# Linear confinement of a scalar and spin-0 particle in a topologically trivial flat Gödel-type space-time

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## Abstract

In the previous work (Ahmed, Eur Phys J C 78(7):598, 2018), we investigated the relativistic quantum effects on a scalar and spin-half particles in a topologically trivial flat Gödel-type space-time. We have found that the energy eigenvalues of the system are influenced by the vorticity parameter characterizing the space-time. In the present work, we investigate the linear confinement of a scalar particle on the Klein–Gordon equation with a linear and Coulomb-type scalar potential in this flat Gödel-type solution. The energy eigenvalues of the system get modifies due to the presence of scalar potentials, and the vorticity parameter. In addition, we study the relativistic quantum motion of spin-0 particles with vector and scalar potentials of Coulomb-type and analyze the effects on the energy eigenvalues.

## 1 Introduction

Investigation of relativistic quantum effects on scalar and spin-half particles in Gödel as well as Gödel-type space-times have been addressed by several authors (see [1] and references therein). Soares et al. [29] first studied this problem, where the Klein–Gordon and Dirac equations in a class of Gödel-type space-times with positive and negative curvatures, were investigated and also in flat Gödel-type space-time. Drukker et al. [21] have investigated the close relationship between the quantum dynamics of a scalar particle in background of general relativity with Gödel solutions and the Landau levels in the flat, spherical and hyperbolic spaces. They solved the Klein–Gordon equation in these class of Gödel-type space-times and observed the similarity of the energy eigenvalues with the Landau levels in curved background (see also [30]). Das et al. [22] also have investigated the same problem by studying the Klein–Gordon equation in flat Gödel-type solution (called Som-Raychaudhuri space-time) and Landau levels in flat spaces. Furtado et al. [31] have studied the Landau levels in the presence of disclination parameter. Carvalo et al. [23] solved the Klein–Gordon equation in a class of Gödel-type solutions with a cosmic string and analyzed the similarity of the energy eigenvalues with the Landau levels in flat, spherical and hyperbolic spaces. They demonstrated there that the presence of a cosmic string, and the vorticity parameter modifies the energy levels and breaks the degeneracy of energy eigenvalues. The quantum influence of topological defects in Gödel-type space-times in flat, spherical and hyperbolic cases, were investigated in [24]. The relativistic quantum dynamics of Dirac particle with the topological defect in a class of Gödel-type space-times with torsion have been investigated in [25]. In Ref. [32] (see also [33, 34]), Weyl fermions in a class of family of Gödel-type geometries with a topological defect, were investigated. In Ref. [35], relativistic wave equation for spin-half particles in the Melvin space-time, a space-time where the metric is determined by a magnetic field, were investigated.

The relativistic quantum dynamics of a scalar particle subject to different confining potentials has been investigated in several areas of physics by various authors [36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50]. The linear confinement of quantum particles has great importance for models of confinement of quarks [51]. It is worth mentioning the linear scalar potential has attracted a great interest in atomic and molecular physics [52, 53, 54, 55, 56, 57, 58] and also in relativistic quantum mechanics [59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79]. Another important case is the confinement of a scalar particle subjected to Coulomb-type potential which were investigated by many authors [80, 81, 82, 83, 84]. It is worth mentioning studies that have dealt with Coulomb-type potential in the propagation of gravitational waves [85], quark models [51], and relativistic quantum mechanics [86, 87, 88, 89]. In Refs. [84, 90], a scalar potential is introduced (non-electromagnetic potential) in the Klein–Gordon equation by making a modification in the mass term in the form : \(m\rightarrow m+S\), where *m* is the mass of the free particle and *S* is the scalar potential (linear or Coulomb-type). This modification in the mass term has been explored in recent decades, for instance, by analysing the behaviour of Dirac particles in the presence of a static scalar and Coulomb potential [91], relativistic scalar particle in the presence of a cosmic string [92]. The quark-antiquark interaction is mapped into a problem of relativistic spinless possessing a position-dependent mass (PDM), where the mass term acquires a contribution given by the interaction potential that consist of linear and harmonic confining potential plus a Coulomb potential term, were investigated in [93]. The Klein–Gordon equation with vector and scalar potentials of Coulomb-types under the influence of non-inertial effects in the cosmic string space-time, were studied by Santos et al. [94]. The relativistic quantum effects of confining potentials on the Klein–Gordon oscillator, were investigated in [77]. Boumali et al. [95] investigated the Klein–Gordon oscillator in the background of cosmic strings in the presence of a uniform magnetic field. Later, the Klein–Gordon oscillator was investigated in the presence of Coulomb-type potential by two ways: (1) via a modification of mass term [89], and (2) via the minimal coupling [78], in the latter case the linear scalar potential was also included. The Klein–Gordon oscillator in curved background within the Kaluza-Klein theory, were investigated in [24]. Recently, Santos et al. [94] investigated the Klein–Gordon oscillator in the background space-time generated by a cosmic string. A scalar quantum particle confined in two concentric thin shells in curved space-time backgrounds with a cosmic string, were investigated in [33]. It is worth mentioning the studies of Dirac oscillator under the influence of non-inertial effects in the background of cosmic string space-time [96], and the rotating effects in the cosmic string space-time [97]. In Ref. [98], two different classes of solution for the Klein–Gordon equation in the presence of a scalar potential under the influence of noninertial effects in the cosmic string space-time, were discussed. The behaviour of a scalar particle within the Yukawa-like potential in Som-Raychaudhuri space-time, were investigated in [99].

## 2 The Klein–Gordon and spin-0 Equation

*m*is described by the Klein–Gordon equation. In its covariant form, this equation takes the following form:

*r*at the time

*t*. By introducing a scalar potential into the Klein–Gordon equation by modifying the mass term in the form : \(m \rightarrow m+S\), where

*S*is the scalar potential [84, 90]. The KG-equation becomes

*e*is the electric charge, and \(A_{\mu }\) is the vector potential. A scalar potential

*S*may be taken into account by making a modification on the mass term: \(m\rightarrow m+S\). Substituting this mass term into Eq. (7) we obtain the following differential equation:

*S*and vector potential \(A_{\mu }\) [83, 88, 101]. In order to simplify solution of the Klein–Gordon equation, the four-vector potential can be written as \(A_{\mu }=(A_{0},0,0,0)\). The first component of the four-vector potential is represented by a vector potential,

*i.e.,*\(A_{0}=V\). To have a bound-state (real) solutions for a relativistic spin-zero particle, the relationship between vector and scalar potential must be \(S\ge V\) [38, 42, 43, 44].

In the present work, we consider a flat Gödel-type space-time and discuss the linear confinement of a relativistic scalar particle with a linear scalar potential. We also study solution of the Klein–Gordon equation in the presence of Coulomb-type scalar potential and analyze the influence of confining potential, and the vorticity parameter on the energy eigenvalues. In addition, we investigate the relativistic quantum motion of spin-0 particles with scalar and vector potentials of Coulomb-type and analyze the effects.

## 3 A flat Gödel-type space-time

*y*coordinate, one can easily show that the space-time display causality violation, namely, closed time-like curves for \( x >\frac{1}{\alpha }\) and closed null curve at \( x =x_0=\frac{1}{\alpha }\). The determinat of the metric tensor for the metric (9) is \(det g=-1\) and the covariant and contravariant components of the metric tensor are

*x*. Here \(\varOmega \) is the vorticity parameter characterising the space-time.

*z*-axis given by \(B_{z}=2\,\varOmega \).

*z*-axis given by \(B_{z}=2\,\varOmega \).

Note that if one takes \(\varOmega =0\) which implies \(\alpha =0\), the study space-time (9) or (13) reduces to four-dimensional Minkowski flat space metric.

### 3.1 Scalar particle with linear and Coulomb-type scalar potentials in flat Gödel-type metric

*t*,

*y*,

*z*. One can choose the following ansatz for the function \(\varPsi \)

*E*, \(p_{y}\) and \(p_{z}\) are constants. Here we have considered the potentials as follow:

**Case A**: The linear scalar potential [84, 90] is given by

**Case B**: The scalar potential of Coulomb-type is given by

*H*(

*r*) is the biconflent Heun function :

*H*(

*r*) as a power series expansion around the origin in Eq. (25), then, the relativistic bound state solutions can be achieved by imposing that the power series expansion (25) or the biconfluent Heun series becomes a polynomial of degree

*n*. Through the recurrence relation (41), we can see that the power series expansion (25) becomes a polynomial of degree

*n*by imposing two conditions [31, 89, 107, 109, 110, 111]:

Note that the Eq. (45) does not represent the general expression for eigenvalue problem. One can obtain the eigenvalues one by one, that is, \(E_{1}\), \(E_{2}\) by imposing the additional condition \(c_{n+1}=0\) in the recurrence relation. The solution with Heun’s Equation makes it possible to obtain the eigenvalues one by one as done in Refs. [109, 110] and not explicited in general form by all eigenvalues *n*. To obtain the eigenvalues and corresponding eigenfunctions explicitly, two special cases corresponding to \(n=1,2\) are study below in some details similar to that in [109, 110]. The remaining cases corresponding to \(n>2\) can be studied in the same way.

### 3.2 Spin-0 particle with vector and scalar potentials of Coulomb-type in flat Gödel-type metric

Here we study solutions of the Klein–Gordon equation with vector and scalar potentials of Coulomb-type.

*t*,

*y*,

*z*, so it is appropriate to choose the ansatz given in (16), we obtain the following differential equation :

*H*(

*r*) is the biconflent Heun function :

*n*. The eigenfunctions and corresponding eigenvalues for two special cases corresponding to \(n=1,2\) are studied below in some details (see Refs. [109, 110]). The remaining cases corresponding to \(n>2\) can be studied in the same way.

## 4 Results

Before presenting our result, let us present the recent studies [28, 58, 77, 78, 107, 114] on the problems involving Linear, Coulomb-type and other interactions on the Klein–Gordon equation and the Heun’s differential equation.

*n*of a scalar particle in the Klein–Gordon equation under the influence of linear scalar potential with a cosmic string.

Note that for solutions investigated in this article, we do not have a general expression for the eigenvalues of energy. For solutions that have discussed in *sub-section* 3.1 and *sub-section* 3.2, the Eqs. (45) and (66) are not represent the general expression for eigenvalues of energy. By imposing the additional condition \(c_{n+1}=0\) in the recurrence relation, one can get solutions for each level individually as done in Refs. [109, 110]. We have evaluated these levels corresponding to \(n=1,2\) by Eq. (49), Eq. (52) and Eq. (71), Eq. (74) for examples. Similarly, the Eq. (91), Eq. (92) are the eigenvalues for each level due to the imposition of the additional condition \(c_{n+1}=0\).

## 5 Conclusions

The relativistic energy levels of a scalar and spin-0 particles with linear scalar potential in Gödel universe, Gödel-type solutions in flat, spherical and hyperbolic spaces with or without cosmic string, were investigated by several authors. In Ref. [28], authors have shown that the vorticity and the topological defect stem from a particular Gödel-type solution called the Som-Raychaudhuri space-time with a cosmic string. By analysing the energy associated with the radial mode, they have shown that both the vorticity and the topology of the cosmic string modifies the energy eigenvalues and give rise to the allowed energies. Note that the line element considered for the Som-Raychaudhuri space-time in [28] is for the cylindrical symmetry system and by solving the Klein–Gordon equation they obtained the energy eigenvalues. In Ref. [99], authors considered the cylindrical symmetry Som-Raychaudhuri space-time in the presence of a topological defect, and analyze the Yukawa-like confinement of a relativistic scalar particle. They have calculated the energy eigenvalues and wave-functions by using the generalized series method. In Ref. [94], spin-0 equation in the presence of a vector and a scalar potential, were investigated. They examined the wave equation on the Klein–Gordon oscillator in a cosmic string space-time and have shown that the potentials allow the formation of bound states, and the topological defect modifies the energy levels of the system. In Ref. [23], the obtained energy eigenvalues of flat Gödel-type space-time (Som-Raychaudhuri metric) is reduced to the energy levels obtained in [21, 30] and to the Landau levels in the presence of cosmic string [31].

We have analysed the influence of a linear scalar potential on the Klein–Gordon equation in a topologically trivial flat Gödel-type space-time. The flat Gödel-type space-time considered here is in cartesian coordinates system and we have solved the Klein–Gordon equation in this spac-time with a linear scalar potential. We have seen that Eq. (31) the energy eigenvalues of the system get modified by the presence of the confining potential, and the vorticity parameter \(\varOmega \) characterising the space-time. In the energy eigenvalues, the ground state is determined by the quantum number \(n=1\) instead of the quantum number \(n=0\). Observe that the present result is different from the one obtained in Ref. [28] because of the cylindrical symmetry Som-Raychaudhuri space-time whereas, the present Gödel-type flat solution is in the cartesian coordinates system. Similarly, the obtained result is different from the one presented in Ref. [99]. Models with confining potentials have been used to describe the spectrum of quarkonia-type systems. In the case of linear confining potential, this would be the confining potential of quarks. This type of potential has been used to investigate quarkonia in heavy quark-model. Thereby, the study of the linear confinement of relativistic scalar particle in the studied space-time can be used as Quarkonia-type systems to investigate heavy quarkonia where, the rotation plays the role of external uniform magnetic field. We have also considered the Coulomb-type scalar potential on the Klein–Gordon equation and seen that the presence of Coulomb-like scalar potential modifies the energy levels and Eq. (45) the \(n^{th}\) degree polynomial of energy spectrum associated with this equation depends on the confining scalar potential, and the vorticity parameter and Eq. (46) the corresponding eigenfunctions. We have evaluated the eigenvalues and eigenfunctions for the special cases corresponding to \(n=1,2\) more explicitly Eqs. (49), (52). Furthermore, we have obtained Eq. (66) the energy eigenvalues for \(n^{th}\) degree polynomial with vector and scalar potentials of Coulomb-type by solving the relativistic Klein–Gordon equation of spin-0 particles. We have seen that the energy eigenvalues of the system get modifies and depend on the vorticity parameter, and the vector (\(\kappa _{0}\)) and scalar potential (\(k_{c}\)) of Coulomb-type. We have evaluated the energy eigenvalues and eigenfunction for the special cases corresponding to \(n=1,2\) in more details Eq. (71), Eq. (74) more explicitly. Note that the Eq. (45) and Eq. (66) are not represent the general expression for eigenvalues problem. One can obtain the eigenvalues one by one, that is, \(E_{1}\), \(E_{2}\) by imposing the condition \(c_{n+1}=0\) in the recurrence relation. The solution with Heun’s Equation makes it possible to obtain the eigenvalues one by one as done in Refs. [109, 110] and not explicited in general form by all eigenvalues *n*. We have evaluated solution for each level individually corresponding to \(n=1,2\) by Eq. (49), Eq. (52), Eq. (71), Eq. (74) and Eq. (91), Eq. (92) for examples.

So, in this paper, we have shown some results about quantum systems where general relativistic effects are taken into account, that in addition with the previous results [21, 22, 28, 35, 94, 98, 99] the present may possess some interesting effects.

## Notes

### Acknowledgements

I would like to thank the annoymous kind referee(s) for the positive suggestions and valuable comments which have greatly improved the present text. I am also very much thankful to the Academic Editor for giving the opportunity for revising this work.

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