ЖЭТФ, 2022, том 162, вып. 5 (11), стр. 673-679
© 2022
GRAVITATIONAL FIELD EFFECTS PRODUCED BY
TOPOLOGICALLY NON-TRIVIAL GEOMETRY AND ROTATING
FRAMES SUBJECT TO A COULOMB-TYPE SCALAR POTENTIAL
F. Ahmeda*
a Department of Physics, University of Science and Technology Meghalaya,
Ri-Bhoi, Meghalaya-793101, India
Received July 17, 2022
revised version August 26, 2022,
Accepted for publication August 26, 2022
DOI: 10.31857/S0044451022110074
like and spiral dislocations [18], on spin-zero scalar mas-
EDN: KYWSPL
sive charged particles subject to Coulomb-type scalar
and vector potentials [19], on spin-1/2 particles with a
field and mixed potential [20], on the Casimir energy
1. INTRODUCTION
in a space-time with one extra compactified dimension
[21], on spin-zero scalar particles in a space-time with
Rotation and rotating frames have always been a
magnetic screw dislocation [22], on the Dirac particles
source of confusion while dealing with the problem of a
in an accelerated reference frame [23], on the Dirac
uniformly rotating disk and its spatial geometry in the
fields in a space-time with spiral dislocation [24], on
context of special theory of relativity (STR) [1]. An in-
spin-zero scalar particles in a space-time with distor-
teresting feature in treating a rotational phenomena is
tion of a vertical line to a vertical spiral [25], on the
the Galilean rotational transformation (GRT) between
Klein-Gordon oscillator in a topologically non-trivial
inertial (laboratory) frames and non-inertial rotating
space-time [26] and in a cosmic string space-time with
frames.
space-like dislocation [27], on spin-zero scalar particles
in a Lorentz symmetry violation environment [28], on
This coordinate transformation {xμ} → {x′μ} is
spin-zero scalar particles induced by the topology as-
defined by (t → t′, r → r′, φ → φ′ + Ω t′, z → z′)
sociated with a time-like dislocation space-time [29],
[2-4], where Ω is the uniform angular speed of the ro-
on spin-zero scalar massive charged particles subject to
tating frame measured by an observer in the inertial
Coulomb-type potential [30], on scalar particles [31,32],
frame. They had showed that the axial coordinate is
and the Klein-Gordon oscillator with scalar potential
restricted by 0 ≤ r <cΩ and others are usual ranges.
[33] in the context of Kaluza-Klein theory.
Rotating frame of reference for various physical sys-
tems have been investigated in literature, for instance,
We are mainly interest on a space-time that is
on free scalar fields [5], on the Dirac particle [6], on
produced by a non-trivial topology defined by the
a neutral particle [7], with quantum states under an
geometry S1 × R3, where R3 represents usual di-
electromagnetic field [8],
rections and S1 is a compact dimension (see fig.
on the Dirac oscillator [9-11], on the Dirac par-
1). The metric in polar coordinates (t′, r′, φ′, θ′) for
ticle subject to a hard-wall confining potential [12],
this topologically non-trivial geometry is given by
on massive scalar fields [13], on spin-1 particles [14],
ds2 = -dt′2 + dr′2 + r′2 dφ′2 + R2 dθ′2 [26].
on quantum fermionic fields inside a cylinder [15], on
scalar bosons subject to Coulomb-type potential [16],
For S1
rotating frame of reference, we per-
on scattering problem of a non-relativistic particle [17],
form the coordinate transformation from in-
on spin-zero scalar particles in a space-time with space-
ertial frame (t, r, φ, θ) to the rotating frame
(t′
= t, r′
= r, φ′
= φ, θ′
= θ + Ωt), one will
* E-mail: faizuddinahmed15@gmail.com
have
673
F. Ahmed
ЖЭТФ, том 162, вып. 5 (11), 2022
(
)
ds2 = - 1 - R2 Ω2 dt2 + dr2 + r2 dφ2 + R2 dθ2
Thus, the quantum dynamics of scalar bosons sub-
ject to a potential S(r) following the first approach is
+2 Ω R2 dt dθ.
(1)
described by the wave equation [19,21-23,30-33,39-44]
[
]
The ranges of the coordinate 0 < θ < 2 π and others are
) (
)2
1
in the usual ranges. Here R is radius of the compact di-
-
(√-g gμν
Dν
+ M + S(r)
Ψ = 0,
√-gDμ
mension S1, and the determinant of the corresponding
(2)
metric tensor gμν is det g = -r2 R2. An interesting fea-
where M is the rest mass of the scalar bosons.
ture one can see in contrast to the rotating Minkowski
In this analysis, we have chosen the electromagnetic
space-time is that the radius of the compact dimension
1
four-vector potential Aμ = (0,
A) [22,27,33,42,44] with
S1 satisfies the condition R <
[26] such that the
Ω
the following components
metric component gtt is always negative otherwise this
rotating system is physically unacceptable for R >1Ω .
ΦB
Ar = 0 = Aθ
,
Aφ =
,
(3)
2π
where ΦB = Φ Φ0 is the Aharnov-Bohm flux which
2. GRAVITATIONAL FIELD EFFECTS UNDER
is a constant, Φ0 =2πe is the amount of quantum
ROTATING FRAME ON SCALAR BOSONS
flux, and Φ is the magnetic flux which is a positive
SUBJECT TO COULOMB-TYPE POTENTIAL
integer. The presence of a magnetic flux in quantum
In this section, we study the relativistic quantum
system shows an analogue of the Aharonov-Bohm eff-
motions of scalar bosons subject to a Coulomb-type
fect [37,38] which is a quantum mechanical phenomena
scalar potential in a topologically non-trivial rotating
that has been studied by many researchers in literature
space-time. There are two ways that one can introduce
[27, 30-33, 41-44].
a potential into the KG-equation. First one being an
The Klein-Gordon equation (2) using (3) in the ro-
electromagnetic four-vector potential Aμ that can be
tating space-time background (1) becomes
introduced through a minimal substitution in momen-
[
)2
)2
tum four-vector via pμ → (pμ - e Aμ) or in the partial
( ∂
∂
1
∂ (
∂ )
1 (
∂
-
-Ω
+
r
+
-iΦ
derivative via ∂μ → (∂μ - i e Aμ) [39], where e is the
∂t
∂θ
r ∂r
∂r
r2
∂φ
]
electric charges. This procedure has been widely used
(
)2
1
∂2
by several authors in literature [16,19,27,30-33,40-44].
+
Ψ = M + S(r)
Ψ.
(4)
R2 ∂ θ2
The second procedure is to introduce a scalar poten-
tial S(t, r) by modifying the mass term in the KG-
Several authors have been studied quantum motions of
equation via transformation M2 → (M + S(t, r))2.
scalar and spin-half particles using potential of differ-
This procedure has also been used by several authors
ent kinds, such as the Cornell-type potential [40,41]. In
to study the effects of potential in quantum systems
this analysis, we are interested on another kind of po-
[16, 19, 27, 30-33, 39-42].
tential proportional to the inverse of the axial distance.
This type of potential is used for short-range interac-
tions and called the Coulomb-type potential given by
1
η
S(r) ∝
⇒ S(r) =
,
(5)
r
r
where η > 0 is a constant characterizes the potential
parameter. This Coulomb-type potential has widely
been studied in literature [41,43,45-57].
The total wave function Ψ(t, r, φ, z) can express in
terms of a radial wave function ψ(r) as follows:
Ψ(t, r, φ, θ) = ei(-Et+lφ+qθ) ψ(r),
(6)
where E is energy of the scalar bosons, l = 0, ±1, ±2, ..
are the eigenvalues of the angular momentum operator
Fig. 1. Representation of the topologically non-trivial geometry
-i∂φ, and q is a constant associated with the operator
S1 × R3 [26]
-i∂θ. Noted that for S1 compact dimension defined by
674
ЖЭТФ, том 162, вып. 5 (11), 2022
Gravitational Field Effects. . .
1
a finite radius R satisfying the condition R <
, the
The radial wave function is given by
Ω
total wave function obeys the following condition
√
ξ
ψn,l(ξ) = ξ
(l-Φ)2+η2 e-
2×
Ψ(θ + 2 π R) = Ψ(θ).
(7)
(
)
γ
1
×1 F1
j+
+
,2j + 1;ξ
(14)
Thereby, substituting the scalar potential (5) and
δ
2
the total wave function Eq. (6) into the Eq. (4), we
have obtained the following radial wave equation
Equation (13) is the relativistic energy eigenvalue
[
]
and Eq. (14) is the radial wave function of the scalar
2
1
j
2γ
ψ′′(r) +
ψ′(r) +
-δ2 -
-
ψ(r) = 0,
(8)
bosons in a topologically non-trivial rotating space-
r
r2
r
time subject to a Coulomb-type external potential. We
can see that the eigenvalue solution is modified by the
where
non-trivial topology of the geometry defined by the ra-
√
2
√
n
dius R, and the Coulomb-type potential. We also see
δ= M2 +
- (E + Ω n)2, j =
(l - Φ)2 + η2,
R2
that the energy levels are shifted by rotating frame of
γ =Mη.
(9)
reference, and hence, these are not equally spaced on
= 0 for constant values of l,q.
either side about En,l,q
Performing a change of variables via ξ = 2 δ r into the
This effect arises due to the coupling between the quan-
Eq. (8), we have
tum number q = 0 and the uniform angular speed Ω of
(
)
2
rotating frame of reference.
1
j
γ 1
1
ψ′′(ξ) +
ψ′(ξ) +
-
-
-
ψ(ξ) = 0. (10)
In Ref. [26], authors studied the Klein-Gordon os-
ξ
ξ2
δ ξ
4
cillator in a non-trivial topological space-time geome-
Suppose, a possible solution for the Eq.
(10) in
try. They solved the wave equation analytically and ob-
terms of a function F (ξ) as:
tained the following energy eigenvalue expression (see
Eq. (28) there and we have replaced n → q)
ψ(ξ) = ξj e-2 F (ξ).
(11)
√
q2
Substituting this solution (11) into the Eq.
(10),
E± = ± M2 +
+ 2M ω (2N′ + |l|),
(15)
R2
we have obtained the following second-order differen-
tial equation:
where N′ = N + 1 = 1, 2, 3, ...
(
)
One can easily show that the presented energy
(
)
γ
1
eigenvalue (13) is completely different from the re-
ξ F′′(ξ)+ 1+2j-ξ F′(ξ)+
-j-
-
F (ξ) = 0.
δ
2
sult (15) obtained in Ref. [26]. This is because, we
(12)
have considered a non-inertial reference frame which
Equation (12) is the well-known confluent hyperge-
rotates with constant angular speed Ω, the Coulomb-
ometric equation form [58, 59]. As state in Refs.
type scalar potential characterise by the parameter η
[16, 19, 22, 26, 43, 51, 56, 58, 59], the solution to the
as well as the magnetic flux Φ which shifts the energy
differential equation of the form (12) can be ex-
levels and the wave function. Thus, our presented re-
pressed in terms(of a confluent hyper)geometric func-
sult in this section is completely new and different from
tion F (ξ) =1F1 j +γδ +12 , 2 j + 1; ξ which is well-
the previous result given in Ref. [26].
behaved for ξ
→ ∞. Then, in searching for the
bound-state solutions of the wave equation, the func-
3. GRAVITATIONAL FIELD EFFECTS UNDER
tion 1F1 must be a finite deg(ee polynom)al in ξ
ROTATING FRAME ON KG-OSCILLATOR
of degree n, and the quantity j +γδ +1
= -n
2
SUBJECT TO COULOMB-TYPE SCALAR
POTENTIAL
[16, 19, 22, 26, 43, 51,56, 58, 59], where n = 0, 1, 2, ... )
After simplifying this condition j +γδ +1
= -n,
2
In this section, we will study the Klein-Gordon os-
one will have the following expression of the energy
cillator [60] subject to an external potential in a topo-
eigenvalues:
logically non-trivial four-dimensional rotating space-
En,l,q = -Ω q±
time. In Ref. [26], authors studied the KG-oscillator in
[
]1/2
2
this topologically non-trivial rotating space-time with-
q
η2
± M2+
-
(
)2
(13)
out any external potential. In this work, we have in-
√
R2
1
n+
(l - Φ)2 + η2 +
serted a Coulomb-type external potential and magnetic
2
675
F. Ahmed
ЖЭТФ, том 162, вып. 5 (11), 2022
flux as stated earlier and analyze their effects on the
As stated earlier the wave function ψ(x) is well-
eigenvalue solution of the oscillator fields. The KG-
behaved and regular everywhere. Suppose, a possible
oscillator analogous to the Dirac oscillator [61] has at-
solution to the above radial wave equation Eq. (21) is
tracted attention among researchers in current times
given by
(see, Refs. [19, 22, 26, 27, 33, 57, 62]). The KG-oscillator
ψ(x) = xj e-
2
H(x),
(22)
is examined by the replacements of the radial momen-
where H(x) is an unknown function.
tum vector [19, 22, 26, 27, 33, 57, 62]
Thereby, substituting the radial wave function Eq.
(22) into the Eq. (21), we have
p → (p - iM ωr),
p† → (p + i M ω r),
(16)
]
[
]
[1+2j
ς
where ω is the frequency of the oscillator fields, and r
H′′(x) +
- 2x H′(x) +
-
+ Ξ H(x) = 0,
x
x
being distance from the particle to the axis of symme-
(23)
try.
where Ξ =ΛMω - 2 (1 + j).
Therefore, the Klein-Gordon oscillator equation is
Equation (23) is the biconfluent Heun differential
given by
equation form [22,32,33,40,42] and H(x) is the Heun
[
function. Substituting a power series expansion
(
)
1
-
Dμ + M ω Xμ ×
∑
√-g
H(x) =
di xi
{
(
)}
√
i=0
×
-g gμν
Dν - M ω Xν
+
]
[59] into the Eq. (23), we have obtained few coefficients
(
)2
(
)
[
]
+ M + S(r)
Ψ = 0,
(17)
ς
1
d1 =
d0, d2 =
ςd1 -Ξd0
1+2j
4 (1 + j)
where Xμ = (0, r, 0, 0) = r δrμ is a four-vector.
with the following recurrence relation
Explicitly witting the KG-oscillator equation (17)
[
]
1
in the rotating space-time background (1) and using
dm+2 =
ς dm+1-(Ξ-2m)dm .
(m + 2)(m + 2 + 2 j)
the electromagnetic potential Eq. (3) and the external
(24)
potential Eq. (5), we have
One can see this power series expansion H(x) becomes
[
)
2
a polynomial of finite degree m by imposing the follow-
(∂
∂
∂2
1
∂
-
-Ω
+
+
-M2ω2r2 -2Mω
ing two conditions [22, 32, 33, 40, 42]
∂t
∂θ
∂r2
r ∂r
]
(
)2
Ξ = 2m (m = 1,2,...)
,
dm+1 = 0.
(25)
2
(
)2
1
∂
1
∂
η
+
-iΦ
+
Ψ= M+
Ψ. (18)
r2
∂φ
R2 ∂θ2
r
By analyzing the first condition, we have obtained fol-
lowing energy eigenvalue Em,l,q expression:
Substituting the wave function (6) into the Eq. (18),
Em,l,q = -Ω q ±
we have obtained the following radial wave equation:
[
[
]
2
± M2 +2Mωm,l ×
1
j
2γ
ψ′′(r) +
ψ′(r) + Λ- M2 ω2 r2 -
-
ψ(r) = 0,
r
r2
r
]1/2
(
√
)
(19)
q2
× m+
(l - Φ)2 + η2 + 2
+
(26)
where j, γ are defined in Eq. (9) and
R2
)2
(q
The corresponding radial wave function is given by
Λ = (E + Ωq)2 - M2 - 2M ω -
(20)
R
√
x2
ψm,l(x) = x
(l-Φ)2+η2 e- 2 H(x),
(27)
Let us now perform a change of variables via
√
x=
M ω r. Then, Eq. (19) can be rewritten as
where H(x) is now a finite degree polynomial of degree
[
]
m.
2
1
Λ
ς
j
Finding solutions of the quantum system still not
ψ′′(x)+
ψ′(x)+
-x2-
-
ψ(x) = 0, (21)
x
Mω
x
x2
complete because one must analyze the second condi-
tion dm+1 = 0 one by one to get the complete infor-
2γ
where ς =
√
mation of a quantum state. As example, for the radial
Mω
676
ЖЭТФ, том 162, вып. 5 (11), 2022
Gravitational Field Effects. . .
mode m = 1, we have Ξ = 2 and d2 = 0 which gives
space-time geometry, and the Coulomb-type potential.
us a constraint on the oscillation frequency ω → ω1,l
One can show that the presented energy eigenvalue gets
given by
modified in comparison to those result obtained in [26]
(
)
due to the presence of the Coulomb-type external po-
2
Mη
tential and the magnetic quantum flux. This Coulomb-
ω1,l =
√
(28)
(l - Φ)2 + η2 +1
type external potential is responsible for the bound-
2
state solutions, and thus, the ground state is defined
Therefore, the ground state energy level associated
by the radial quantum number n = 1 instead of n = 0.
with the radial mode m = 1 is given by
E1,l,q = -Ω q ±
-
(√
)
)2
(l - Φ)2 + η2 + 3
( q
4. CONCLUSIONS
±M
√1 + 2 η2
√
+
.(29)
(l - Φ)2 + η2 +1
MR
2
In this analysis, we have determined solutions of
And the ground state radial wave function is given by
the wave equation under the effects of the gravitational
√
2
field produced a topologically non-trivial geometry sub-
ψ1,l(x) = x
(l-Φ)2+η2 e-x2
×
(
)
ject to a Coulomb-type external potential in a rotating
x
frame of reference. We have seen that the non-trivial
× 1+
d0.
(30)
√√
topology of the geometry defined by the radius R of
(l - Φ)2 + η2 +1
2
the compact dimension, and the Coulomb-type external
potential modified the eigenvalue solutions. Further-
Similarly, for the radial mode m = 2, we have Ξ = 4
more, the presence of the magnetic flux caus(s a chang)
and d3 = 0 which gives us another constraint on the os-
cillation frequency ω → ω2,l given by
in the angular quantum number l → l0 = l -eΦB
2π
(
)
which shows that the energy eigenvalue depends on
2
1
Mη
the geometric quantum phase. This dependence of the
ω2,l =
√
,
(31)
2
(l - Φ)2 + η2 + 1
eigenvalue on the geometric quantum phase gives us
the gravitational analogue to the Aharonov-Bohm ef-
Therefore, the first excited state energy level of the
fect [37, 38]. Several authors have been investigated
bound-states solution defined by the radial mode m = 2
this quantum mechanical effect in literature (e. g.,
is given by
[27, 30, 31, 33]). Also, we have seen a coupling between
the angular quantum number q and the uniform angu-
E2,l,q = -Ω q ±
lar speed Ω of the rotating frame of reference. This
-
(√
)
)2
coupling causes asymmetry in the relativistic energy
(l - Φ)2 + η2 + 3
( q
±M
√1 + η2
√
+
. (32)
levels, and hence, are not equally spaced on either side
(l - Φ)2 + η2 + 1
MR
about En/m,l,q = 0 for constant values of l, q.
And the corresponding radial wave function is given
We has seen that the presence of Coulomb-type po-
by
tential allowed the formation of bound-state solutions
√
2
and causes difference in results with those obtained in
ψ2,l(x) = x
(l-Φ)2+η2 e-x2 (d0 + d1 x + d2 x2),
(33)
Ref. [26]. Another point we have noticed is that the ro-
1
tating frames restricted the radius of compact circle S
where
in the range R <1Ω , and an analogous to the Sagnac-
⎛√√
⎞
3
type effect [6,10,27,33] is observed due to the coupling
(l - Φ)2 + η2 +
4
d1 = 2⎝
√
⎠d0,
between the quantum number q and uniform angular
(l - Φ)2 + η2 +1
2
speed Ω of rotating frames. This coupling causes asym-
(
)
metry in the energy levels and therefore, they are not
1
d2 =
√
d0.
(34)
equally spaced on either side about En,l,q = 0 for con-
(l - Φ)2 + η2 +1
2
stant values of l, q.
We can see that the energy eigenvalues and the wave
The full text of this paper is published in the English
function are modified by the non-trivial topology of the
version of JETP.
677
5
ЖЭТФ, вып. 5 (11)
F. Ahmed
ЖЭТФ, том 162, вып. 5 (11), 2022
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