# Comment on “Linear confinement of a scalar particle in a Gödel-type space-time [Eur. Phys. J. C (2018) **78** : 44]”

## Abstract

We point out an incorrect expression in a recent paper published in this journal (Vitória et al. Eur Phys J C, 78:44, 2018) regarding solution for the Klein–Gordon equation without potential in the background of Som–Raychaudhuri space-time with the cosmic string. The obtained eigenvalue of energy there is not similar to the result in Carvalho (Eur Phys J C 74: 2935, 2014).

In a recent paper in this journal, Vitória et al. [1] have studied relativistic quantum dynamics of a scalar particle subject to a linear scalar potential in the background of Som–Raychaudhuri space-time with the cosmic string. The authors obtained the bound states energy eigenvalues and the corresponding wavefunctions. The obtained expression Eqs. (5)–(21) in Ref. [1] are correct. After that, they have obtained two expressions Eqs. (22)–(23) for the same metric without any potential which are seem to be incorrect. The eigenvalue of energy Eq. (23) is not similar to the result Eq. (14) obtained in Ref. [2]. In this paper, we derive the final form of the Klein-Gordon equation in the background of Som–Raychaudhuri space-time with the cosmic string without any potential, and solve it using the method of Nikiforov–Uvarov [3]. We then obtain the correct expression of the energy eigenvalue which is similar to the result obtained in Ref. [2].

*M*is described by the Klein–Gordon equation:

*g*is the determinant of the metric tensor with \(g^{\mu \nu }\) its inverse.

*M*as \(M\rightarrow M'=M+ S\) where,

*S*is the scalar potential. Thereby, the KG-equation (2) becomes

*E*is the energy of the scalar particle, \(l=0,\pm 1, \pm 2, \pm 3, \ldots .\) are the eigenvalues of the

*z*-component of the angular mpmentum operator, and

*k*are the eigenvalues of the

*z*-component of the linear momentum operator.

*H*(

*x*) is the solution to the biconfluent Heun equation which has the form

*H*(

*x*) is achieved by imposing the following two conditions

For \(\alpha =1\), the energy eigenvalue reduces to the result obtained in Refs. [4, 5]. For \(M=0=k\), the energy eigenvalue (19) becomes \(E_{n,l}=2\,\varOmega \,(2\,n+1+\frac{l}{\alpha }+\frac{|l|}{\alpha })\). This is similar to the energy eigenvalue obtained for the Landau levels in the presence of cosmic string in Ref. [6].

In summary, we have solved the Klein-Gordon equation without interactions in the Som–Raychaudhuri space-time with the cosmic string using the Nikiforov–Uvarov method. We then have obtained the energy eigenvalue (19) and the corresponding eigenfunctions (20) of the system. We have seen that our eigenvalue of energy (19) is similar to the result obtained in Ref. [2] (see Eq. (14) in Ref. [2]) but different from the result in Ref. [1] (see Eq. (23) in Ref. [1]). This fact indicates that the eigenvalue of energy obtained in Ref. [1] is incorrect.

## Notes

### Acknowledgements

The author acknowledge valuable comments and suggestions from the anonymous kind referee(s).

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