturbations for various mode analy-

sis results. If the mode frequencies are signiﬁcantly mod-

iﬁed with perturbations in the potential of inﬁnitesimal en-

ergy costs, associated claims need to be treated with extra

caution due to the susceptibility to external perturbers, or

even internal variations due to nonlinearities. A robust claim

should have converging measures with respect to small per-

turbations. In this work, we focus on the spectral stability

of near-extremal black holes, which in general host a class

of zero-damping modes (ZDMs) with decay rates approach-

ing zero in the extremal limit [19,20]. For diﬀerent kinds of

background near-extremal spacetime, these modes have been

used to demonstrate parametric nonlinear instability in con-

nection with turbulence [21], possible breakdown of strong

cosmic censorship (SCC) [22], and near-horizon critical be-

havior that leads to the instability of extremal black holes

[23,24], as possibly connected to gravitational critical col-

lapse [25]. Using the example of near-extremal Kerr, we will

discuss whether these modes will be unstable against small

perturbations and under what conditions the mode spectrum

becomes unstable. We also study the ZDMs for near-extremal

Reissner-Nordstr¨

om-de Sitter (RNdS) black holes, as an ex-

ample for nonasymptotically ﬂat spacetimes, and comment

on whether the spectral stability aﬀects the divergence on the

Cauchy horizon. Throughout the analysis, we adopt the natu-

ral unit that G=c=1.

arXiv:2210.01724v2 [gr-qc] 6 Mar 2023

Re r

r-r+

FIG. 1. An illustration for the contour in the complex r-plane that

can be used for the perturbation analysis of black hole quasinormal

modes. There is a branch cut extending from r+to inﬁnity. The QNM

wavefunctions generally behave as eiωr∗and converge to zero at both

open ends of C, making the boundary terms hχ|Hηi−hHχ|ηivanish.

II. SPECTRAL STABILITY AND ENERGETIC

PSEUDOSPECTRUM

The eigenvalue spectrum of a non-self-adjoint operator

could be unstable under inﬁnitesimal perturbations of the op-

erator, which, in Ref. [15], is used to explain the signiﬁ-

cant migration of high-overtone QNM frequencies by small-

amplitude yet short-wavelength perturbations of the poten-

tial. Nevertheless, perturbative analysis of the spectrum can

be made posssible if one replaces the usual inner product with

a carefully constructed bi-linear form. Considering an eigen-

value problem with H(ω0)ψ0=0 and a perturbation in the

potential H(ω)→H(ω)+ δV, we may formally expand the

eigenfrequency and eigenfunction as ω=ω0+ ω1+O(2)

and ψ=ψ0+ ψ1+O(2), respectively. For the unperturbed

eigenfunction χ, if one can construct a bi-linear form such that

hχ|Hηi=hHχ|ηi, then we have

ω1=−hψ0|δV|ψ0i

hψ0|∂ωH|ψ0i.(1)

Such bi-linear form is explicitly constructed in Ref. [17]. For

Kerr black holes it is given by

hχ|ηi=ZC

(r−r+)s(r−r−)sdr Zsin θdθ χ(r, θ)η(r, θ),(2)

where sis the spin of the ﬁeld, r±are the radius of outer and

inner horizon, respectively, and Cis the contour as shown in

Fig. 1. For example, a Kerr black hole QNM wavefunction

can be written as [26]

R(r)=riωr(r−r+)−s−iσ+(r−r−)−1−s+iω+iσ+∞

n=1

dn r−r+

r−r−!n

(3)

with σ+=(ωr+−am)/(r+−r−). The (r−r+)−s−iσ+term, which

is associated with the horizon boundary condition, leads to

a branch cut starting at r=r+as we analytically continue

the wavefunction to the complex r plane. The other branch

cut starting at r=r−is not relevant for the discussion here.

Having in mind H(ω) taking the form of

H(ω)= ∆−sd

dr ∆s+1d

dr !−V(ω, r) (4)

with (∆ = (r−r+)(r−r−)), the boundary terms are

∆s+1χdη

dr−dχ

drη¯

C. Here ¯

Cdenotes the open ends of the con-

tour. The QNM wave functions generally have eiωrbehavior

for large r[26], where the mode frequency ωhas a positive

real part. It is straightforward to see that the mode wave func-

tion quickly converges to zero at both open ends of C, and the

boundary terms vanish identically. The corresponding eigen-

value analysis is applied for computing the QNM frequency

of weakly charged Kerr-Newman black holes [1], showing no

instability of spectrum. The results obtained by the pertur-

bative analysis in Ref. [1] were later shown to be fully con-

sistent with the numerical quasinormal mode frequencies in

Ref. [27]. Also see Ref. [28] for a construction of bi-linear

form with the focus on the orthogonality of QNMs.

The high-frequency and the distant Gaussian-bump pertur-

bations considered in Refs. [15,29] can all be consistently

treated within the above formalism, although the normaliza-

tions of these operators are dramatically diﬀerent. In fact, sim-

ilar perturbation analysis has been performed in Ref. [30] for

a scalar ﬁeld in Schwarzschild spacetime, in which case the

mode frequency perturbation is shown to scale as e2iω0xV/x2

for a bump centered around xV. Since Im(ω0)<0, the impor-

tance of the potential perturbation is exponentially ampliﬁed

by its distance xV. Moreover, this formalism produces useful

insights for results from pseudospectrum analysis. Assuming

the QNM frequency shift is ω1(rV) for a potential perturbation

δ(r−rV), the frequency shift would be RdrVω1(rV)δV(rV)

for a general perturbation δV(r), as long as is controlled to

ensure a small shift. For δV∼eikr as discussed in Ref. [15],

the integral contributed by potential perturbations near the

horizon scales as k−2Im(ω)−1as ω1(r)∝ψ2

0∝(r−r+)−2iωnear

the horizon at r+. So high-overtone modes are more suscepti-

ble to high-kperturbations.

Despite not making the operator self-adjoint, the inner

product deﬁned in Ref. [15] closely ﬁts the intuitive expec-

tation of the magnitude (or normalization) of a potential. The

phenomena observed are physically relevant. To reconcile

with the above discussion and to be precise with the termi-

nology, we shall follow the convention to refer the analysis

in Ref. [15] as “the energetic pseudospectrum” to emphasize

the special choice of inner product [18]. For a generic mode

analysis, it is necessary to address the spectral stability un-

der energetically small perturbations, as it characterizes the

robustness of the mode spectrum and the associated implica-

tions, such as mode stability and SCC. In the following, we

shall consider distant perturbations as examples of energeti-

cally small perturbations, and investigate the migration of the

ZDMs of near-extremal Kerr and RNdS black holes.

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Spectralstabilityofnear-extremalspacetimesHuanYang1,2,andJunZhang3,4,y1PerimeterInstituteforTheoreticalPhysics,Ontario,N2L2Y5,Canada2UniversityofGuelph,Guelph,OntarioN1G2W1,Canada3InternationalCentreforTheoreticalPhysicsAsia-Pacic,UniversityofChineseAcademyofSciences,100190Beijing,China4TaijiLabor...

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Spectral stability of near-extremal spacetimes Huan Yang1 2and Jun Zhang3 4y 1Perimeter Institute for Theoretical Physics Ontario N2L 2Y5 Canada

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