环形量子黑洞的引力轴向扰动和准模态
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Eur. Phys. J. C (2019) 79:157
https://doi.org/10.1140/epjc/s10052-019-6565-2
Regular Article - Theoretical Physics
Gravitational axial perturbations and quasinormal modes of loop
quantum black holes
M. B. Cruz
1,a
,C.A.S.Silva
2,b
,F.A.Brito
1,3,c
1
Departamento de Física, Universidade Federal da Paraíba, João Pessoa, Paraíba 58051-970, Brazil
2
Instituto Federal de Educação Ciência e Tecnologia da Paraíba (IFPB), Campus Campina Grande-Rua Tranquilino Coelho Lemos 671, Jardim
Dinamérica I, João Pessoa, Brazil
3
Departamento de Física, Universidade Federal de Campina Grande, Caixa Postal 10071, Campina Grande 58429-900, Paraíba, Brazil
Received: 3 November 2018 / Accepted: 3 January 2019 / Published online: 21 February 2019
© The Author(s) 2019
Abstract Loop quantum gravity (LQG) is a theory that pro-
poses a way to model the behavior of the spacetime in sit-
uations where its atomic characteristic arises. Among these
situations, the spacetime behavior near the Big Bang or black
hole’s singularity. The detection of gravitational waves, on
the other hand, has opened the way to new perspectives in the
investigation of the spacetime structure. In this work, by the
use of a WKB method introduced by Schutz and Will (Astro-
phys J 291:L33, 1985), and after improved by Iyer and Will
(Phys Rev D 35:3621, 1987), we study the gravitational wave
spectrum emitted by loop quantum black holes, which corre-
spond to a quantized version of the Schwarzschild spacetime
by LQG techniques. From the results obtained, loop quantum
black holes have been shown stable under axial gravitational
perturbations.
1 Introduction
One of most exciting predictions of general relativity is the
existence of black holes, objects from which no physical bod-
ies or signals can get loose of their drag due to its strong grav-
itational field. Going far beyond astrophysics, black holes
appear as objects that may help us to clarify one of the most
intriguing bone of contention in the current days, the quan-
tum nature of gravity. It is because, in the presence of a black
hole’s strong gravitational field, quantum features of space-
time must be manifested [3–9].
Loop quantum gravity, on the other hand, is a theory that
has given ascent to models that provide a portrait of the quan-
tum features of spacetime unveiled by a black hole. In partic-
a
e-mail: messiasdebritocruz@gmail.com
b
e-mail: carlos.silva@ifpb.edu.br
c
e-mail: fabrito@df.ufcg.edu.br
ular, in the context of this theory, a black hole metric known as
loop quantum black hole (LQBH), or self-dual black hole, has
been proposed [10,11]. This solution corresponds to a quan-
tum corrected Schwarzschild solution and possess the inter-
esting property of self-duality. From this property, the black
hole singularity is removed and replaced by another asymp-
totically flat region, which is an expected effect in a quantum
gravity regime. Moreover, LQBHs have been pointed as a
possible candidate for dark matter [11,12] and as the building
blocks of a holographic version of loop quantum cosmology
[13].
In order to move black holes from a simple mathemati-
cal solution of the gravitational equations to objects whose
existence in nature is possible, a key point consists in to inves-
tigate black hole’s stability under perturbations. It is due to
the fact that an isolated black hole would never be found
in nature. In fact, complex distributions of matter such as
accretion disks, galactic nuclei, strong magnetic fields, other
stars, etc are always present around black holes, which in turn
actively interact with their surroundings. Even if all macro-
scopic objects and fields in space have been removed, a black
hole will interact with the vacuum around it, creating pairs
of particles and evaporating due to Hawking phenomena.
Besides, in the first moments after its formation, a black hole
is in a perturbed state due to gravitational collapse of matter.
In this way, a real black hole will be always in a perturbed
state.
A black hole’s response to a perturbation occurs by emit-
ting gravitational waves whose evolution corresponds, firstly,
to a relatively short period of initial outburst of radiation fol-
lowed by a phase where the black hole get going to vibrat-
ing into exponentially decaying oscillations, “quasinormal
modes”, whose frequencies and decay times depend only
on the intrinsic characteristics of the black hole itself, being
indifferent to the details of the collapse. Finally, at a very large
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