Investigation of spin-$0$ massive charged particle subject to a homogeneous magnetic field with potentials in a topologically trivial flat class of G\"{o}del-type space-time

In this paper, we investigate relativistic quantum dynamics of spin- $0$ massive charged particle subject to a homogeneous magnetic field in the G\"{o}del-type space-time with potentials. We solve the Klein-Gordon equation subject to a homogeneous magnetic field in a topologically trivial flat class of G\"{o}del-type space-time in the presence of a Cornell-type scalar and Coulomb-type vector potentials and analyze the effects on the energy eigenvalues and eigenfunctions.


Introduction
The first solution to the Einstein's field equations containing closed time-like curves is the cylindrical symmetry Gödel rotating Universe [1]. Reboucas et al. [2,3,4] investigated the Gödel-type solutions characterized by vorticity, which represents a generalization of the original Gödel metric with possible sources and analyzed the problem of causality. The line element of Gödeltype solution is given by tic quantum dynamics of spin-zero particles in 4D curved space-time with the cosmic string subject to a homogeneous magnetic field was studied in [19].
Furthermore, Dirac and Weyl fermions in the background of the Som-Raychaudhuri space-times in the presence of topological defects with torsion was studied in [26]. Weyl fermions in the background of the Som-Raychaudhuri space-times in the presence of topological defects was studied in [27] (see, Refs. [28,29]). The relativistic wave-equations for spin-half particles in the Melvin space-time, a space-time where the metric is determined by a magnetic field was studied in [30]. The fermi field and Dirac oscillator in the Som-Raychaudhuri space-time was studied in [31]. The Fermi field with scalar and vector potentials in the Som-Raychaudhuri space-time was investigated in [32]. The Dirac particles in a flat class of Gödel-type space-time was studied in [5]. Dirac fermions in (1+2)-dimensional rotational symmetry space-time background was investigated in [33].
The relativistic quantum dynamics of a scalar particle subject to different confining potentials have been studied in several areas of physics by various authors. The relativistic quantum dynamics of scalar particles subject to Coulomb-type potential was investigated in [34,35,36,37]. It is worth mentioning studies that have dealt with Coulomb-type potential in the propagation of gravitational waves [38], quark models [39], and relativistic quantum mechanics [40,41,42,43]. Linear confinement of scalar particles in a flat class of Gödel-type space-time, were studied in [44]. The Klein-Gordon equation with vector and scalar potentials of Coulomb-type under the influence of non-inertial effect in cosmic string space-time was studied in [45]. The Klein-Gordon oscillator in the presence of Coulomb-type potential in the background space-time generated by a cosmic string was studied in [43,46]. Other works on the relativistic quantum dynamics are the Klein-Gordon scalar field subject to a Cornell-type potential [47], and survey on the Klein-Gordon equation in a Gödel-type space-time [48].
Our aim in this paper is to investigate the quantum effects on bosonic massive charged particle by solving the Klein-Gordon equation subject to a homogeneous magnetic field in the presence of a Cornell-type scalar and Coulomb-type vector potentials in Gödel-type space-time. We see that the presence of magnetic field as well as various potential modifies the energy spectrum.
2 Bosonic charged particle : The KG-Equation The relativistic quantum dynamics of a charged particle of modifying mass m → m + S, where S is the scalar potential is described by the following equation [42] 1 where g is the determinant of metric tensor with g µν its inverse, D µ = ∂ µ − i e A µ is the minimal substitution, e is the electric charge and A µ is the electromagnetic four-vector potential, and ξ is the non-minimal coupling constant with the background curvature.
We choose the electromagnetic four-vector potential such that the constant magnetic field is along the axis B = ∇ × A = −B 0ẑ . Consider the following stationary space-time [50] (see, [5,12,13,44,51]) in the Cartesian coordinates (x 0 = t, x 1 = x, x 2 = y, x 3 = z) is given by where α 0 > 0 is a real positive constant. In Ref. [5], we have discussed different classes of Gödel-type space-time. For the space-time geometry (4), it belongs to a linear or flat class of Gödel-type metrics. The parameter α 0 = 2 Ω where, Ω characterize the vorticity parameter of the space-time.
For Ω → 0, the study space-time reduces to four-dimensional flat Minkowski metric.
The determinant of the corresponding metric tensor g µν is The scalar curvature of the metric is For the space-time geometry (4), the equation (2) becomes Since the metric is independent of t, y, z. One can choose the following ansatz for the function Ψ where E is the total energy of the particle, l = 0, ± 1, ± 2, ... are the eigenvalues of the y-component operator, and −∞ < k < ∞ is the eigenvalues of the z-component operator.

Interaction with Cornell-type and Coulomb-type potentials
Here we study a spin-0 massive charged particle by solving the Klein-Gordon equation in the presence of an external fields in a flat class of Gödel-type space-time subject to a Cornell-type scalar and Coulomb-type vector potentials. We obtain the energy eigenvalues and eigenfunctions and analyze the effects due to various physical parameters. The Cornell-type potential contains a confining (linear) term besides the Coulomb interaction and has been successfully accounted for the particle physics data [52]. This type of potential is a particular case of the quarkantiquark interaction, which has one more harmonic-type term [53]. The Coulomb potential is responsible by the interaction at small distances and the linear potential leads to the confinement. The quark-antiquark interaction potential has been studied in the ground state of three quarks [54], and systems of bound heavy quarks [55,56,57]. This type of interaction has been studied by several authors ( [11,23,42,58,59,60,61,62,63,64]). We consider the scalar S to be Cornell-type [42] where η c , η L are the Coulombic and confining potential constants, respectively.
Another potential that we are interest here is the Coulomb-type potential which we discussed in the introduction. Therefore, the Coulomb-type vector potential is given by where ξ c is the Coulombic potential constants. Substituting the potentials (10) and (11) into the Eq. (9), we obatin the following equation: where we have defined is called the cyclotron frequency of the particle moving in the magnetic field. Let us define a new variable r = √ ω x, Eq. (12) becomes where We now use the appropriate boundary conditions to investigate the bound states solution in this problem. It is known in relativistic quantum mechanics that the radial wave-functions must be regular both at r → 0 and r → ∞.
Then we proceed with the analysis of the asymptotic behavior of the radial eigenfunctions at origin and in the infinite. These conditions are necessary since the wave-functions must be well-behaved in these limit, and thus, the bound states of energy eigenvalues for this system can be obtained. Suppose the possible solution to the Eq. (14) is Substituting the solution Eq. (16) into the Eq. (14), we obtain where γ = 1 + 2 j, Equation (17) is the biconfluent Heun's differential equation [42,43,46,59,60,61,62,63,65,66] with H(r) is the Heun polynomials function.
Writing the function H(r) as a power series expansion around the origin [49]: Substituting the series solution into the Eq. (17), we obtain the following recurrence relation: Few coefficients of the series solution are As the function H(r) has a power series expansion around the origin in Eq. (19), then, the relativistic bound states solution can be achieved by imposing that the power series expansion becomes a polynomial of degree n and we obtain a finite degree polynomial for the biconfluent Heun series. Furthermore, the wave-function ψ must vanish at r → ∞ for this finite degree polynomial of power series otherwise the function diverge for large values of r. Therefore, we must truncate the power series expansion H(r) a polynomial of degree n by imposing the following two conditions [11,14,42,43,44,46,59,60,61,62,63,67,68]: By analyzing the condition Θ = 2 n, we have the second degree eigenvalues equation The corresponding eigenfunctions is given by Note that Eq. (23) does not represent the general expression of the eigenvalue problem. One can obtain the individual energy eigenvalues one by one, that is, E 1 , E 2 , E 3 ,.. by imposing the additional recurrence condition c n+1 = 0 on the eigenvalues. The solution with Heun's equation makes it possible to obtain the individual eigenvalues one by one as done in [11,14,42,43,44,46,59,60,61,62,63,67,68]. In order to analyze the above conditions, we must assign values to n. In this case, consider n = 1, that means we want to construct a first degree polynomial to H(r). With n = 1, we have Θ = 2 and c 2 = 0 which implies from Eq. (21) a constraint on the physical parameter ω 1,l . The relation given in Eq. (25) gives the possible values of the parameter ω 1,l that permit us to construct first degree polynomial to H(r) for n = 1 [42,43,46,59]. Note that its values changes for each quantum number n and l, so we have labeled ω → ω n,l . In this way, we obtain the following energy eigenvalue E 1,l : Then, by substituting the real solution ω 1,l from Eq. (25) into the Eq. (26) it is possible to obtain the allowed values of the relativistic energy levels for the radial mode n = 1 of a position-dependent mass system. We can see that the lowest energy state is defined by the real solution of the algebraic equation Eq. (25) plus the expression given in Eq. (26) for the radial mode n = 1, instead of n = 0. This effect arises due to the presence of Cornell-type potential in the system. Note that, it is necessary physically that the lowest energy state is n = 1 and not n = 0, otherwise the opposite would imply that c 1 = 0 which is not possible.
The corresponding radial wave-function for n = 1 is given by where

Interaction without potential
Here we study a spin-0 massive charged particle by solving the Klein-Gordon equation in the presence of an external fields in a Gödel-type space-time without potential and obtain the relativistic energy eigenvalue. We choose here zero scalar and vector potentials, S = 0 = V . In that case, Eq (9) becomes The above equation can be expressed as Let us define a new variable r = (x + l ω ), Eq. (30) becomes where Again introducing a new variable ρ = √ ω r into the Eq. (31), we obtain which is similar a harmonic-type oscillator equation. Therefore, the energy eigenvalues equation is The energy eigenvalues associated with n th modes is E n = (2 n + 1) Ω + (2 n + 1) 2 Ω 2 + m 2 + k 2 + 2 ξ Ω 2 + 2 m ω c (2 n + 1) = (2 n + 1) Ω + (2 n + 1) 2 Ω 2 + m 2 + k 2 + 2 ξ Ω 2 + |e B 0 | (2 n + 1), where n = 0, 1, 2, .. We can see that the energy eigenvalues (35) depend on the parameter Ω characterizing the vorticity parameter of the space-time geometry, and the external magnetic field B 0 as well the non-minimal coupling constant ξ with the background curvature.
Equation (36) is the energy eigenvalues of spin-0 particle in the background of a flat class of Gödel-type space-time and consistent with the result in [12].
Thus we can see that the energy eigenvalues (35) in comparison to the result in [12] get modify due to the presence of external fields and the non-minimal coupling constant with the background curvature. Therefore, the individual energy levels for n = 0, 1 using (35) are follows: A special case corresponds to m = 0 = k, the energy eigenvalues (35) reduces to E n = (2 n + 1) Ω + (2 n + 1) 2 Ω 2 + 2 ξ Ω 2 + |e B 0 | (2 n + 1). (38) The individual energy levels for n = 0, 1 in that case are follows: And others are in the same way. We can see that the presence of external magnetic field B 0 as well as the non-minimal coupling constant ξ causes asymmetry in the energy levels and hence, the energy levels are not equally spaced. The eigenfunctions is given by where |N| = 1 2 n n! √ π is the normalization constant and H n (ρ) are the Hermite polynomials and define as

The Klein-Gordon Oscillator
Here we study a spin-0 massive charged particle by solving the Klein-Gordon equation of the Klein-Gordon oscillator in the presence of an external fields in a Gödel-type space-time subject to a Cornell-type scalar and Coulomb-type vector potentials. We analyze the effects on the relativistic energy eigenvalue and corresponding eigenfunctions due to various physical parameters.
To couple Klein-Gordon field with oscillator [69,70], the generalization of Mirza et al. prescription [71], in which the following change in the momentum operator is taken: where m is the particle mass at rest, ω 0 is the frequency of the oscillator and X µ = (0, x, 0, 0), with x being the distance of the particle. In this way, the Klein-Gordon oscillator equation becomes Using the space-time (2), we obtain the following equation Using the ansatz (8) into the above Eq. (44), we arrive at the following equation Substituting the potentials Eq. (3), (10) and (11) into the equation (45), we obtain the following equation: where we have defined Let us define a new variable r = √ω x, Eq. (46) becomes Suppose the possible solution to the Eq. (48) is Substituting the solution Eq. (50) into the Eq. (48), we obtain where γ is given earlier andΘ =β +θ Equation (51) is the biconfluent Heun's differential equation [42,43,46,59,60,61,62,63,65,66]. Substituting the series solution Eq. (19) into the Eq. (51), we obtain the following recurrence relation: Few coefficients of the series solution are The power series expansion becomes a polynomial of degree n by imposing following two conditions [11,14,42,43,44,46,59,60,61,62,63,67,68]: Using the first condition we obtain the following energy eigenvalues : The corresponding eigenfunctions is given by As done earlier, we obtain the individual energy levels by imposing the recurrence condition c n+1 = 0. For n = 1, we have c 2 = 0 which implies from Eq. (54) a constraint on the physical parameterω 1,l . The relation given in Eq. (58) gives the possible values of the parameterω 1,l that permit us to construct first degree polynomial to H(r) for n = 1 [42,59,43,46]. In this way, we obtain the following second degree algebraic equation for E 1,l : Then, by substituting the real solutionω 1,l from Eq. (58) into the Eq. (59) it is possible to obtain the allowed values of the relativistic energy levels for the radial mode n = 1 of a position-dependent mass system. We can see that the lowest energy state is defined by the real solution of algebraic equation Eq.
(58) plus the expression given in Eq. (59) for the radial mode n = 1, instead of n = 0. This effect arises due to the presence of Cornell-type potential in the system. Note that, it is necessary physically that the lowest energy state is n = 1 and not n = 0, otherwise the opposite would imply that c 1 = 0 which is not possible.
The corresponding radial wave-function for n = 1 is given by where

Conclusions
The relativistic quantum system of scalar and spin-half particles in Gödeltype space-times was investigated by several authors (e. g., [7,8,9,11,12,13,14,44,59]). They demonstrated that the energy eigenvalues of the relativistic quantum system get modified and depend on the global parameters characterizing the space-times. In this work, we have investigated the influence of vorticity parameter on the relativistic energy eigenvalues of a relativistic scalar particle in a Gödeltype space-time subject to a homogeneous magnetic field with potentials.
We have derived the radial wave-equation of the Klein-Gordon equation in a class of flat Gödel-type space-time in the presence of an external fields with(-out) potentials by choosing a suitable ansatz of the wave-function. In sub-section 2.1, we have introduced a Cornell-type scalar and Coulomb-type vector potentials into the considered relativistic system and obtained the energy eigenvalue Eq. (23) and corresponding eigenfunctions Eq. (24). We have seen that the presence of a uniform magnetic field and potential parameters modifies the energy spectrum in comparison those result obtained in [12].
By imposing the additional recurrence condition c n+1 = 0, we have obtained the ground state energy levels Eq. (26) and wave-functions Eq. (27)- (28) for n = 1. In sub-section 2.2, we have considered zero potential into the relativistic system and solved the radial wave-equation of the Klein-Gordon equation in the presence of an external field. We obtained the energy eigenvalues Eq. (35) and compared with the results obtained in [12]. We have seen that the relativistic energy eigenvalues Eq. (35) get modify in comparison to those in [12] due to the presence of a homogeneous magnetic field. In section 3, we have solved the Klein-Gordon equation of the Klein-Gordon oscillator in a Gödel-type space-time subjected to a homogeneous magnetic field in the presence of a Cornell-type scalar and Coulomb-type vector potentials.
We have obtained the energy eigenvalue Eq. (56) and corresponding eigenfunctions Eq. (57). We have seen that the presence of a uniform magnetic field and potential parameters modifies the energy spectrum in comparison to those in [12]. By imposing the additional recurrence condition c n+1 = 0, we have obtained the ground state energy levels Eq. (59) and wave-function Eq. (60) for n = 1 and others are in the same way. So, in this paper we have some results which are in addition to the previous results obtained in [7,8,9,11,12,13,44,51,59] present may interesting effects. This is the fundamental subject in physics and the connection between these theories (quantum mechanics and gravitation) are not well understood.

Data Availability
No data has been used to prepare this paper.

Conflict of Interest
Author declares that there is no conflict of interest regarding publication this paper.