Physics Letters B 798 (2019) 135026
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Superradiant stability of the Kerr black holes
Jia-Hui Huang
a
, Wen-Xiang Chen
a
, Zi-Yang Huang
a
, Zhan-Feng Mai
b,∗
a
Institute of quantum matter, School of Physics and Telecommunication Engineering, South China Normal University, Guangzhou 510006, China
b
Center for Joint Quantum Studies, School of Science, Tianjin University, Tianjin 300350, China
a r t i c l e i n f o a b s t r a c t
Article history:
Received
23 July 2019
Received
in revised form 8 October 2019
Accepted
10 October 2019
Available
online 14 October 2019
Editor:
M. Cveti
ˇ
c
We study the superradiant stability of the system of a Kerr black hole and a massive scalar perturbation.
It was proved previously that this system is superradiantly stable when μ ≥
√
2m
H
, where μ is the
proper mass of the scalar, m is the azimuthal number of the scalar mode, and
H
is the angular velocity
of the Kerr black hole horizon. Our study is a complementary work of this result. We analytically prove
that in the complementary parameter region μ <
√
2m
H
, when the parameters of scalar perturbation
and Kerr black hole satisfy two simple inequalities, ω <
μ
√
2
,
r
−
r
+
< 0.802, the system is also superradiantly
stable.
© 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license
(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP
3
.
1. Introduction
Black holes are important and peculiar objects predicted by
general relativity. Aspects of black hole physics have been stud-
ied
extensively. One interesting phenomenon is the superradiant
scattering of black holes [1–5], e.g., when a charged bosonic wave
is impinging upon a charged rotating black hole, the wave is am-
plified
by the black hole if the wave frequency ω obeys
ω < m
H
+e, (1)
where e and m are the charge and azimuthal number of the
bosonic wave mode,
H
is the angular velocity of black hole hori-
zon
and is the electromagnetic potential of the black hole hori-
zon.
This amplification is the superradiant scattering, which was
studied long time ago [6–12], and has broad applications in various
areas of physics (for a recent review, see [4]). Through the super-
radiant
process, the rotational energy or electromagnetic energy of
a black hole can be extracted. Due to the existence of superradiant
modes, a black hole bomb mechanism was proposed by Press and
Teukolsky [13]. If there is a mirror between the black hole hori-
zon
and space infinity, the amplified wave can be scattered back
and forth and grows exponentially, which leads to the superra-
diant
instability of the background black hole geometry [14–16].
Superradiant (in)stability of various kinds of black holes have been
studied extensively in the literature.
*
Corresponding author.
E-mail
address: zhanfeng.mai@gmail.com (Z.-F. Mai).
For charged Reissner-Nordstrom (RN) black holes, it has been
proved that they are superradiantly stable against charged massive
scalar perturbation [17–20]. The reason is that when the superradi-
ant
modes exist in such a system of a RN black hole with a charged
massive scalar wave, there is no effective trapping potential/mir-
ror
outside the black hole horizon, which reflects the superradiant
modes back and forth [18,19]. However, when a mirror or a cav-
ity
is imposed outside a charged RN black hole horizon, this black
hole is superradiantly unstable in certain parameter spaces [15,16,
21–23].
Charged black holes in curved backgrounds, such as anti-
de
Sitter/de Sitter (AdS/dS) space, are proved to be superradiantly
unstable because these backgrounds provide natural mirror-like
boundary conditions [24–28]. There is a similar case for stringy RN
black holes. The stringy RN black hole is shown to be superradi-
antly
stable against charged massive scalar perturbation [29]. But if
a mirror is introduced, superradiant modes are supported and the
stringy RN black hole becomes superradiantly unstable [30–32]. It
is also found that extra coupling between the scalar field and the
gravity can result in superradiant instability of RN/RN-AdS black
holes [33,34].
For
rotating Kerr black holes, if the incoming scalar perturba-
tion
has a nonzero mass, this mass term will act as a natural mir-
ror
and lead to superradiant instability of Kerr black holes when
the parameters of the Kerr black holes and the scalar fields are
in certain parameter spaces [35–43]. Beyond the massive scalar
perturbations, superradiant instability of Kerr black holes that are
impinged upon by a massive vector field is also discussed [44,45].
The superradiant instability of rotating black holes in curved space,
such as Kerr-AdS black holes, has also been reported [46–52].
https://doi.org/10.1016/j.physletb.2019.135026
0370-2693/
© 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by
SCOAP
3
.
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