#### Rotating effects on the scalar field in the cosmic string spacetime, in the spacetime with space-like dislocation and in the spacetime with a spiral dislocation

Eur. Phys. J. C
Rotating effects on the scalar field in the cosmic string spacetime, in the spacetime with space-like dislocation and in the spacetime with a spiral dislocation
R. L. L. Vitória 0
K. Bakke 0
0 Departamento de Física, Universidade Federal da Paraíba , Caixa Postal 5008, 58051-900 João Pessoa, PB , Brazil
In the interface between general relativity and relativistic quantum mechanics, we analyse rotating effects on the scalar field subject to a hard-wall confining potential. We consider three different scenarios of general relativity given by the cosmic string spacetime, the spacetime with space-like dislocation and the spacetime with a spiral dislocation. Then, by searching for a discrete spectrum of energy, we analyse analogues effects of the Aharonov-Bohm effect for bound states and the Sagnac effect.
1 Introduction
Two interesting points raised by Landau and Lifshitz [
1
]
are the effects of rotation in the Minkowski spacetime. One
of them is the singular behaviour at larges distances for a
system in a uniformly rotating frame, while the other is
the effect on the clocks on the rotating body. In
particular, this singular behaviour at larges distances means that
there exists a restriction on the spatial coordinates due to
the effects of rotation. For instance, by taking the line
element of the Minkowski spacetime in cylindrical
coordinates, we have: ds2 = − c2 dt 2 + dr 2 + r 2 dϕ2 + d z2;
thus, by taking ϕ → ϕ + ωt , where ω is the constant
angular velocity of the rotating reference frame, then, the
line element of the Minkowski spacetime becomes: ds2 =
− c2 1 − ω2 r 2/c2 dt 2 + 2ω r 2dϕ dt + dr 2 +r 2 dϕ2 + d z2.
Therefore, we can see that the radial coordinate becomes
determined in the range: 0 ≤ r < c/ω. This restriction
on the radial coordinate in a uniformly rotating frame has
drawn attention to the interface between general relativity
and relativistic quantum mechanics [
2–5
].
In recent years, aspects of the uniformly rotating frame
have been investigated in the cosmic string spacetime [
6
]. It
has been shown that the topology of the cosmic string
spacetime also determines the restriction of the values of the radial
coordinate. Then, rotating effects on relativistic quantum
systems have been investigated in the background of the cosmic
string spacetime, such as the Dirac oscillator [
6
], the Klein–
Gordon oscillator [
7
], scalar bosons [
8
], Duffin–Kemmer–
Petiau equation [
9
] and nonrelativistic topological quantum
scattering [
10
]. In view of these studies, a point that has not
been raised is the aspect of the scalar field inside a
cylindrical shell, where rotation is present and the corresponding
background is the cosmic string spacetime. In this interface
between general relativity and quantum mechanics, it is
interesting to search for the effects associated with rotation and
the topology of the spacetime. Cosmic strings [
11–14
] are
linear topological defects in the spacetime characterized by
a conical singularity [15]. This singularity is determined by
the curvature concentrated on the symmetry axis of the
cosmic string. Recently, a great deal of work has investigated the
influence of the cosmic string spacetime on quantum systems
[
16–28
] and in Gödel-type spacetimes [
29–33
].
Another branch of research is topological defects in
spacetime associated with torsion. A well-known example is the
spacetime with a space-like dislocation [
34,35
], which is an
analogue of the screw dislocation in solids [
36–38
]. This kind
of topological defect background has been used in studies of
the Aharonov–Bohm effect for bound states [39],
noninertial effects [
40
], relativistic position-dependent mass systems
[
41,42
] and Kaluza–Klein theories [43]. One point that has
not been dealt with in the literature is the rotating effects on
the scalar field by considering the cosmic string spacetime,
the spacetime with space-like dislocation and the spacetime
with a spiral dislocation as backgrounds. Therefore, in this
work, we deal with a scalar field subject to a hard-wall
confining potential in the cosmic string spacetime. We also consider
a uniformly rotating frame. Then, we analyse a particular
case where a discrete spectrum of energy can be obtained.
We show that there exists the influence of the topology of
the spacetime and rotation on the relativistic energy levels.
Further, we consider two spacetime backgrounds with the
presence of torsion [
34
] and analyse rotating effects on the
scalar field subject to a hard-wall confining potential.
The structure of this paper is: in Sect. 2, we start by
introducing the line element of the cosmic string spacetime in a
uniformly rotating frame. Then, we analyse the scalar field
subject to a hard-wall confining potential; in Sect. 3, we
obtain the line element of the spacetime with a space-like
dislocation in a uniformly rotating frame, and thus,
investigate the scalar field subject to a hard-wall confining potential;
in Sect. 4, we consider a spacetime with a spiral dislocation.
Thus, we discuss the scalar field subject to a hard-wall
confining potential in both nonrotating and uniformly rotating
frame; in Sect. 5, we present our conclusions.
2 Rotating effects in the cosmic string spacetime
In studies of linear topological defects in the spacetime, the
cosmic string is the most known example [
12,14
]. It is
characterized by a line element that possesses a parameters related
to the deficit of angle: α = 1 − 4Gμ, with μ as being the
dimensionless linear mass density of the cosmic string and
G is the gravitational Newton constant. By working with the
units h¯ = c = 1 from now on, the line element of the cosmic
string spacetime is written in the form:
ds2 = −dt 2 + dr 2 + α2 r 2 dϕ2 + d z2.
Note that this topological defect has a curvature
concentrated only on the cosmic string axis, i.e., the curvature
tensor is given by Rρρ,,ϕϕ = 14−αα δ2(r), where δ2(r) is the
twodimensional delta function [
15
]. Besides, the parameter α is
defined in the range 0 < α < 1. In order to study the aspects
of the uniformly rotating frame, let us perform a coordinate
transformation given by ϕ → ϕ + ω t , where ω is is the
constant angular velocity of the rotating frame. Thereby, the line
element (1) becomes [
6
]
ds2 = − 1 − ω α r
2 2 2
dt 2 + 2ωα2 r 2dϕdt
+ dr 2 + α2r 2dϕ2 + d z2.
Hence, in the rotating reference frame, the line element
(2) show us that the radial coordinate in the cosmic string
spacetime is restrict to the range:
1
0 ≤ r < , (3)
ωα
otherwise, we would have a particle placed outside of the
light-cone. Note that by taking α → 1, we recover the
discussion made in Ref. [
1
] in the Minkowski spacetime. Note
that the restriction on the radial coordinate is determined by
(1)
(2)
the angular velocity and by the parameter related to the deficit
of angle.
Henceforth, let us study the effects of rotation on the scalar
field in the cosmic string spacetime. In curved spacetime, the
Klein–Gordon equation is written in the form [
41,44
]:
m2φ = √
1
−g
∂μ
√−g gμν ∂ν φ,
where g = det gμν . Then, with the line element (2), the
Klein–Gordon equation (4) becomes
∂2φ ∂2φ 2 ∂2φ ∂2φ
m2φ = − ∂t 2 + 2ω ∂ϕ∂t − ω ∂ϕ2 + ∂r 2
1 ∂φ 1 ∂2φ ∂2φ
+ r ∂r + α2r 2 ∂ϕ2 + ∂ z2 .
With the cylindrical symmetry, we have that φ is an
eigenfunction of the operators pˆz = − i ∂z and Lˆ z = − i ∂ϕ .
Therefore, a solution to Eq. (5) can be given in terms of the
eigenvalues of the operator pˆz and Lˆ z as follows:
φ (t, ρ , ϕ, z) = e−i E t ei l ϕ ei k z f (r ) ,
where l = 0, ± 1, ± 2, . . . and −∞ < k < ∞. And then,
by substituting the solution (6) into Eq. (5), we obtain the
following radial equation:
f
1 l2
+ r f − α2r 2 f + λ2 f = 0,
where we have defined the parameter λ in the form:
λ2 = (E + lω)2 − m2 − k2.
Hence, Eq. (7) is the well-known the Bessel differential
equation [
45
]. The general solution to Eq. (7) is given in
the form: f (r ) = A J |l| (λ r ) + B N |l| (λ r ), where J |l| (λ r )
α α α
and N |l| (λ r ) are the Bessel function of first kind and second
α
kind [
45
], respectively. With the purpose of having a regular
solution at the origin, we must take B = 0 in the general
solution since the Neumann function diverges at the origin.
Thus, the regular solution to Eq. (7) at the origin is given by:
f (r ) = A J |l| (λ r ) .
α
As we have pointed out in Eq. (3), the restriction on the
radial coordinate imposes that the scalar field must vanish at
r → ρ0 = 1/αω. This means that the radial wave function
(9) must satisfy the boundary condition:
f (r → ρ0 = 1/αω) = 0.
This corresponds to the scalar field subject to a hard-wall
confining potential. In other words, the geometry of the
spacetime plays the role of a hard-wall confining potential [
6
].
Next, let us consider a particular case where λ ρ0 1. In
this particular case, we can write [
40,45
]:
(4)
(5)
(6)
(7)
(8)
(9)
(10)
J |l| (λ ρ0) →
α
2 |l| π
π λ ρ0 cos λ ρ0 − 2α
π
− 4
Hence, by substituting (11) into (9), we obtain from the
boundary condition (10) that
m2 + α2ω2π 2 n + 2|lα| + 43 2
Equation (12) gives us the spectrum of energy of a scalar
field subject to a hard-wall confining potential determined
by the topology of the cosmic string spacetime in a
uniformly rotating frame. The contributions to the relativistic
energy levels (12) that stem from the topology of the cosmic
string are given by the effective angular momentum given by
leff = |αl| , and by the presence of the fixed radius ρ0 = α1ω .
Since there is no interaction between the scalar field and
the topological defect, the presence of the effective angular
momentum in the relativistic energy levels means that there
exists an analogue of the Aharonov–Bohm effect for bound
states [
39,43,46,47
]. Besides, we can observe a Sagnac-type
effect [
2,48–50
] by the presence of the coupling between the
angular velocity ω and angular momentum quantum
number l. Furthermore, by taking the limit α → 1, we have that
ρ0 → ω1 and the allowed energies of the system (12) become
m2 + ω2π 2 n + |2l|
which corresponds to the allowed energies of the scalar field
subject to a hard-wall confining potential in the Minkowski
spacetime in a uniformly rotating frame.
3 Rotating effects in the spacetime with a space-like
dislocation
In Ref. [
34
], examples of topological defects in the spacetime
associated with torsion are given. We start this section by
considering a spacetime with a space-like dislocation, whose
line element is given by
ds2 = − dt 2 + dr 2 + r 2 dϕ2 + (d z + χ dϕ)2 ,
(14)
where χ is a constant (χ > 0) associated with a dislocation
in the spacetime.1 Next, let us follow the previous section
and make the coordinate transformation ϕ → ϕ + ω t . Then,
the line element (14) becomes [
40
]:
1 Note that the spatial part of the line element (14) is known in the
context of Condensed Matter Physics as screw dislocation [
38–40
].
ds2 = − 1 − ω2 r 2 − χ 2ω2
ω
Observe that the restriction on the radial coordinate is
determined by the angular velocity and the parameter associated
with the torsion of the defect in contrast to that given in Eq. (3)
for the cosmic string spacetime. In this case, if r ≥ √1−ωχ2ω2
we would have a particle is placed outside of the light-cone.
Note that by taking χ = 0, we recover the discussion made
in Ref. [
1
] in the Minkowski spacetime.
Let us go further by writing the Klein–Gordon equation
(4) in the spacetime described by the line element (15):
∂2φ ∂2φ
m2 φ = − ∂t 2 + 2ω ∂ϕ ∂t − ω
2 ∂2φ ∂2φ
∂ϕ2 + ∂ρ2
2
1 ∂φ 1
+ ρ ∂ρ + ρ2
∂ ∂
∂ϕ − χ ∂ z
∂2φ
φ + ∂ z2 .
The solution to Eq. (17) has the same form of Eq. (6),
therefore, we obtain the following radial equation:
f
1
+ r f −
(l − χ k)2
r 2
where λ has been defined in Eq. (8). Note that Eq. (18) is
also the Bessel differential equation [
45
]. By following the
steps from Eqs. (9) to (11), we have that f (r ) = A J|l| (λ r )
and the boundary condition (10) is replaced with
f r → ρ¯0 =
1 − ω2χ 2
ω
= 0,
which also corresponds to the scalar field subject to a
hardwall confining potential as in the previous section. Let us also
consider λ ρ¯0 1, then, we obtain
En, l, k ≈
− lω ±
ω2 π 2
m2 + 1 − ω2χ 2
1 3 2
n + 2 |l − χ k| + 4
+ k2.
(15)
(16)
(17)
(18)
(19)
(20)
Hence, the relativistic energy levels (20) are obtained
when a scalar field is subject to a hard-wall confining
potential under the effects of rotation in the spacetime with a
space-like dislocation. The contributions to the relativistic
energy levels (20) that stem from the topology of the defect
are given by the effective angular momentum leff = l − χ k,
and by the presence of the fixed radius ρ¯0 = 1−ωω2χ2 . Due
to the effects of the torsion in the spacetime, we have a shift
in the angular momentum that yields the effective angular
momentum leff = l − χ k. As discussed in Refs. [
35,41
],
this shift in the angular momentum corresponds to an
analogue effect of the Aharonov–Bohm effect [
46,47
]. Note that
for χ = 0, the contribution of torsion in the spacetime
vanishes, and thus, we recover the relativistic energy levels (13)
in the Minkowski spacetime in a uniformly rotating frame.
Again, we can also observe a Sagnac-type effect [
2,48–50
]
by the presence of the coupling between the angular velocity
ω and angular momentum quantum number l.
4 Rotating effects in the spacetime with a spiral
dislocation
In Ref. [38], examples of topological defects in solids
associated with torsion are given. We start this section by
considering a generalization of a topological defect in gravitation.
Let us consider the distortion of a circle into a spiral, that
corresponds to a spiral dislocation. Then, the
corresponding line element of this topological defect in the context of
gravitation is [
38,51
]:
ds2 = − dt 2 + dr 2 + 2β dr dϕ +
β2 + r 2 dϕ2 + d z2,
(24)
(25)
(26)
(27)
(28)
where θ 2 = E 2 − m2 − k2. With the purpose of solving this
radial equation, let us write
f (r ) = exp i l tan−1
× u (r ) ,
r
β
then, we obtain the following equation for u (r ):
We proceed with a change of variables given by x =
θ r 2 + β2, and thus, we obtain the following equation:
Hence, Eq. (26) is the Bessel differential equation [
45
]. We
also follow the steps from Eqs. (9) to (11), then, we can
write u (x ) = A J|l| (x ) and the boundary condition (10) is
replaced with
2
u x → x0 = θ r0 + β2
= 0,
which also corresponds to the scalar field subject to a
hardwall confining potential (r0 is a fixed radius). Let us also
consider x0 1, and then, by using the relation (11), we
obtain
4.1 Scalar field subject to a hard-wall confining potential
Let us investigate the topological effects of the spacetime
with a spiral dislocation on the scalar field when it is subject to
a hard-wall confining potential. The Klein–Gordon equation
(4) in the spacetime described by the line element (21) is
given by
∂2φ
m2 φ = − ∂t 2 +
∂φ
∂r
2β ∂2φ β ∂φ 1 ∂2φ ∂2φ
− r 2 ∂r ∂ϕ + r 3 ∂ϕ + r 2 ∂ϕ2 + ∂ z2 .
1 β2
f + r − r 3 −
l2
f − r 2 f
f + θ 2 f = 0,
where β is a constant (β > 0) associated with the distortion
of the defect.
En, l, k ≈ ±
m2 + k2 +
π 2
2
r0 + β2
|l| 3 2
n + 2 + 4
.
The solution to Eq. (22) has the same form of Eq. (6),
therefore, from Eq. (22) we obtain the radial equation:
4.2 Rotating effects on the scalar field subject to a
hard-wall confining potential
(21)
(22)
(23)
Therefore, we have obtained in Eq. (28) the relativistic
spectrum of energy for the scalar field subject to the hard-wall
confining potential in the spacetime with a spiral dislocation.
The effect of the topology of this spacetime can be viewed by
the presence of the parameter β that gives rise to an effective
radius ζ0 = r02 + β2. In contrast to Eqs. (12) and (20), we
have that the angular momentum remains unchanged, i.e.,
there is no effect of the topology of the spacetime that yields
an effective angular momentum. Hence, there is no analogue
of the Aharonov–Bohm effect for bound states [
39,43,46,
47
] in this sense. Note that, by taking β = 0 we obtain the
spectrum of energy in the Minkowski spacetime.
Henceforth, let us consider a uniformly rotating frame as in
the previous sections. For this purpose, we also perform the
coordinate transformation ϕ → ϕ + ω t , and then, the line
element (21) becomes:
ds2 = − 1 − ω2 β2 − ω r
2 2 dt 2 + 2β ω dr dt
+ 2ω β2 + r 2 dϕ dt
+ dr 2 + 2β dr dϕ +
β2 + r 2 dϕ2 + d z2.
In this case, we have that the radial coordinate is restricted
by range:
Hence, the restriction on the radial coordinate is determined
by the angular velocity and the parameter associated with the
distortion of a circle into a spiral (torsion). This restriction on
the radial coordinate differs from the cosmic string spacetime
given in Eq. (3), but it is analogous to that of the spacetime
with space-like dislocation given in Eq. (16). Therefore, if
√
r ≥ 1−ωβ2ω2 . we would have a particle is placed outside
of the light-cone. Note that by taking β = 0, we also recover
the discussion made in Ref. [
1
] in the Minkowski spacetime.
Further, the Klein–Gordon equation (4) in the spacetime
described by the line element (29) is given by
∂2φ ∂2φ
m2 φ = − ∂t 2 + 2ω ∂ϕ ∂t +
The solution to Eq. (31) has the same form of Eq. (6),
therefore, the radial equation becomes
where we have also defined λ in Eq. (8). By following the
steps from Eqs. (24) to (26), but with the change of variables
given by y = λ r 2 + β2, we obtain
1 l2
u + y u − y2 u + u = 0.
have that u (y) = A J|l| (y). With ρ0 =
the boundary condition (10) becomes
Hence, by following the steps from Eqs. (9) to (11), then, we
√
1−β2ω2 , therefore,
ω
λ
u y → y0 = ω
= 0,
which also corresponds to the scalar field subject to a
hardwall confining potential as in the previous sections. Let us
also consider y0 1, then, we obtain
(29)
(30)
(31)
(32)
(33)
(34)
m2 + k2 + ω2 π 2 n + |2l| + 43 2
In Eq. (35), we have the relativistic spectrum of energy
for the scalar field subject to a hard-wall confining potential
under the effects of rotation in the spacetime with a spiral
dislocation. In contrast to Eqs. (12), (20) and (28), there is no
influence of the topology of the spacetime on the relativistic
energy levels in the uniformly rotating frame. Despite having
effects of the topology of the spacetime on the radial
coordi√
nate as we have seen in Eq. (30), for ρ0 = 1−ωβ2ω2 , we have
only effects of rotation on the relativistic energy levels. The
Sagnac-type effect [
2,48–50
] can be observed in Eq. (35) due
to the presence of the coupling between the angular
velocity ω and angular momentum quantum number l. Besides,
there is no effect analogous to the Aharonov–Bohm effect
for bound states [
39,43,46,47
], since the angular momentum
remains unchanged. Therefore, the spectrum of energy (35)
is analogous to that obtained in Eq. (13) in the Minkowski
spacetime.
5 conclusions
We have investigated rotating effects on the scalar field
confined to a hard-wall confining potential in three different
topological defect spacetimes: the cosmic string spacetime, the
spacetime with a space-like dislocation and the spacetime
with a spiral dislocation. We have started our discussion in
all these cases through the restriction on the radial coordinate
that arises from the uniformly rotating frame and the
topology of the spacetime. Then, we have used this information
about the restriction on the radial coordinate to impose the
boundary condition that corresponds to a hard-wall
confining potential. We have shown in all these cases that a
discrete spectrum of energy can be achieved. In the case of the
cosmic string spacetime, we have seen that the spectrum of
energy depends on the fixed radius ρ0 = α1ω and the
effective angular momentum leff = |αl| . On the other hand, in the
case of the spacetime with a space-like dislocation, we have
seen a dependence of the spectrum of energy on the fixed
√
radius ρ¯0 = 1−ωω2χ2 and the effective angular momentum
leff = l − χ k. Hence, due to this presence of the effective
angular momentum, we have an analogue of the Aharonov–
Bohm effect for bound states [
39,43,46,47
]. Furthermore,
in these two cases we have also seen a Sagnac-type effect
[
2,48–50
].
The spacetime with a spiral dislocation corresponds to a
generalization of a topological defect (the distortion of a
circle into a spiral [
38
]) in gravitation. With this background,
we have first analysed the confinement of the scalar field
to a hard-wall confining potential without rotating effects.
We have seen that a discrete spectrum of energy can also
be obtained in this topological defect background, where the
topology of the spacetime yields a contribution that gives rise
to an effective radius ζ0 = r02 + β2. On the other hand,
no analogue effect of the Aharonov–Bohm effect for bound
states exists. In the second case analysed, we have considered
the uniformly rotating frame. We have also seen that there
exists a restriction on the radial coordinate that arises from
the uniformly rotating frame and the topology of the
spacetime. By investigating the confinement of the scalar potential
to a hard-wall confining potential, we have also obtained a
discrete spectrum of energy. In this case, even though there
√
exists the presence of the fixed radius ρ0 = 1−ωβ2ω2 which
determines the boundary condition of the hard-wall
confining potential, we have seen no dependence of the
relativistic energy levels on the parameter associated with the
spiral dislocation spacetime. Besides, no analogue effect of the
Aharonov–Bohm effect for bound states exists. Due to the
effects of rotation, a contribution to the relativistic energy
levels that gives rise to a Sagnac-type effect exists.
Therefore, the relativistic energy levels obtained in the spacetime
with a spiral dislocation in a uniformly rotating frame are
analogous to the case of the Minkowski spacetime.
Acknowledgements The authors would like to thank the Brazilian
agencies CNPq and CAPES for financial support.
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