Eur. Phys. J. C (2018) 78:858
https://doi.org/10.1140/epjc/s10052-018-6352-5
Regular Article - Theoretical Physics
Greybody factors and quasinormal modes for a nonminimally
coupled scalar field in a cloud of strings in (2 + 1)-dimensional
background
Ángel Rincón
1,a
, Grigoris Panotopoulos
2,b
1
Instituto de Física, Pontificia Católica Universidad de Chile, Av. Vicuña Mackenna 4860, Santiago, Chile
2
Centro de Astrofísica e Gravitação, Departamento de Física, Instituto Superior Técnico-IST, Universidade de Lisboa-UL, Av. Rovisco Pais,
1049-001 Lisboa, Portugal
Received: 24 August 2018 / Accepted: 16 October 2018 / Published online: 25 October 2018
© The Author(s) 2018
Abstract We study the propagation of a probe massless
nonminimally coupled scalar field in a fixed gravitational
background of a cloud of strings in (2 + 1) dimensions. We
obtain exact analytical expressions for the reflection coef-
ficient, the absorption cross section, the decay rate as well
as the quasinormal frequencies. The impact of the nonmini-
mal coupling is investigated in detail. Our results show that
universality is not respected in general, and that scalar per-
turbations are stable.
1 Introduction
Black holes (BHs hereafter), a generic prediction of Ein-
stein’s General Relativity, are objects of paramount impor-
tance both for classical and quantum gravity. Greybody fac-
tors and quasinormal modes are two topics related to black
hole physics of particular interest. On the one hand, Hawking
radiation [1–3], since it is as a manifestation of a quantum
effect in curved spacetime, has always attracted a lot of inter-
est although it has not been detected in the Universe yet. The
emitted particles feel an effective potential that back scat-
ters part of the emitted radiation back into the black hole.
The greybody factor is a frequency dependent quantity that
measures the deviation from the original black body radi-
ation spectrum, and provides us with valuable information
about the black hole horizon structure [4]. The propagation
and relativistic scattering of fields has been investigated both
in asymptotically flat spacetimes and in background with a
non-vanishing cosmological constant. For a partial list see
e.g. [5–24] and references therein.
a
e-mail: arrincon@uc.cl
b
e-mail: grigorios.panotopoulos@tecnico.ulisboa.pt
On the other hand, LIGO historical direct detection of
gravitational waves [25–27] from black hole mergers has
opened a completely new window to our Universe, and allows
us to test gravity and probe strong gravitational fields. Con-
sequently, lately there is an increasing interest in black hole
perturbations [28–33] and quasinormal modes of black holes,
intimately related to the ring down phase after the formation
of the distorded object during the merging of two black holes.
When a black hole is perturbed the geometry of spacetime
undergoes dumped oscillations, which are characterized by
the quasinormal modes with a non-vanishing imaginary part.
Chandrasekhar’e monograph provides us with a comprehen-
sive overview of black hole perturbations [34]. Quasinormal
modes of black holes have been extensively studied, and for
excellent reviews see e.g. [35–37].
The Bañados, Teitelboim and Zanelli (BTZ) black hole
solution [38–40] in (1 + 2) dimensions marked the birth
of the interest in lower-dimensional gravity. The absence of
propagating degrees of freedom as well as its deep connec-
tion to the Chern–Simons term only [41–43] make three-
dimensional gravity special, and at the same time a frame-
work which allow us to get insight into realistic black holes in
four dimensions by studying a mathematically simpler three-
dimensional system. The BTZ black hole is sourced by a
negative cosmological constant, but other possibilities, such
as scalar or electromagnetic fields [44–47], also exist. What
is more, looking for a complete theory of quantum gravity,
black hole solutions that admit scale-dependent couplings,
have been recently investigated. For an incomplete list see
[48–57] and references therein.
One option less studied in the literature, which leads to
a black hole solution alternative to the BTZ one, is a cloud
of strings [58]. The matter contribution is described by the
Nambu–Goto action, which is well-known both from string
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