Scalar Perturbation Around Rotating Regular Black Hole Superradiance Instability and Quasinormal Modes Zhen Li

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Scalar Perturbation Around Rotating Regular Black Hole: Superradiance Instability

and Quasinormal Modes

Zhen Li∗

DARK, Niels Bohr Institute, University of Copenhagen, Jagtvej 128, 2200 Copenhagen Ø, Denmark

(Dated: January 25, 2023)

Black holes provide a natural laboratory to study particle physics and astrophysics. When black

holes are surrounded by matter ﬁelds, there will be plenty of phenomena which can have observa-

tional consequences, from which we can learn about the matter ﬁelds as well as black hole spacetime.

In this work, we investigate the massive scalar ﬁeld in the vicinity of a newly proposed rotating reg-

ular black hole inspired by quantum gravity. We will especially investigate how this non-singular

spactime will aﬀect the superradiance instability and quasinormal modes of the scalar ﬁled. We

derive the superradiant conditions and the ampliﬁcation factor by using the Matching-asymptotic

Method, and the quasinormal modes are computed through Continued Fraction Method. In the

Kerr limit, the results are in excellent agreements with previous research. We also demonstrate

how the quasinormal modes will change as a function of black hole spin, regularity described by a

parameter kand scalar ﬁeld mass respectively, with other parameters taking speciﬁc values.

I. INTRODUCTION

Our current best understanding on gravitational inter-

action is described by general relativity (GR). The recent

observation of gravitational waves [1–3] and black hole

shadows[4,5] provide even more evidences on this fasci-

nating theory. However, GR also faces several challenges,

such as, the incompatibility between GR and quantum

theory [6], the singularities [7,8], the late time accelera-

tion of the universe and so on [9–11]. Among these, the

singularities in classical GR are most severe. Because it

is widely belief that singularities do not exist in nature,

rather they reveal the limitations of GR. Therefore, the

idea of regular black holes may provide a solution or a

trial to the singularity problem. The regular black holes

are the solutions that have horizons and non-singular

at the origin, and their curvature invariants are regular

everywhere[12–16]. A novel spherical symmetric regu-

lar black hole proposed in [17–19] and reformulated in

[20] is a very promising solution to the singularity prob-

lem. Later it has also been generalized to the rotating

axisymmetric scenario[21–23]. The exponential conver-

gence factor is used in these regular black holes, which is

also used in formulation of the quantum gravity[24].

Scalar ﬁled play a crucial role in the fundamental

physics as well as astrophysics, like the inﬂation ﬁeld

[25–27] and also in the dark energy models[28]. Dark

matter could also be a kind of scalar ﬁeld, especially,

the ultralight scalar ﬁeld dark matter could have some

advantages over the standard Lambda cold dark matter

model[29]. When the Compton wavelength of the scalar

ﬁeld particles are comparable to the characteristic size

of the black hole horizon, they can eﬃciently extract ro-

tational energy from rotating black holes through super-

radiance instabilities and form macroscopic quasinormal

condensates[30,31]. This provide a unique way and nat-

∗zhen.li@nbi.ku.dk

ural laboratory to detect the ultralight scalar ﬁeld par-

ticles through black hole observations, for example, they

will leave imprints on the gravitational waves[32,33]. Be-

cause of this and its importance in black hole physics, su-

perradiance recently attracts plenty of attention from sci-

ence community, and physicists have performed investi-

gation in many diﬀerent aspects and scenarios[34–50]. It

is also worth to mention that there are alternative mech-

anisms for energy extraction from a rotating black hole,

such as Penrose process[51,52], the Blandford-Znajek

process[53], magnetic reconnection process[54–56] and so

on, which may also produce (charged) scalar ﬁeld parti-

cles.

Thus, to study the phenomenology of scalar ﬁeld

around rotating regular black holes will provide us much

more insights on both gravity, astrophysics and parti-

cle physics. Usually, the scalar ﬁled will be taken as a

test ﬁeld or perturbation ﬁled such that it will not shift

the black hole background spacetime. There are some

related works on this topic but with diﬀerent focus or

regular spacetime[57–59]. In this work, we will study

the superradiance instabilities and quasinormal modes of

scalar ﬁeld around the newly proposed rotating regular

black hole[21–23]. We will demonstrate how the regu-

lar parameter aﬀects the superradiance and quasinormal

modes.

The structure of this paper is as follows: In Section.II,

we will introduce the rotating regular black hole space-

time. In Section.III, we will solve the massive Klein-

Gordon equation in this spacetime, and obtained the ra-

dial and angular equations. Then, in Section.IV, we will

analysis the superradiance instabilities and compute the

ampliﬁcation factor. Then, in Section.V, we will compute

the quasinormal modes by Continued Fraction method,

we also demonstrate how the quasinormal modes will

change as a function of black hole spin, regular parame-

ter and scalar ﬁeld mass respectively. In Section.VI, we

will make a conclusion and discussion.

arXiv:2210.14062v2 [gr-qc] 23 Jan 2023

II. ROTATING REGULAR BLACK HOLE

The metric of non-singular rotating black hole men-

tioned in the introduction could be written in the

Boyer–Lindquist coordinates as [21–23],

ds2=−1−2Mre−k/r

Σdt2+Σ

∆dr2+ Σdθ2

−4aMre−k/r

Σsin2θdtdφ

+r2+a2+2Mra2e−k/r

Σsin2θsin2θdφ2

(1)

with Σ = r2+a2cos2θ, ∆ = r2+a2−2Mre−k/r . and

M,a, and kare three parameters, which were assumed

to be positive. The Kerr metric could be reduced when

set k/r = 0.

To show the regularity of this metric, it is conve-

nient to study the spacetime invariants, for example, the

Kretschmann invariant K = RabcdRabcd (Rabcd is the Rie-

mann tensor).

K=4M2e−2k

r6Σ6Σ4k4−8r3Σ3k3+Ak2+Bk +C(2)

where A,B, and Care functions of rand θ, given by

A=−24r4Σ−r4+a4cos4θ

B=−24r5r6+a6cos6θ−5r2a2cos2θΣ

C= 12r6r6−a6cos6θ

−180r8a2cos2θr2−a2cos2θ

(3)

For M6= 0, they are regular everywhere.

The solutions of equation

∆ = r2+a2−2Mre−k/r = 0 (4)

will give us the event horizons. The numerical results

of horizon structure with diﬀerent parameters were dis-

cussed in [18]. However, there are no analytical solutions.

Despite this, we can use approximation method to

solve (4) analytically as long as k/M 1, and it also

satisﬁes the condition for (4) to have two distinct real

solutions (see [18]), i.e, less than the critical value kEH

which decreases with the increase in a, for a= 0.9M,

kEH

c≈0.1M, for a= 0.95M,kEH

c≈0.05M. In the

Kerr limit, ∆kerr =r2+a2−2Mr = (r−r+)(r−r−),

where r+and r−are called event and inner horizon of

Kerr black hole respectively. They can be seen as the

zeroth order (with respect to k/r) solution to equation

(4). Because the equation (4) can be written as

r2+a2−2Mr = 2Mr(e−k/r −1) (5)

where the right-hand side is much smaller than the left-

hand side if k/r 1, so the right-hand side is the small

perturbation. Therefore, if we brought the zeroth order

solutions r±into the right-hand side of (5), we will get

high order approximation solutions, there are

∆kerr −2Mr+(e−k/r+−1) = (r−rI

+)(r−˜r−)

∆kerr −2Mr−(e−k/r−−1) = (r−˜r+)(r−rI

−)(6)

where rI

+and rI

−could be seen as the ﬁrst order approx-

imate solutions to (4), i.e,

∆≈(r−rI

+)(r−rI

−) (7)

where ˜r+and ˜r−are the two extra roots because we are

solving two quadratic equations, and they are numeri-

cally less accurate compared to rI

+and rI

−. The explicit

forms for rI

+and rI

−are given by

+=M+qM2−a2+ 2Mr+(e−k/r+−1) (8)

−=M−qM2−a2+ 2Mr−(e−k/r−−1) (9)

For better accuracy, we can carry rI

+and rI

−back to the

right-hand side of (5) and repeat the process above to get

more accurate second order solutions of (4).

rII

+=M+qM2−a2+ 2MrI

+(e−k/rI

+−1) (10)

rII

−=M−qM2−a2+ 2MrI

−(e−k/rI

−−1) (11)

even third order solutions

rIII

+=M+qM2−a2+ 2MrII

+(e−k/rII

+−1) (12)

rIII

−=M−qM2−a2+ 2MrII

−(e−k/rII

−−1) (13)

they could be seen as the event horizon and inner hori-

zon of metric (1). One could repeat the approximation

steps to get more higher order solutions, but third order

rIII

+and rIII

−are suﬃcient in this work, see Appendix.A.

Here after we will deﬁne ˆr+≡rIII

+and ˆr−≡rIII

−for

simplicity.

III. DECOUPLED MASTER EQUATIONS FOR

MASSIVE SCALAR FIELD

The dynamics of a massive scalar ﬁeld Φ in the space-

time (1) is governed by the Klein-Gordon equation

∇a∇a−µ2Φ=(√−g)−1∂µ√−ggµν ∂νΦ−µ2Φ=0

(14)

where g=det(gµν ) and µis the mass of the scalar ﬁeld.

We can rewrite it more explicitly in the Boyer–Lindquist

coordinates as

r2+a22

∆−a2sin2θ!∂t∂tΦ + 4M are−k/r

∆∂t∂φΦ

+a2

∆−1

sin2θ∂φ∂φΦ−∂r(∆∂rΦ)

−1

sin θ∂θ(sin θ∂θΦ) + µ2ΣΦ = 0 (15)

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ScalarPerturbationAroundRotatingRegularBlackHole:SuperradianceInstabilityandQuasinormalModesZhenLi*DARK,NielsBohrInstitute,UniversityofCopenhagen,Jagtvej128,2200Copenhagen,Denmark(Dated:January25,2023)Blackholesprovideanaturallaboratorytostudyparticlephysicsandastrophysics.Whenblackholesaresurround...

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Scalar Perturbation Around Rotating Regular Black Hole Superradiance Instability and Quasinormal Modes Zhen Li

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