Physics Letters B 798 (2019) 135011
Contents lists available at ScienceDirect
Physics Letters B
www.elsevier.com/locate/physletb
Black holes quasinormal modes, Loop Quantum Gravity Immirzi
parameter and nonextensive statistics
Everton M.C. Abreu
a,b,c
, Jorge Ananias Neto
b
, Edésio M. Barboza Jr.
d
, Bráulio B. Soares
e
a
Departamento de Física, Universidade Federal Rural do Rio de Janeiro, 23890-971, Seropédica, RJ, Brazil
b
Departamento de Física, Universidade Federal de Juiz de Fora, 36036-330, Juiz de Fora, MG, Brazil
c
Programa de Pós-Graduação Interdisciplinar em Física Aplicada, Instituto de Física, Universidade Federal do Rio de Janeiro, 21941-972, Rio de Janeiro, RJ, Brazil
d
Departamento de Física, Universidade do Estado do Rio Grande do Norte, 59610-210, Mossoró-RN, Brazil
e
Departamento de Ciência e Tecnologia, Universidade do Estado do Rio Grande do Norte, Natal, RN, Brazil
a r t i c l e i n f o a b s t r a c t
Article history:
Received
8 April 2019
Received
in revised form 12 September
2019
Accepted
6 October 2019
Available
online 9 October 2019
Editor:
N. Lambert
Keywords:
Loop
Quantum Gravity
Immirzi
parameter
Tsallis
statistics
It is argued that, using the black hole area entropy law together with the Boltzmann-Gibbs statistical
mechanics and the quasinormal modes of the black holes, it is possible to determine univocally the
lowest possible value for the spin j in the context of the Loop Quantum Gravity theory which is j
min
= 1.
Consequently, the value of Immirzi parameter is given by γ = ln 3/(2π
√
2). In this paper, we have shown
that if we use Tsallis microcanonical entropy rather than Boltzmann-Gibbs framework then the minimum
value of the label j depends on the nonextensive q-parameter and may have values other than j
min
= 1.
© 2019 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
.
Loop Quantum Gravity (LQG) [1]is a theory of quantum gravity
that proposes to unify quantum field theory and general relativity.
The main attractive point in LQG is, in principle, the possibility to
describe the quantum spacetime in a nonperturbative background-
independent
form. The Hilbert space of LQG is formed by spin
networks which are graphs with edges that carry labels such as
j = 0, 1/2, 1, 3/2, .... In LQG, the area of a given region of space
has a discrete spectrum in such a way that, if a surface is inter-
sected
or punctured by the spin network edge that carries the
label j, then the surface carries an area element written as [2–4]
a( j) = 8πl
2
p
γ
j( j +1), (1)
where l
p
is the Planck length and γ is the so-called Immirzi pa-
rameter
[5]. Eq. (1)is an relevant prediction of LQG. However,
this important relation is weakened by the fact that the Immirzi
parameter is, in principle, undetermined. By definition, the Im-
mirzi
parameter carries the measure of the size of the quantum
E-mail addresses: evertonabreu@ufrrj.br (E.M.C. Abreu), jorge@fisica.ufjf.br
(J.A. Neto),
edesiobarboza@uern.br (E.M. Barboza), brauliosoares@uern.br
(B.B. Soares).
of area in Planck’s units. One way to compute the Immirzi pa-
rameter,
by solving the problem mentioned above, can be carried
out with the help of quasinormal modes in the black holes the-
ory.
Quasinormal modes are damped oscillations that appear in
the perturbation equations of the Schwarzchild geometry. These
solutions were initially found by Regge and Wheeler [6]. This pro-
cedure,
as we will see, connects the relation between area and
mass of a Schwarzschild black hole to the area produced by the
spin network in the context of LQG [7]. On the other hand, the
interested reader can notice that an interesting work [8], using a
conformal gauge structure in a novel generalized Holst action, ob-
tains
non-fixed values of the Immirzi parameter, i.e., the Immirzi
parameter may be not beset by ambiguities in this approach.
In
this paper, following the work of Dreyer [7], we will use the
quasinormal modes to obtain a new equation for the minimum
value of the spin j that appears in Eq. (1)as a consequence of
a nongaussian statistics, namely, Tsallis’ statistics. It is important
to mention that in Ref. [7]the author has considered Boltzmann-
Gibbs
(BG) statistics.
Tsallis’
formalism [9], which is an extension of BG statistical
theory, defines a nonextensive (NE), i.e., a nonadditive entropy
such as
https://doi.org/10.1016/j.physletb.2019.135011
0370-2693/
© 2019 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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