-
HEP/123-qed
Solution of Dirac equation and greybody radiation around a
regular Bardeen black hole surrounded by quintessence
Ahmad Al-Badawi
Department of Physics, Al-Hussein Bin Talal University,
P. O. Box: 20, 71111, Ma’an, Jordan. and
[email protected]
İzzet Sakallı
Physics Department, Arts and Sciences Faculty,
Eastern Mediterranean University, Famagusta,
North Cyprus via Mersin 10, Turkey. and
[email protected]
Sara Kanzi
Physics Department, Arts and Sciences Faculty,
Eastern Mediterranean University, Famagusta,
North Cyprus via Mersin 10, Turkey. and
[email protected]
(Dated: []; Received)
1
arX
iv:1
907.
1014
4v2
[gr
-qc]
9 N
ov 2
019
-
Abstract
The exact solutions of the Dirac equation that describe a
massive, non-charged particle with
spin−12 in the curved spacetime geometry of regular Bardeen
black hole surrounded by quintessence
(BBHSQ) are investigated. We first derive the Dirac equation in
the BBHSQ background using
a null tetrad in the Newman-Penrose formalism. Afterwards, we
separate the Dirac equation
into ordinary differential equations for the radial and angular
parts. The angular equations are
solved exactly in terms of standard spherical harmonics. The
radial part equations are transformed
into one-dimensional Schrödinger like wave equations with
effective potentials. The effect of the
quintessence on the regular Bardeen black hole is analyzed via
the physical behaviors of the effective
potentials. We also exhibit the potential graphs by changing the
quintessence parameters, magnetic
monopole charge parameter, and the frequency of the particle in
the physically acceptable regions.
Finally, we study the greybody factors of bosons and fermions
from the BBHSQ.
2
-
Contents
I. INTRODUCTION 3
II. Regular BBHSQ Space-time 5
III. Dirac Equation in BBHSQ 7
IV. Solution of Angular and Radial Equations 9
V. Greybody Radiation of Bosons from BBHSQ 15
VI. Greybody Radiation of Fermions from BBHSQ 17
VII. Conclusion 18
References 20
I. INTRODUCTION
At astrophysics scale, observations confirm the accelerating
expansion of the universe
[1]. To explain the expansion, it is suggested that the matter
content in the universe has
a negative pressure called dark energy [2]. There are two kinds
of negative pressure, first
the cosmological constant [3, 4] and the second is the so-called
quintessence that causes
the acceleration of the universe [5–7]. Quintessence is
characterized by the state equation:
p = wqρq where p is the pressure, ρq is the energy density, and
wq is the state parameter.
In addition, the scalar fields are also hypothetical forms of
dark energy where a broad types
of scalar field models have been suggested such as quintessence
[8–15], phantom models [16–
20], K-essence [21, 22], quintom [23, 24], and so on.
Ultimately, the difference between these
models is due to the magnitude of wq and for quintessence −1 ≤
wq ≤ −13 . The quintessencemodel refers to a minimally coupled
scalar field with a potential which decreases as the field
increases. Quintessence is a scalar field with an equation of
state where wq is given by the
potential energy and a kinetic term. Hence, quintessence is
dynamic and generally has a
density, and wq parameter varies with time. By contrast, a
cosmological constant is static,
possessing a fixed energy density, and it has wq = −1.
3
-
Usually, black holes (BHs) have singularity inside the horizon.
However, Bardeen BH
(BBH) is a regular BH which does not have singularity inside the
horizon. It was first
introduced by Bardeen [25]. Since Bardeen introduced his model,
many other models of
spherical symmetric regular BHs were presented in the literature
[26–36]. Later, in Ref. [37,
38] the authors have shown that BBH model is explained as the
gravitational field produced
by a nonlinear magnetic monopole. This explanation was extended
so that it includes
nonlinear electric charge. Moreover, regular BHs surrounded by
quintessence have received
major attention. Kiselev [8] obtained the first analytical
solutions with spherical symmetry
with quintessence surrounding the static BHs. Later, the
generalization of Kiselev solutions
was obtained by constructing their rotating counterpart [39,
40]. On the other hand, the
effect of quintessence on BHs have received considerable
attention and their thermodynamics
has been investigated. For example, in Ref. [41] the
thermodynamic properties of the
Bardeen black hole surrounded by quintessence (BBHSQ) was
thoroughly studied.
In this paper, we consider the Dirac equation in regular BBHSQ
space-time. Recall that
analytical solutions to the Dirac equation can be obtained in
several backgrounds [42–48]. A
reader is referred to see the complete analytical solutions to
the Dirac equation on de-Sitter
and anti de-Sitter space-time [49–53]. The Dirac equation that
we consider in this study
describes a massive and non-charged particle with spin-12. To
this end, we choose a null
tetrad in order to apply the Newman-Penrose (NP) formalism.
Next, we separate the Dirac
equation into ordinary differential equations and the get
coupled radial and angular parts.
The angular part equations are solved exactly in terms of
standard spherical harmonics.
The radial equations are transformed into one dimensional
Schrödinger like differential wave
equations with effective potentials. In addition, we investigate
the behavior of the effective
potentials by plotting them as a function of radial distance and
expose the effect of the
quintessence parameters, magnetic monopole charge parameter, and
the frequency of the
particle on them. Finally, we study the outcome of scattering a
wave off a BBHSQ in terms
of an absorption cross-section. The absorption cross-section,
which is a measure for the
probability of an absorption process, is directly connected to
the greybody factor. Then,
we compute the greybody factor, which is nothing but the
transmission probability for
an outgoing wave emitted from the event horizon of the BBHSQ to
reach the asymptotic
region [54–57]. Our main motivation in the present paper paves
the way to study the quasi-
normal modes associated to a field of spin-12
on the BBHSQ background. Further, the given
4
-
analytical expressions of the solution could be useful for the
study of the thermodynamical
properties of the spinor field in same background.
The plan of the paper is as follows. In the next section, we
give a brief discussion on the
regular BBHSQ space-time. In Sec. 3, we present the Dirac
equation in BBHSQ geometry
and decouple the equations into ordinary differential equations
for having the radial and
angular parts. We then obtain solutions of the angular and
radial equations in Sec. 4. The
influence of the quintessence parameter is investigated through
the behavior of the effective
potentials by plotting them as a function of radial distance in
the physically acceptable
region. Sections 5 and 6 are devoted to the studies of greybody
factors of the BBHSQ for
the spin-0 and spin-12
particles, respectively. Finally, we present our conclusions in
Sec. 7.
II. REGULAR BBHSQ SPACE-TIME
In this section, we shall give a brief introduction to BBHSQ
which was obtained by Kiselev
[8], who assumed a spherically symmetric static gravitational
field with the following energy-
momentum tensor:
T tt = Trr = ρq,
T θθ = Tφφ = −
ρq2
(3wq + 1) , (1)
where wq is the quintessence state parameter with range −1 ≤ wq
≤ −1/3 and ρq is thedensity of the quintessence matter given by
ρq = −3cwq
2r3(1+wq), (2)
where c is the positive normalization factor (c ≥ 0). The metric
of the regular BBHSQ canbe expressed as [41]
ds2 = −f (r) dt2 + f−1(r)dr2 + r2(dθ2 + sin2 θdφ2
)(3)
where f (r) has the following form
f (r) = 1− 2Mr2
(r2 + β2)3/2− cr3wq+1
. (4)
in which M is the mass of the BH and β can represent the
monopole charge of a self-
gravitating magnetic field described by a nonlinear
electrodynamics source or an electric
5
-
source with a field that does not behave as the Coulomb field
[58]. In fact, c term is related
to the density of quintessence:
ρq =−3cwq
2r3(wq+1).
The curvature of the metric (3) has the form of
R = 2T µµ = 2ρq (3wq − 1) , (5)
which admits a singularity at r = 0 if wq 6= {0, 13 ,−1}.
Therefore, the metric in (3) representsa spherically symmetric
solutions for the Einstein equations describing BBHSQ with the
energy-momentum tensors given in (1). This metric satisfies all
the required limits: when
(c = 0 = β), we have Schwarzschild BH metric; as (c = 0, β 6=
0), we get Bardeen BH; and(c 6= 0, β 6= 0) yields the BBHSQ.
We use the NP formalism [59, 60] to write and solve the Dirac
equation in the spacetime
of (3). Therefore let us define the complex null tetrad vectors
(l, n,m,m) for the metric (3)
where they satisfy the orthogonality conditions, (l.n = −m.m =
1) as
lµ = dt−dr
f (r),
nµ =1
2f (r) dt+
1
2dr,
mµ =−r√
2(dθ + i sin θdφ),
mµ =−r√
2(dθ − i sin θdφ), (6)
and
lµ = f (r) dt+ dr,
nµ =1
2dt− 1
2f (r) dr,
mµ =1√2r
(dθ +i
sin θdφ),
mµ =−r√
2(dθ − i sin θdφ), (7)
6
-
We determine the nonzero NP complex spin coefficients [60] as
follows
ρ = −1r, µ =
1
2r− Mr
(r2 + β2)3/2− c
2r3wq+2,
γ =Mr
2
[r2 − 2β2
(r2 + β2)5/2
]+
(3w + 1) c
4r2r3w, α = −β1 =
− cot θ2√
2r. (8)
III. DIRAC EQUATION IN BBHSQ
We write the Dirac equations in the NP formalism by using the
standard notation for
the spin coefficients [59, 60] as
(lµ∂µ + �− ρ)F1 + (mµ∂µ + π − α)F2 = iµ0G1, (9)
(nµ∂µ + µ− γ)F2 + (mµ∂µ + β1 − τ)F1 = iµ0G2, (10)
(lµ∂µ + �− ρ)G2 − (mµ∂µ + π − α)G1 = iµ0F2, (11)
(nµ∂µ + µ− γ)G1 −(mµ∂µ + β1 − τ
)G2 = iµ0F1. (12)
where F1, F2, G1and G2 represent the components of the wave
functions ”Dirac spinors”, the
mass of the particle µ0 =√
2µp and �, ρ, π, α, µ, γ, β1, τ are the spin coefficients and
bar over
a quantity denotes complex conjugation. We now study the Dirac
equations (9-12) in the
background of metric (3). To solve the Dirac equations, we will
consider the corresponding
Compton wave of the Dirac particle as in the form of F = F (r,
θ) ei(kt+mφ) , where k is the
frequency of the incoming wave and m is the azimuthal quantum
number of the wave. For
separable solutions, we assume [59],
rF1 = R1 (r)A1 (θ) exp [i (kt+mφ)] ,
F2 = R2 (r)A2 (θ) exp [i (kt+mφ)] ,
G1 = R2 (r)A1 (θ) exp [i (kt+mφ)] ,
rG2 = R1 (r)A2 (θ) exp [i (kt+mφ)] . (13)
Substituting the appropriate spin coefficients (8) and the
spinors (13) into the Dirac equa-
tions (9-12), we obtain the following set of equations
A1
(d
dr+ i
k
f
)R1 +
1√2R2LA2 = iµ0rR2A1,
7
-
r2fA2
(d
dr− ik
f+
2f + rf ′
2rf
)R2 −
√2R1L
†A1 = −2iµ0rR1A2,
A2
(d
dr+ i
k
f
)R1 −
1√2R2L
†A1 = iµ0rR2A2,
r2fA1
(d
dr− ik
f+
2f + rf ′
2rf
)R2 +
√2R1LA2 = −2iµ0rR1A1. (14)
where L and L†
are the angular operators, which are known as the laddering
operators:
L =d
dθ+
m
sin θ+
cot θ
2, L
†=
(d
dθ− m
sin θ+
cot θ
2
)(15)
From (14) and 15), we get (d
dr+ i
k
f
)R1 − iµ0rR2 = −λ1R2,
r2f
(d
dr− ik
f+
2f + rf ′
2rf
)R2 + 2iµ0rR1 = λ2R1,(
d
dr+ i
k
f
)R1 − iµ0rR2 = λ3R2,
r2f
(d
dr− ik
f+
2f + rf ′
2rf
)R2 + 2iµ0rR1 = −λ4R1, (16)
LA2 = λ1A1, L†A1 = λ2A2,
L†A1 = λ3A2, LA2 = λ4A1, (17)
The constants λ1, λ2, λ3, and λ4 are called the separation
constants. To obtain the radial
and the angular pair equations, we assume (λ4 = λ1 = −λ, λ2 = λ3
= λ), therefore (16) and(17) reduce to
(d
dr+ i
k
f
)R1 = (λ+ iµ0r)R2, (18)
(d
dr− ik
f+
2f + rf ′
2rf
)R2 =
1
r2f(λ− 2iµ0r)R1, (19)
LA2 = −λA1, L†A1 = λA2. (20)
8
-
IV. SOLUTION OF ANGULAR AND RADIAL EQUATIONS
Angular equations (20) can be rewritten as
dA1dθ
+
(cot θ
2− m
sin θ
)A1 = −λA2, (21)
dA2dθ
+
(cot θ
2+
m
sin θ
)A2 = λA1. (22)
which lead to the spin-weighted spheroidal harmonics whose
solution is given in terms of
standard spherical harmonics [60–62] as
A1,2 = Yml (θ) , (23)
with λ2 =(l + 1
2
)2.
The radial equations (18) and (19) can be rearranged as(d
dr+ i
k
f
)R1 = (λ+ iµ∗r)R2, (24)
r√f
(d
dr− ik
f+
2f + rf ′
2rf
)r√fR2 = (λ− iµ∗r)R1, (25)
where µ∗ is the normalized rest mass of the spin-12
particle.
Our task now is to put the radial equations (24) and (25) in the
form of one dimensional
wave equations. To this end, we follow the method applied by
Chandrasekhar’s book [59].
We start by making the following transformations
P1 = R1, P2 = r√fR2. (26)
Hence, (24) and (25) transform to
dP1dr
+ ik
fP1 =
1
r2f(λ+ iµ∗r)P2, (27)
dP2dr− ik
fP2 +
2f + rf ′
2rfP2 =
1
r2f(λ− iµ∗r)P1, (28)
Assumingdu
dr=
1
f, (29)
then, (27) and (28), in terms of the new independent variable u,
become
dP1du
+ ikP1 =
√f
r(λ+ iµ∗r)P2, (30)
9
-
dP2du− ikP2 +
2f + rf ′
2rfP2 =
√f
r(λ− iµ∗r)P1. (31)
where
u = r −√M2 − a2M2 − 2Mr tan−1
(r√
M2 − a2M2 − 2Mr
). (32)
Let us apply another transformation:
P1 = φ1 exp
[−i2
tan−1(µ∗rλ
)], P2 = φ2 exp
[i
2tan−1
(µ∗rλ
)], (33)
and then changing the variable u into r̂ as r̂ = u − 12k
tan−1(µ0rλ
), then (30) and (31) can
be written in the alternative forms:
dφ1dr̂
+ ikφ1 = Wφ2, (34)
dφ2dr̂− ikφ2 = Wφ1, (35)
where
W =2k√f (λ2 + µ2∗r
2)3/2
2kr (λ2 + µ2∗r2) + rfλµ∗
. (36)
To put (34) and (35) into one dimensional wave equations, we
define
2φ1 = ψ1 + ψ2, 2φ2 = ψ1 − ψ2. (37)
Hence, (34) and (35) becomedψ1dr̂−Wψ1 = −ikψ2, (38)
dψ2dr̂
+Wψ2 = −ikψ1. (39)
Finally, we end up with the following pair of one dimensional
wave equations
d2ψ1dr̂2
+ k2ψ1 = V+ψ1, (40)
d2ψ2dr̂2
+ k2ψ2 = V−ψ2, (41)
where the effective potentials can be obtained from
V± = W2 ± dW
dr̂. (42)
We calculate the effective potentials as
10
-
V± =r2B3
D2
(1− 2Mr
2
(r2 + β2)3/2− cr3wq+1
)
± rD2
√B3 − 2Mr
2B3
(r2 + β2)3/2− cB
3
r3wq+1
((r −M)B + 3r3µ2∗ −
6r5µ2∗M
(r2 + β2)3/2− 3r
3µ2∗c
r3wq+1
)
∓ r3B5/2
D3
(1− 2Mr
2
(r2 + β2)3/2− cr3wq+1
)3/2 [(2rB + 2r3µ2∗
)+
(r −M)λµ∗k
], (43)
where
B =(λ2 + µ2∗r
2), D = r2B +
λµ∗r2
2k
(1− 2Mr
2
(r2 + β2)3/2− cr3wq+1
). (44)
Let us note that the effective potentials for the case of
massless Dirac particle (neutrino)
can be obtained by setting µ∗ = 0 in (43) namely
V± = λ2
(1
r2− 2M
(r2 + β2)3/2− cr3wq+3
)± λ (r −M)
r3
√√√√(1− 2Mr2(r2 + β2)3/2
− cr3wq+1
)
∓ 2λr2
(1− 2Mr
2
(r2 + β2)3/2− cr3wq+1
)3/2, (45)
whereas substituting c = 0, β 6= 0 reduces to the effective
potentials for BBH
V± =r2B3
D2
(1− 2Mr
2
(r2 + β2)3/2
)± rD2
√B3 − 2Mr
2B3
(r2 + β2)3/2
((r −M)B + 3r3µ2∗ −
6r5µ2∗M
(r2 + β2)3/2
)
∓ r3B5/2
D3
(1− 2Mr
2
(r2 + β2)3/2
)3/2 [(2rB + 2r3µ2∗
)+
(r −M)λµ∗k
], (46)
where
B =(λ2 + µ2∗r
2), D = r2B +
λµ∗r2
2k
(1− 2Mr
2
(r2 + β2)3/2
). (47)
The complete solution of (40) and (41) can be obtained by the
WKB approximation method
(for more details, a reader is referred to [63, 64]). To study
the asymptotic behavior of the
potentials (43), we can expand it up to order O(1r
)3. The potentials (43) for BBHSQ (here,
we choose wq = −13 ) behave as
V± ' µ2∗ (1− c)−2Mµ2∗ ± µ∗c
√1− c
r+ η±
(1
r
)2+O
(1
r
)3, (48)
11
-
where
η± =
[λ2 (1− c)− λµ∗
k
(c2 − 2c+ 1
)± Mµ∗ (3c∓ 4)√
1− c ± 5Mµ∗√
1− c], (49)
In case of neutrino, the potentials simplified to
V± ' λ2(
1
r
)2+O
(1
r
)3. (50)
The first term in (48) represents the constant value of the
potential at the asymptotic infinity.
The second term corresponds to the monopole type potential,
while the third term represents
the dipole type potential. As seen from the asymptotic expansion
of the potentials (48), the
effect of adding the quintessence term(
cr3wq+1
)to the lapse function f (r) in the metric (3)
is observed at all terms. From (43), we notice that the
potentials V± depend on the c, β,
and k parameters. We would like to remind that the effect of the
quintessence on BH has
received considerable attention in General Relativity [65, 66].
Therefore, it is interesting to
investigate the quintessence affect the massive and non-charged
Dirac particle.
It is obvious from (43) that the potentials become singular when
D = 0. They also
have local extrema when (dV±/dr) = 0, however, these local
extrema are very complicated
algebraic equation to be solved. To understand the physical
behavior of the potentials (43)
in the physical region and to expose the effect of the
quintessence parameters c, magnetic
monopole charge parameter β and the frequency k of the
particle,we make two-dimensional
and three-dimensional plots of the potentials for massive
particles. In all plots, we choose
the fermion’s mass µ∗ = 0.12 and the frequency k = 0.2 such that
k > µ∗. The effective
potential V± (43) versus r for different values of c is depicted
in Fig. 1. It is observed that
the potentials have sharp peaks for all values of c. We notice
that when the normalization
factor c increases, the sharpness of the potential peaks also
increases. We deduse From Fig.
1 that, a massive Dirac particle in the presence of quintessence
matter (c 6= 0) meets with ahigh potential barrier, which causes a
decreasing in their kinetic energies. However, without
quintessence (c = 0) the particle encounters a low potential
barrier, which means that the
Dirac particle’s kinetic energy would increase.
In Fig. 2, we investigate the behavior of the effective
potentials by obtaining potential
curves for some specific values of the frequency k while keeping
the normalization factor con-
stant (c = 0.01). We can see that the potentials have peaks for
all values of k. Again, while
the frequency increases, the potentials barrier increases, and
potentials behave similarly in
the sufficiently large distances.
12
-
1 2 3 4r
0.1
0.2
0.3
0.4
0.5
0.6
0.7
V
V
V
c = 0.05 c = 0.03 c = 0.0
FIG. 1: Family of potential graphs V± for different values of c
with µ∗ = 0.12,
k = 0.2,M = 0.5, λ = 1, β = 0.25, and wq = −56 .
1 2 3 4r
0.1
0.2
0.3
V
VV
k = 1.0 k= 0.5 k = 0.2
FIG. 2: Family of potential graphs V± for different values of
frequency k with µ∗ = 0.12,
c = 0.01, M = 0.5, λ = 1, β = 0.25, and wq = −1.
The effect of the quintessence term can be observed explicitly
by making a three-
dimensional plot of the potential with respect to the
normalization factor c and the radial
distance r. In Fig. 3, we observe a three-dimensional small peak
for values of normalization
factor c. As the value of radial distances increases, potentials
level off. Fig. 4 represents
the three-dimensional plot of potential with respect to
frequency k and the radial distance.
It is seen from Fig. 4 that; sharp peaks are clear for high
frequencies. The effect of the
monopole charge of a self-gravitating magnetic field β on the
the potentials for the massive
charged spin-12
particle can be observed in Fig. 5. We deduce that; high
potential barriers
are observed for small values of β whereas for large values the
potential barriers decrease.
13
-
0.5
1
1.5
2
r
0
0.02
0.04
0.06
c
0
0.1
0.2V
0.5
1
1.5
2
r
FIG. 3: Three-dimensional plots of potential graphs V+ for
different values of c with
µ∗ = 0.12, M = 1, k = 0.4, λ = 0.4, β = 0.25, and wq = −13 .
Again, the potential levels decrease for large values of
distance r and asymptote behavior is
manifested. Finally, the three-dimensional plot of potential
with varying the state parameter
wq is shown in Fig. 6. We see from Fig. 6 that the potential
levels are constant for different
values of wq which implies that the state parameter wq does not
affect the potentials.
2
3
4
r0.2
0.4
0.6
0.8
1
k
0
0.05
0.1
0.15
V
2
3
4
r
FIG. 4: Three-dimensional plots of potential graphs V+ for
different values of k with
µ∗ = 0.12, M = 1, c = 0.05, λ = 0.4, β = 0.25, and wq = −1.
14
-
2
3
4
r
0
0.25
0.5
0.75
1
0
0.05
0.1
0.15
V
2
3
4
r
FIG. 5: Three-dimensional plots of potential graphs V+ for
different values of β with
µ∗ = 0.12, M = 1, c = 0.05, λ = 0.4, k = 0.4, and wq = −13 .
1
2
3
4
r
-1
-0.8
-0.6
-0.4
0
0.2
0.4V
1
2
3
4
wq
FIG. 6: Three-dimensional plots of potential graphs V+ for
different values of wq with
µ∗ = 0.12, M = 1, c = 0.05, λ = 0.4, k = 0.4, and β = 0.25.
V. GREYBODY RADIATION OF BOSONS FROM BBHSQ
In this section, we evaluate the greybody factor of BBHSQ for
spin−0 particles. For thesake of simplicity, we consider the
massless Klein-Gordon equation [67]
�Ψ(t, r, θ, φ) = 0, (51)
15
-
where the box symbol denotes the Laplacian operator [68]:
� =1√−g∂µ
√−ggµν∂ν . (52)
By considering the metric (3) of BBHSQ, then (51) reads
− f−1∂2t Ψ + r−2∂r(r2f∂r
)Ψ +
r−2
sin θ∂θ (sin θ∂θ) Ψ +
r−2
sin2 θ∂2φΨ = 0. (53)
Where the scalar field can be defined as
Ψ = p (r)A (θ) exp (−iωt) exp (imφ) , (54)
here ω denotes the frequency of the wave. Substituting the the
scalar field in (53), one
obtainsd2p(r)
dr2+
(f−1
df
dr+ 2r−1
)dp(r)
dr+(ω2f−2 − λ̂r−2f−1
)p (r) = 0, (55)
where λ̂ = l (l + 1) is the eigenvalue coming from the physical
solution of the angular
equation of A (θ), which is nothing but the standard spherical
harmonics [60–62]. By chang-
ing the variable in a new form as p = ur, the radial wave
equation (55) recasts into a one
dimensional Schrödinger like equation as follows
d2u
dr2∗+(ω2 − Veff
)u = 0, (56)
where the effective potential for BBHSQ is given by
Veff = f
(λ
r2+
1
r
df
dr
). (57)
To evaluate the greybody factor we use [59, 69]
σl (ω) ≥ sech2(∫ +∞−∞
℘dr∗
), (58)
in which r∗ is the tortoise coordinate:dr∗dr
= 1f(r)
,and
℘ =1
2h
√(dh
dr
)2+ (ω2 − Veff − h2)2. (59)
In (59), h is a particular positive function that satisfies the
following conditions: h(r?) > 0
and h(−∞) = h(∞) = ω [70]. Here, without loss of generality, we
simply set it as h = ω[69, 70], resulting in that the integration
of (58) becomes
σl (ω) ≥ sech21
2ω
∫ +∞rh
(λ
r2+
1
r
df
dr
)dr. (60)
16
-
After making a straightforward calculation, one finds
σl (ω) ≥ sech2 1
2ω
l (l + 1)rh
+c (3wq + 1)
(3wq + 2) r3wq+2h
− 2Mβ2
+2M
β2√
1 + β2
r2h
+2M
r2h
(1 + β
2
r2h
)3/2 ,
(61)
where rh represents the event horizon.
σl(ω)
ω
wq = −1, rh ' 0.92
wq = −0.75, rh ' 0.88
wq = −0.5, rh ' 0.77
wq = −0.6, rh ' 0.83
wq = −0.35, rh ' 0.42
wq = −0.85, rh ' 0.89
FIG. 7: σl (ω) versus ω graph. The plots are governed by Eq.
(61). For different wq values,
the corresponding event horizons (i.e, f(rh) = 0) are
illustrated. The physical parameters
for this plot are chosen as M = l = c = 1, and β = 2.
We depict the greybody factors of the BBHSQ in Fig. 7. As seen
from Fig. 7, the
values of wq and event horizon (rh) are linearly proportional to
each other. It is obvious
that greybody radiation strictly depends also on the state
parameter wq. According to the
information we obtained from the graph, a similar radiation
emission occurs around critical
wq values (−13 and −1). However, while wq value moves away from
those critical values,then radiation may decrease or increase
depending on wq.
VI. GREYBODY RADIATION OF FERMIONS FROM BBHSQ
In this section, we shall derive the fermionic greybody factors
of the neutrinos emitted
from BBHSQ. To this end, we consider the case of wq = −13 in
order to obtain analytical
17
-
results from Eq. (58). In the case of wq = −13 , the potentials
(45) can be rewritten as
V± =λ
r2f ± λ (r −M)
r3
√f ∓ 2λ
r2f 3/2, (62)
in which
f = 1− 2Mr2
(r2 + β2)3/2− c. (63)
Following the procedure described in the previous section [see
Eqs. (58)-(62)], one can
get
σ±l (ω) ≥ sech2(
1
2ω
∫ +∞rh
[λ
r2±(λ
r2− λM
r3
)1√f∓ 2λr2
√f
]dr
), (64)
in which σ+l (ω) and σ−l (ω) stand for the greybody factors of
the spin-up and spin-down
fermions, respectively. After performing some tedious
computations, the greybody factors
of the fermions can be obtained as follows
σ±l (ω) ≥ sech2{
1
2ω
[λ
r±(
λ
rh√
1− c +Mλ
2 (1− c)3/2 r2h+
M2λ
2 (1− c)5/2 r3h+
−3λMβ2(1− c)2 + 5M3λ8r4h(1− c)3
√1− c +
−9M2β2λ(1− c)2 + 354M4λ
10r5h (1− c)4√
1− c− λM
2√
1− cr2h− λM
2
3 (1− c)3/2 r3h−
3λM3
8r4h (1− c)5/2− −3λM
2β2(1− c)2 + 5λM410r5h (1− c)3
√1− c
− −9λM3β2(1− c)2 + 35
4λM5
12r6h (1− c)4√
1− c
)
∓(
2λ√
1− crh
− λM√1− cr2h
− λM2
3 (1− c)3/2 r2h+λ√
1− c (3Mβ2(1− c)2 −M3)4r4h (1− c)3
+λ√
1− c (12M2β2(1− c)2 − 5M4)20r5h (1− c)4
)]}( for 0 ≤ c < 1) , (65)
As it can been seen from Figs. (8) and (9), the greybody factors
of spin-up and spin-down
fermions exhibit almost the same behaviors as c changes.
VII. CONCLUSION
In this paper, we have investigated the exact solutions of the
Dirac equations that de-
scribe a massive, non-charged particle with spin-12
in the curved space-time geometry of
BBHSQ, using NP (null tetrad) formalism. By employing an axially
symmetric ansatz for
the Dirac spinors, we decouple equations into angular and radial
parts. The angular equa-
tion leads to the spin-weighted spheroidal harmonics with
eigenvalue λ2 =(l + 1
2
)2. The
18
-
c = 0 rh ∼= 0.33
c = 0.05 rh ∼= 0.31
σ+l (ω
(a) For c ≈ 0 values.
σ+l (ω)
c = 0.98, rh ∼= 99.99
c = 0.8, rh ∼= 9.96
c = 0.5, rh ∼= 3.90
c = 0.1, rh ∼= 2.04
(b) For c > 0 values.
FIG. 8: σ+l (ω) versus ω graph for the case of wq = −13 . The
plots are governed by Eq.(65). For different c values, the
corresponding event horizons (i.e, f(rh) = 0) are
illustrated. The physical parameters for these plots are chosen
as M = l = 1, and β = 0.5.
σl ω( )−
c = 0, rh ∼= 0.33
c = 0.05, rh ∼= 0.31
(a) For c ≈ 0 values.
c = 0.1, rh ∼= 2.04
c = 0.5, rh ∼= 3.90
c = 0.8, rh ∼= 9.96
c = 0.98, rh ∼= 99.99
σl ω( )−
(b) For c > 0 values.
FIG. 9: σ−l (ω) versus ω graph for the case of wq = −13 . The
plots are governed by Eq.(65). For different c values, the
corresponding event horizons (i.e, f(rh) = 0) are
illustrated. The physical parameters for these plots are chosen
as M = l = 1, and β = 0.5.
radial equations were reduced to pair of one-dimensional
Schrodinger-like wave equations
with effective potentials for the Dirac particle. We then
studied the potentials by plotting
them as a function of radial distance. Thus, the effect of the
quintessence term on the BBH
are unfolded. We revealed that potentials barriers having
quintessence matter become more
higher than the potentials without the quintessence. We also
showed that, as the frequency
increases, potentials levels increase as well. However, as the
magnetic monopole charge
19
-
parameter β increases, the potential levels decrease whereas the
potentials do not change
for varying the state parameter wq. Remarkably, we depicted how
the greybody factors of
bosons and fermions vary with the quintessence state parameters
wq (see Fig. 7) and c (see
Figs. 8 and 9), respectively.
In future work, we will extend our analysis to the Dirac
equation of charged massive
fermionic waves propagating in the rotating geometry of the
BBHSQ. In this way, we plan
to analyze the effect of quintessence on the stationary
spacetimes using fermions. For this
purpose, we shall also consider the Janis-Newman algorithm [71]
for the static BBHSQ (3).
[1] P. J. E. Peebles and B. Ratra, Rev. Mod. Phys. 75, 559
(2003).
[2] P.A. Ade et al., Astron. Astrophys. 571, A1 (2014).
[3] R. Caldwell, Braz. J. Phys. 30, 215 (2000).
[4] J.A.S. Lima, Braz. J. Phys. 34(1A), 194 (2004).
[5] S. Perlmutter et al., Astrophys. J. 517, 565 (1999).
[6] A. G. Riess et al., Astron. J. 116, 1009 (1998).
[7] P. M. Garnavich et al., Astrophys. J. 509, 74 (1998).
[8] V. V. Kiselev, Class. Quantum Grav. 20, 1187 (2003).
[9] Z. Shuang-Yong, Phys. Lett. B 660, 7 (2008).
[10] C. Wetterich, Nucl. Phys. B 302, 668 (1988).
[11] B. Ratra and P. J. E. Peebles, Phys. Rev. D 37, 3406
(1988).
[12] R. R. Caldwell, R. Dave, and P. J. Steinhardt, Phys. Rev.
Lett. 80, 1582 (1998).
[13] C. Armendariz-Picon, V. Mukhanov, and P. J. Steinhardt,
Phys. Rev. Lett. 85, 4438 (2000).
[14] C. Armendariz-Picon, V. Mukhanov, and P. J. Steinhardt,
Phys. Rev. D 63, 103510 (2001).
[15] T. Chiba, T. Okabe, and M. Yamaguchi, Phys. Rev. D 62,
023511 (2000).
[16] A. E. Schulz and M. White, Phys. Rev. D 64, 043514
(2001).
[17] R. R. Caldwell, Phys. Lett. B 545, 23 (2002).
[18] S. Nojiri, and S. D. Odintsov, Phys. Lett. B 562, 147
(2003).
[19] P. Singh, M. Sami, and N. Dadhich, Phys. Rev. D 68, 023522
(2003).
[20] L. P. Chimento and R. Lazkoz, Phys. Rev. Lett. 91, 211301
(2003).
[21] T. Chiba, T. Okabe, and M. Yamaguchi, Phys. Rev. D 62,
023511 (2000).
20
-
[22] R. J. Scherrer, Phys. Rev. Lett. 93, 011301 (2004).
[23] Z.K. Guo, Y.-S. Piao, X. Zhang, and Y.Z. Zhang, Phys. Lett.
B 608,177 (2005).
[24] J.-Q. Xia, B. Feng, X. Zhang, Phys. Rev. D 74, 123503
(2006).
[25] J. M. Bardeen, Non-singular general-relativistic
gravitational collapse, Proceedings of GR5,
Tiflis, Georgia, U.S.S.R. page 174 (1968).
[26] A. Borde, Phys. Rev. D 50, 3692 (1994).
[27] A. Borde, Phys. Rev. D 55, 7615 (1997).
[28] C. Barrabes and V. P. Frolov, Phys. Rev. D 53, 3215
(1996).
[29] A. Cabo and E. Ayon-Beato, Int. J. Geom. Methods Mod. Phys.
A14, 2013 (1999).
[30] S. A. Hayward, Phys. Rev. Lett. 96, 031103 (2006).
[31] E. Ayon-Beato and A. Garcia, Phys. Rev. Lett. 80, 5056
(1998).
[32] E. Ayon-Beato and A. Garcia, Phys. Lett. B 464, 25
(1999).
[33] E. Ayon-Beato and A. Garcia, Gen. Relativ. Gravit. 37, 635
(2005).
[34] M. S. Ma, Ann. Phys. (Amsterdam) 362, 529 (2015).
[35] E. Ayon-Beato and A. Garcia, Phys. Lett B 493, 149
(2000).
[36] Z.Y. Fan and X. Wang, Phys. Rev. D 94, 124027 (2016).
[37] E. Ayen-Beato and A. Garcia, Gen. Rel. and Grav. 31, 629,
(1999).
[38] C. Bambi and L. Modesto, Phys. Lett. B 721, 329 (2013).
[39] B. Toshmatov, Z. Stuchĺık, and B. Ahmedov, Eur. Phys. J.
Plus 132, 98 (2017).
[40] S. G. Ghosh, Eur. Phys. J. C 76, 1 (2016).
[41] K. Ghaderi and B. Malakolkalami, Astrophys. Space Sci. 361,
161 (2016).
[42] B. Mukhopadhyay, Class. Quantum Grav. 17, 2017 (2000).
[43] A. Zecca, Int. J. Theor. Phys. 45, 12 (2006).
[44] H. Cebeci and N. Ozdemir, Class. Quantum Grav. 30, 175005
(2013).
[45] T. Birkandan and M. Hortacsu, J. Math. Phys. 48, 092301
(2007).
[46] A. Al-Badawi and I. Sakalli, J. Math. Phys. 49, 052501
(2008).
[47] A. Al-Badawi, Gen. Rel. and Grav. 50, 16 (2018).
[48] A. Al-Badawi and M .Q. Owaidat, Gen. Rel. and Grav. 49, 110
(2017).
[49] C. A. Sporea, Mod. Phys. Lett. A 30, 1550145 (2015).
[50] I. I. Cotaescu, Phys. Rev. D 60, 124006 (1999).
[51] I. I. Cotaescu, Phys. Rev. D 65, 084008 (2002).
21
-
[52] I. I. Cotaescu and C. Crucean, Mod. Phys. Lett. A 22, 3707
(2008).
[53] I. I. Cotaescu, R. Racoceanu, and C. Crucean, Mod. Phys.
Lett. A 21, 1313 (2006).
[54] Y. S. Myung and H. W. Lee, Class. Quantum Grav. 20, 3533
(2003).
[55] T. Harmark, J. Natario, and R. Schiappa, Adv. Theor. Math.
Phys. 14, 727 (2010).
[56] I. Sakalli and O. A. Aslan, Astropar. Phys. 74, 73
(2016).
[57] H. Gursel and I. Sakalli, Adv. High Energy Phys. 2018,
8504894 (2018).
[58] P. Gaete and I. Schmidt, Int. J. Mod. Phys. A 19, 3427
(2004).
[59] S. Chandrasekhar, The Mathematical Theory of Black Holes
(Clarendon, London, 1983).
[60] E. T. Newman and R. Penrose, J. Math. Phys. 3, 566
(1962).
[61] S. K. Chakrabarti, Proc. R. Soc. A 391, 27 (1984).
[62] J. N. Goldberg, A. J. Macfarlane, E. T. Newman, F.
Rohrlich, and E. C. G. Sudarsan, J.
Math. Phys. 8, 2155 (1967).
[63] A. M. Davydov, Quantum Mechanics (2 nd Edition) (Oxford,
Pergamon, 1976).
[64] J. Mathews and R. L. Walker, Mathematical Methods of
Physics (2 nd Edition) (W.A. Ben-
jamin, New York, 1970).
[65] K. Ghaderi and B. Malakolkalami, Nucl. Phys. B 903, 10
(2016).
[66] S. Mahamat and B. B. Thomas, Eur. Phys. J. C 78, 325
(2018).
[67] R. M. Wald, General Relativity (The University of Chicago
Press, Chicago and London, 1984).
[68] S. H. Mazharimousavi, I. Sakalli, and M. Halilsoy, Phys.
Lett. B 672, 177 (2009).
[69] S. Kanzi and I. Sakalli, arXiv:1905.00477 (to be appeared
in Nuclear Physics B).
[70] Y. G. Miao and Z. M. Xu, Phys. Lett. B 772, 542 (2017).
[71] H. Erbin, Universe 3, 19 (2017).
22
http://arxiv.org/abs/1905.00477
ContentsI INTRODUCTIONII Regular BBHSQ Space-timeIII Dirac
Equation in BBHSQIV Solution of Angular and Radial EquationsV
Greybody Radiation of Bosons from BBHSQVI Greybody Radiation of
Fermions from BBHSQVII Conclusion References
