#### Yukawa-like confinement potential of a scalar particle in a Gödel-type spacetime with any l

Eur. Phys. J. C
Yukawa-like confinement potential of a scalar particle in a Gödel-type spacetime with any l
Mahdi Eshghi 2
Majid Hamzavi 0 1
0 Department of Mathematics and Statistics, University of Texas at El Paso , El Paso, TX , USA
1 Physics Department, Zanjan University , Zanjan , Iran
2 Department of Physics, Imam Hossein Comprehensive University , Tehran , Iran
In this paper, we investigate the behavior of a scalar particle within the Yukawa-like potential in a Gödeltype space-time for any l. The behavior of a spinless particle is analyzed in the presence of a topological defect with analytical solutions to the Klein-Gordon equation. By using the generalized series and Nikiforov-Uvarof (NU) methods, we deduce the energy eigenvalues and eigenfunctions. We have discussed on the obtained results.
1 Introduction
Physically space-time is a mathematical model. This model
combines space and time into a single continuum. All
physical quantities can be described by a space-time. In fact, we
can mean a four-dimensional continuum having spatial and
temporal coordinates. For example, Albert Einstein could
describe the gravity of a space-time as a geometric property
and linked with the geometric quantity “curvature” in general
relativity. In space-times, topological defects and the
curvature can be presented by the Riemann curvature tensor or
Riemann–Christoffel tensor. Also, in quantum gravity,
topological defects can be found in the global monopole [
1
], and
the cosmic string [
2–4
]. In fact, a quantum system can be
affected by the topology and local curvature of the
spacetime in a gravitational field. In this regard, see the energy
levels of a hydrogen atom placed in the gravitational fields
of a cosmic string or a global monopole [
5
], and the
correction of energy levels depends on the features of the
specific space-time. Also, during the last decades, some authors
have widely studied the influences of topological defects on
the non-relativistic or relativistic quantum systems such as
bound states of electrons and holes to a disclination [
6
], the
influence of the screw dislocation on the energy levels of
nonrelativistic electrons [
7
], screw dislocations driven growth of
nanomaterials [
8
], topological defects space times [
9
], the
Landau levels in the presence of topological defects [
10,11
],
the influence of Coulomb potential and quantum oscillator
in conical spaces [12], exact solutions of the Klein–Gordon
equation in cosmic string space-time [
13,14
] and others [
15–
17
].
In this discussion, the Gödel space-time is very
important in understanding the nature of time, rotation and
gravitation. Also, we know that the Kurt Gödel cosmological
solution has been obtained by Gödel for the rotating body in
1949 [
18
]. This stationary solution was calculated by
considering a cylindrically symmetric for the Einstein equations
of general relativity. In fact, it has interesting properties in
area of physic. Further, the most appealing property of the
Gödel space-time is the existence of closed time-like curves,
that makes time travel a theoretical possibility and has
certified an enduring appeal in the model [
19–21
]. Moreover, the
Gödel space-time is not a pragmatic model for our universe,
but, in various gravity theories, it can be a significant
theoretical laboratory for considering a range of general properties
of the space-time structure. In compact regions, researches
of quantum computation have presented that the presence of
closed time-like curves (CTCs) in space-time provides for a
physical model in this computation [
22,23
]. In string
theories, a few of researches of the possibility of CTCs have been
also investigated by the Gödel space-time [
24–28
].
Later, in this geometry, the physical properties of the
presence of CTCs was investigated by Hawking [
29
]. Beside,
Reboucas et al. instituted three classes of solutions that are
explained by the properties as: (a) solutions where there are
no CTCs; (b) solutions with a sequence of alternating causal
regions and not causal regions; (c) solutions explained by
only one non-causal region [
30,31
]. In 2004 and 2005,
Barrow et al. analyzed the other properties such as the stability
and dynamic of the Gödel model [
32,33
].
Summary, some of the authors have investigated the
application of the Gödel space-time in matter. For example,
Cavalcante et al, have studied the quantum dynamics of
quasi-particles in a continuum limit investigating an
explanation of fullerene in a spherical solution of the Gödel-type
space-time with a topological defect [
34
]. Garcia et al. have
analyzed Dirac fermions in Gödel-type background
spacetimes with torsion [
35
] and also they investigated a
rotating fullerene molecule and applied a geometric theory to
explain the fullerene as a two-dimensional spherical space
with topological defects submitted to a non-Abelian gauge
field [
36
]. In other work, they studied the Weyl fermions
in a family of Gödel-type geometries in Einstein general
relativity [
37
].
However, the Gödel’s model is a special case of
homogeneous models of the universe in the 1960s [
38,39
]. The Som–
Raychaudhuri space-time is a particular case of the
Gödeltype space-time. Namely, the particular solution of the
Gödeltype space-time is known the Som–Raychaudhuri space-time
(Gödel flat solution) with the cosmic string [
14,40
].
Several researchers have studied the physical properties of a
series of backgrounds with the Som–Raychaudhuri
spacetime. For example, Paiva et al. have investigated the
properties of the rotating Som–Raychaudhuri homogeneous
spacetime [
41
], Wang et al. have studied the relativistic quantum
dynamics of a spinless particle in the Som–Raychaudhuri
space-time under the influence of the gravitational field
produced by a topology [
42
], Carvalho et al. have worked
about the Klein–Gordon oscillator in a Som–Raychaudhuri
space-time with a cosmic dispiration [
14
] and Vitoria et
al. have investigated Linear confinement of a scalar
particle in a Som–Raychaudhuri space-time [
43
]. Also, there
are a series of backgrounds with the cosmic string in Kerr
space-time [
44
], Schwarzschild space-time [
45,46
],
Gödeltype space-time [40] and AdS space [
47
].
On the other hand, the Yukawa potential is known a
screened Coulomb potential in particle and atomic physics.
In the 1930s, Hideki Yukawa presented that such a potential
obtain from the exchange of a scalar field such as the field
of a massive boson [
48
]. The potential is monotone
increasing in r and it is negative, implying the force is attractive. A
form of Yukawa potential has been earlier applied by Taseli in
obtaining a generalized Laguerre basis for hydrogen-like
systems [
49
]. Also, Kermode et al. have applied various forms
of the Yukawa potential to calculate the effective range
functions [
50
]. Napolitano et al. have presented the analysis of
expanded stellar kinematics of elliptical galaxies where a
Yukawa-like correction to the Newtonian gravitational
potential derived from f (R)-gravity is investigated as an
alternative to dark matter in cosmological models [
51
]. Recently, a
new other form of the Yukawa-like have been used by Ikhdair
et al. by solution of the Klein–Gordon equation for a particle
and anti-particle [
52
].
In here, we use Yukawa-like potential as Yukawa plus
exponential with constant potentials for analyzing a
relativistic scalar particle in Gödel’s flat space-time using the Klein–
Gordon equation. However, this equation can be solved in
flat, spherical and hyperbolic spaces under the various
interactions.
In this work, we consider a Som–Raychaudhuri
spacetime (Gödel’s flat) in the presence of a topological defect,
and then analyze the Yukawa-like confinement of a
relativistic scalar particle. We calculate the energy eigenvalues and
wave functions by using the generalized series method. We
discuss on the obtained results with more details. Although,
we try that write the Klein–Gordon equation in spherical and
hyperbolic spaces.
2 Theory and calculations
Now, we will consider a scalar quantum particle dynamic in
a Gödel-type space-time.
2.1 Scalar particle in Som–Raychaudhuri space-time
In here, we would like that work to the string theory
analog of Gödel’s space-time that the general Gödel-type
metric [
14,30
] can be given in polar coordinates (t, ρ , ϕ, z) in
the presence of cosmic string as below:
ds2 = −
dt + α
sinh2lρ
l2
dϕ
2
+ α
2 sinh22lρ
4l2
dϕ2
+dρ2 + d z2,
where ρ ∈ [0, +∞) , in the coordinate system, (t, z) ∈
(−∞, ∞), the angular variable ϕ ∈ [0, 2π ] and the
parameter of α, 0 < α < 1, is given in terms of the linear mass
density of the string, λ as α = 1 − 4λ, while characterizes
the vorticity of the space-time and l is as l = 0, ±1, ±2, . . ..
In this section, we investigate the quantum dynamics of
a relativistic scalar particle under the Yukawa-like
potential in the Gödel-type space-time when l → 0 of metric
Eq. (1), which is also called as the Som–Raychaudhuri
spacetime [
53
] with an angle deficit.
In the Som–Raychaudhuri space-time, the metric tensor
is defined by using cylindrical coordinates with the cosmic
string with the line element below (with c = h¯ = 1):
ds2 = gμν d xi d x j = dρ2 + α2ρ2dϕ2
2
− dt + α ρ2dϕ
In here, we choose the curvilinear coordinate system as the
circular cylindrical coordinate system, x 0, x 1, x 2, x 3 =
(t, ρ , ϕ, z). If we lead α → 1, then, the line element, Eq. (2),
can be turned to the form as without planar angle deficit.
(1)
(2)
⎛
⎛
gμν = ⎜⎜
⎝
gμν
= ⎜⎜
⎝
1
√−g ∂μ
Also, α → 1 and → 1, in which the Som–Raychaudhuri
space-time will be turned to the flat Minkowski spacetime.
And, in this case, the matric tensor is obtained as below:
and, the contravariant components of metric tensor, gμν , is
written as follows:
and, we obtain the determinant of metric tensor as g =
det gμν = −α2ρ2.
The Klein–Gordon equation with the Laplace-Beltrami,
∇2 = √1−g ∂μ √−ggμν ∂ν , under the scalar potential is
written in the Som–Raychaudhuri space-time
√−ggμν ∂ν − (M + V (r ))2 φ (r, t ) = 0, (5)
where M is the mass of free particle, ∂μ, ∂ν are derivatives
with respect to the cylindrical coordinates.
In here, we choose the potential model as the
spatiallydependent Yukawa-like potential that is given as follows:
V (ρ) = −V1 + V2e−μρ + V3
e−2μρ
ρ
,
where V1, V2, V3, and μ ≥ 0 are constant parameters of the
potential. If we set the values of V1 and V2 as zero, V1 = V2 =
0, then, the potential will be reduced as Yukawa potential.
After the substituting Eq. (7) in Ref. [
42
], using the
Taylor’s series expansion of the exponential functions to the first
approximation and substituting it into Eq. (6), we have:
∂2 (r, t)
∂ρ2
Now, we choose the wave function as Φ (r, t ) = e−iεt
eilϕ ei pz z u (ρ), where u (ρ) is the radial part of the wave
function, z (in the interval [−∞, +∞]) and the angular
momentum operator, pz , is as pz = const .
Substituting the selected wave function into Eq. (7), we
have:
∂2u (ρ)
∂ρ2
1 ∂u (ρ)
+ ρ ∂ρ
+
+ ε2 −
Now, the change of the dependent variable u (ρ) = √1ρ G (ρ),
Eq. (8) can be transformed to the compact Liouville’s normal
form
d2
dρ2 −
2
i=−2
δi ρi
G (ρ) = 0,
where l2 3
δ−2 = α2 − V32 − 4 , δ−1 = −2 (M + V1 + V2 + 2V3μ) V3,
2lε
α
δ0 = ε2 −
+ 2V2V3μ − pz − (M + V1 + V2 + 2V3μ)2 ,
2
δ1 = 2V2μ (M + V1 + V2 + 2V3μ) , δ2 = ε2 2 + V22μ2,
In this stage, to solve Eq. (9), we use the generalized
series method by Eshghi et al. in Refs. [
54,55
], parameters
δ0, δ1, δ2, δ−1 and δ−2 given in Eq. (10), and by a little
algebraic calculation, we can calculate the eigenvalues equation
as below:
Finally, we can calculate the wave function as
(r, t ) = e−iεt eilϕ ei pz z √1ρ ρ
× exp 21 ε2 2 + V22μ2
4 ε2 − 2lαε − pz2 − (M + 2V3μ)2 = 0 for the Yukawa
interaction.
In Figs. 1 and 2, we plot the energy levels versus the
vorticity parameter and the parameter of α, respectively.
−
⎛
⎝ 3 +
V2μ (M + V1 + V2 + 2V3μ)
where
Hn(3+)1(A):=
aa010((...01)) aaa0(12−((...011))) aa1(20−(...01)) a2(00−...1) ··· ···.···.···.··· ··· 000... 000... 000...
... ... ... ... . . . ... ... ...
... ... ... ... . . . ... ... ...
00 00 00 00 ·· ·· ·· ·· an(01−)1 aan(n(0−1))1 aan(n−(−011))
, n ∈ Z+.
is known the Hessenberg determinant [
56
] of order n + 1.
In fact, the Hessenberg determinants help us this ability to
obtain the solution of the problem explicitly and analytically.
Using the Eq. (25) Ref. [
55
], the elements of the Hessenberg
determinant are obtained as below:
ai( j) := (i + 1) ⎝⎛ i + 1 + 1 + 4 αl22 − V32 − 43 ⎠⎞ ,
a( j)
i
:= i − 2V2μ (M + V1 + V2 + 2V3μ)
−ε2 −
2lε
α
2.2 Scalar particle in spherical symmetrical Gödel
space-time
Now, we investigate the limit of Eq. (1) where we may
calculate a class of solutions of spherical symmetrical Gödel-type
space-time by substituting l2 < 0 in Eq. (1). Therefore, the
metric tensor can be written in the presence of the cosmic
string as below [
49
]:
4R2
ds2 = 4R2
+ ρ2
dρ2 + α2ρ4dϕ2
4R2α
− dt + 4R2
+ ρ2
ρ2dϕ
2
and the inverse matric is obtained as follows:
2 − 1
0
ρ2
1 + 4R2 αρ2
0
0
16R4
(4R2+ρ2)2
0
0
ρ2
− 1 + 4R2 αρ2 0 ⎞
0 0 ⎟
⎟⎟ .
0 ⎟⎟
⎠
16R4
(4R2+ρ2)2α2ρ4
0
1
And g = det gμν = − α2ρ4245R6R2+8ρ2 4 .
Therefore, in this space, the Klein–Gordon equation under
the Yukawa-like potential by changing of the dependent
vari(16)
⎞
where
and
−
δ¯−2
4
By using the generalized NU method [
57
], the energy levels
and wave functions are obtained as
⎡ ⎛
n2 + ⎣ −1 + 2 + 2 ⎝
δ¯2
4 +
δ¯−42 − R2δ¯0 −
⎞ ⎤
δ¯−2
4 ⎠ ⎦ n
⎛
δ¯−2
4 ⎝ 1 + 2
− 2 −
δ¯2
4 +
⎞
δ¯−42 − R2δ¯0 ⎠
δ¯2
4 +
δ¯−42 − R2δ¯0 +
R2δ¯0 − δ¯−22
= 0.
(22)
2.3 Scalar particle in hyperbolic coordinates
Now, we investigate the limit of Eq. (1) where we may obtain
the hyperbolic solution for l2 > 0 in Eq. (1). Therefore, the
metric tensor may be given in the presence of the cosmic
string as below [
51
]:
ds2 =
Now, by assuming V2 = V3 = 0 and applying the
transformation as ρ = 1l tanh (lθ ) onto Eq. (26), then, defining
y = cosh (lθ ) and ζ = y2 − 1, we have
ζ − γ¯ 4−2 u (ζ ) = 0.
By using the generalized NU method [
57
] as
n2 +
!
3 + 2
γ¯ 2
− 4 +
γ¯ −2
2 −
γ¯ −2
4
n
+
+
and
= 0.
γ¯−42 (1 + ζ )− − γ¯42 + γ¯−22
(25)
(26)
(27)
(28)
(29)
× Pn
2 γ¯ −42 , 2
In Gödel-type space-time, the solutions of the corresponding
Klein–Gordon equation present us that the behavior of the
scalar particle is affected by in the presence of a topological
defect. However, this equation was solved in flat, spherical
and hyperbolic spaces under the various interactions with
more details.
Although, the expressions for the energy of the scalar
particle in the three cases (flat, spherical and hyperbolic spaces)
tell us that effect of the topological defect is to increase the
energy levels under the Yukawa-like confinement. But, we
only plotted the energy levels versus the vorticity
parameter and the parameter of α as Figs. 1 and 2 for the case of
Som–Raychaudhuri space-time, respectively. Also, we
investigated the limit of Eq. (1) for cases l2 > 0 and l2 < 0 in Eq.
(1) with assuming V2 = V3 = 0 in the Yukawa-like
confinement for Eqs. (19) and (26) due to simplification the solution
of our problems.
In Fig. 1, the energy levels have been plotted versus
the vorticity parameter with n = 1, 2, 3 (l = 0) and
V1 = 0.8, V2 = 0.4, V3 = 0.02, μ = 0.2, α = 0.5, pz =
1 and M = 2. Figure 1 shows that the energy of the scalar
particle decreases with increasing the vorticity parameter
as linearly at different the quantum numbers. In fact, the level
energies begin to decrease as approximately exponentially,
but, with increasing l, at the beginning, it decreases with
the increasing the vorticity parameter , but then it reaches
a minimum at around ≈ 0.1, then it increases with the
increasing the vorticity parameter as fast. We notice that,
for a fixed value of the vorticity parameter , the energy
of the scalar particle increases, in Fig. 1, when the quantum
numbers are increasing.
Also, In Fig. 2, the energy levels of the scalar particle have
been plotted versus the parameter α with l = −1, 0, 1 and
V1 = 0.8, V2 = 0.4, V3 = 0.02, μ = 0.2, = 0.05, pz =
1, n = 2 and M = 2. We observe that the energy of system
decreases with increasing the parameter α as exponentially
for l = 1. Also, in l = 0, the level energies are as constant,
but, with l = −1, the level energies begin to increase as
exponentially with the increasing the parameter α.
4 Conclusion
In this paper, we considered a Gödel-type space-time for
any l in the presence of a topological defect, and then
analyzed the Yukawa-like confinement of a relativistic scalar
particle. The corresponding wave functions were written in
terms of the Hessenberg determinant by using the
generalized series method. Moreover, the similarities between our
results and the Landau levels in this spacetime background
is beneficial and the Landau levels in condensed matter
systems of interacting particles can be drown out [
23
]. In
addition, Quarkonia-type systems can be described by confining
potentials Model. Therefore, studying the Yukawa potential
in the Som–Raychaudhuri spacetime in the presence of a
topological defect is an advantage and the results of the
present study can be used to investigate heavy quarkonia
within a strong magnetic field.
Acknowledgements The authors would like to thank the kind
referee(s) for the positive suggestions which have greatly improved the
present text.
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