# Noninertial effects on a scalar field in a spacetime with a magnetic screw dislocation

## Abstract

We investigate rotating effects on a charged scalar field immersed in spacetime with a magnetic screw dislocation. In addition to the hard-wall potential, which we impose to satisfy a boundary condition from the rotating effect, we insert a Coulomb-type potential and the Klein–Gordon oscillator into this system, where, analytically, we obtain solutions of bound states which are influenced not only by the spacetime topology, but also by the rotating effects, as a Sagnac-type effect modified by the presence of the magnetic screw dislocation.

## 1 Introduction

In the context of condensed matter physics, Katanaev and Volovich [1] formulated a description of defects in a three-dimensional continuous elastic solid medium, where such defects may be associated with curvature or torsion of the continuous medium. Then, Puntigam and Soleng [2] went further with this formulation through the generalization of the Volterra distortions, considering the temporal coordinate, that is, \((3+1)\) dimensions, in order to adapt these defects in a Einstein–Cartan gravity and introduce the concept of distorted spacetimes. According to this formulation, disclinations are associated with curvature, while dislocations are associated with torsion. In particular, the dislocations can be typified as spiral and screw [3]. Effects associated with the topology of a medium with dislocations have been investigated in crystal structures through differential geometry [4].

In an analysis by Landau and Lifshitz on the effects of rotation in the Minkowski spacetime with cylindrical symmetry, they showed that the radial coordinate becomes restricted in an interval, where this restriction is an effect directly related to the uniform rotation [21]. This restriction from the effects of uniform rotation has been widely used for studies in a relativistic quantum mechanics system, for example, in a Dirac particle [22], in a relativistic Landau–He–McKellar–Wilkens quantization [23], on the Dirac oscillator [24], on a scalar field in the spacetime with space-like dislocation and in the spacetime with a spiral dislocation [6], on the quantum dynamics of scalar bosons [25], in the relativistic quantum motion of spin-0 particles under in the cosmic string spacetime [26], in the Duffin–Kemmer–Petiau equation with magnetic cosmic string background [27]. In the nonrelativistic case, this restriction has been studied in a Dirac particle in the spacetime with a screw dislocation [16] and on nonrelativistic topological quantum scattering [28]. However, a point that has not been analyzed in the literature is the rotating effect on the scalar field by considering the spacetime with a magnetic screw dislocation as background, that is, the screw dislocation has in the core a magnetic field with magnetic quantum flux \(\Phi _B\) and outside the topological defect this magnetic field vanishes [7, 11].

*w*is the constant angular frequency of the rotating frame, which gives us the metric

In this paper, we investigate the relativistic Aharonov–Bohm effect for bound states [29, 30] on a scalar field in a spacetime with a screw dislocation, where this field is subject to confinement potentials in a uniformly rotating frame. We begin our analysis with the hard-wall potential. After, we consider a scalar field with position-dependent mass interacting with a Coulomb-type central potential. And, finally, we inserted the Klein–Gordon oscillator [31] and investigated the harmonic effects coming from this model of relativistic oscillator. In all these cases, we obtain analytical solutions, where they are not only influenced by the topology of the spacetime, but also the effects of rotation.

*m*is the rest mass of the scalar field and \(A_{\mu }=(0,0,A_{\varphi },0)\) is the electromagnetic 4-vector potential, where \(A_{\varphi }\) is given by [18, 33, 34]

The structure of this paper is as follows: in the Sect. 2, for a particular case, we investigate the effects of the spacetime topology and of rotation on an electrically charged scalar field subject to the hard-wall potential, where it is possible to obtain the energy levels of this system; in the Sect. 3, we inserted a Coulomb-type potential in the Klein–Gordon equation via the mass term and, for a particular case, extracted the energy profile of this system; in the Sect. 4, through a non-minimal coupling in the Klein–Gordon equation, we insert a relativistic oscillator model and analyze the harmonic effects on the scalar field in a uniformly rotating frame in the spacetime with a magnetic screw dislocation, where we determine two energy profiles for the system; in the Sect. 5, we present our conclusions.

## 2 Hard-wall confining potential

*w*, giving us, then, a Sagnac-type effect [22, 49, 50]. For \(\Phi _{B}=0\) and \(\chi \ne 0\) we recover the result obtained in the Ref. [6]; for \(\Phi _{B}\ne 0\) and \(\chi =0\) we obtain the relativistic energy levels of a charged scalar field subject to the Aharonov–Bohm effect in the Minkowski spacetime in a uniformly rotating frame.

## 3 Coulomb-type potential

The standard procedure of inserting the Coulomb potential into relativistic wave equations is through the minimum coupling \({\hat{p}}_{\mu }\rightarrow {\hat{p}}_{\mu }-qA_{\mu }\) via temporal component \(A_{0}\) [51]. Another procedure of inserting central potentials is by modifying the mass term of the relativistic wave equations via transformation \(m\rightarrow m+V(r)\), where *V*(*r*) is a scalar potential. The latter procedure entails a feature which is known in the literature as a position-dependent mass system. This type of system has been studied in atomic physics [52], in the rotating cosmic string spacetime [53, 54], on a two-dimensional Klein–Gordon particle [55], quark–antiquark interaction [56] and on a scalar particle in a Gödel-type spacetime [57].

*a*is a constant that characterizes the Coulomb-type potential. The Coulomb-type potential has been studied in propagation of gravitational waves [58], in a magnetic quadrupole moment [59], in a neutral particle with permanent magnetic dipole moment [60] and in Lorentz symmetry violation scenarios [61, 62]. Then, by substituting the Eq. (15) into the Eq. (6), we obtain from the solution (7) the radial differential equation

*w*and the effective angular momentum quantum number \(l_{\text {eff}}=l-\frac{q\Phi _B}{2\pi }\). By making \(w=0\) we recover the result obtained in the Ref. [19]. For \(\Phi _{B}=0\) and \(\chi \ne 0\), the Eq. (23) represents the relativistic energy levels of scalar field of position-dependent mass under effects of a Coulomb-type potential in a uniformly rotating frame in the spacetime with a screw dislocation; for \(\Phi _{B}\ne 0\) and \(\chi =0\), the Eq. (23) represents the relativistic energy levels of charged scalar field of position-dependent mass under effects of a Coulomb-type potential in the Minkowski spacetime.

## 4 Klein–Gordon oscillator

It is possible to discuss two energy profiles for this system. One for any value of the angular frequency of rotation which induces a hard-wall potential (general case) and another for very small values of the angular frequency of rotation (particular case).

### 4.1 General case

For an arbitrary value of the angular frequency of rotation implies in a similar case seen in Sect. 2, that is, the wave function must vanish in \(\varrho _0=\frac{m\omega (1-\chi ^2 w^2)}{w^2}\), restriction imposed by the rotation. This means that the charged scalar field is restricted in a region where this restriction is characterized by the presence of a hard-wall potential induced by the effects of rotation in a spacetime with a magnetic screw dislocation. This kind of confinement is described by the boundary condition given in the Eq. (12).

### 4.2 Particular case

*n*imposing that \({\bar{b}}=-n\), with \(n=0,1,2,\ldots \). Then, by following the discussion made in the Eq. (22) to the (23), we obtain the expression

## 5 Conclusion

We have investigated the effects of a uniformly rotating frame on a charged scalar field in the spacetime with a magnetic screw dislocation. Due to the rotating effects in this background, we can note that the radial coordinate is restricted and this restriction is determined by the spacetime topology. Through this restriction in the radial coordinate, we determine solutions of bound states, hence, we extract the energy profiles for the systems analyzed. Our investigation starts with the hard-wall potential, which, from the mathematical point of view, is a Dirichlet boundary condition imposed by the rotating effects, and with that we determine the relativistic energy profile of this system.

Through the definition of position-dependent mass system, we inserted a Coulomb-type potential into the Klein–Gordon equation by modifying the mass term. Analytically, for very small values of the frequency of rotation, we determine solutions of bound states and we can note that the presence of the Coulomb-type potential modifies the energy spectrum of the system.

We also investigate the Klein–Gordon oscillator, where, for well-defined rotating frequency scales, it is possible to determine two energetic profiles for this system. First, we determine the energy profile of the Klein–Gordon oscillator in a uniformly rotating frame for arbitrary values of the frequency of rotation, which induces a hard-wall potential. Next for very small values of the frequency of rotation. We can see that the two energy profiles are totally different.

It is worth mentioning that in all cases we can note the influence of the spacetime topology by redefining the angular momentum eigenvalue which is described in terms of the parameter associated with the screw dislocation and the parameter associated with the internal quantum flux of the defect. Consequently, the Sagnac-type effect, which arises at all energy levels due to rotation, is also influenced by the internal quantum flux of the topological defect.

## Notes

### Acknowledgements

The author would like to thank CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico, Brazil). Ricardo L. L. Vitória was supported by the CNPq project No. 150538/2018-9.

## References

- 1.M.O. Katanaev, I.V. Volovich, Ann. Phys.
**216**, 1 (1992)ADSCrossRefGoogle Scholar - 2.R.A. Puntigam, H.H. Soleng, Class. Quantum Gravity
**14**, 1129 (1997)ADSCrossRefGoogle Scholar - 3.K.C. Valanis, V.P. Panoskaltsis, Acta Mech.
**175**, 77 (2005)CrossRefGoogle Scholar - 4.H. Kleinert,
*Gauge Fields in Condensed Matter*, vol. 2 (World Scientific, Singapore, 1989)zbMATHCrossRefGoogle Scholar - 5.A.V.D.M. Maia, K. Bakke, Phys. B
**531**, 213 (2018)ADSCrossRefGoogle Scholar - 6.R.L.L. Vitória, K. Bakke, Eur. Phys. J. C
**78**, 175 (2018)ADSCrossRefGoogle Scholar - 7.C. Furtado, F. Moraes, J. Phys. A Math. Gen.
**33**, 5513 (2000)ADSCrossRefGoogle Scholar - 8.M.J. Bueno, C. Furtado, K. Bakke, Phys. B Condens. Matter
**496**, 45 (2016)ADSCrossRefGoogle Scholar - 9.A.L. Silva Netto, C. Furtado, J. Phys. Condens. Matter
**20**, 125209 (2008)ADSCrossRefGoogle Scholar - 10.C. Furtado, F. Moraes, Europhys. Lett.
**45**(3), 279 (1999)ADSCrossRefGoogle Scholar - 11.G.A. Marques, C. Furtado, V.B. Bezerra, F. Moraes, J. Phys. A Math. Gen.
**34**, 5945 (2001)ADSCrossRefGoogle Scholar - 12.K. Bakke, Phys. B Condens. Matter
**537**, 346 (2018)ADSCrossRefGoogle Scholar - 13.K. Bakke, L.R. Ribeiro, C. Furtado, Cent. Eur. J. Phys.
**8**(6), 893 (2010)Google Scholar - 14.K. Bakke, Ann. Phys. (NY)
**346**, 51 (2014)ADSCrossRefGoogle Scholar - 15.J. Carvalho, C. Furtado, F. Moraes, Phys. Rev. A
**84**, 032109 (2011)ADSCrossRefGoogle Scholar - 16.K. Bakke, C. Furtado, Ann. Phys. (NY)
**336**, 489 (2013)ADSCrossRefGoogle Scholar - 17.J. Carvalho, A.M. de M. Carvalho, E. Cavalcante, C. Furtado, Eur. Phys. J. C
**76**, 365 (2016)ADSCrossRefGoogle Scholar - 18.R.L.L. Vitória, K. Bakke, Int. J. Mod. Phys. D
**27**, 1850005 (2018)ADSCrossRefGoogle Scholar - 19.R.L.L. Vitória, K. Bakke, Eur. Phys. J. Plus
**133**, 490 (2018)CrossRefGoogle Scholar - 20.R.L.L. Vitória, K. Bakke, Gen. Relativ. Gravit.
**48**, 161 (2016)ADSCrossRefGoogle Scholar - 21.L.D. Landau, E.M. Lifshitz,
*The Classical Theory of Fields, Course of Theoretical Physics*, vol. 2, 4th edn. (Elsevier, Oxford, 1980)Google Scholar - 22.F.W. Hehl, W.-T. Ni, Phys. Rev. D
**42**, 2045 (1990)ADSCrossRefGoogle Scholar - 23.K. Bakke, Ann. Phys. (Berlin)
**523**, 762 (2011)ADSCrossRefGoogle Scholar - 24.P. Strange, L.H. Ryder, Phys. Lett. A
**380**, 3465 (2016)ADSMathSciNetCrossRefGoogle Scholar - 25.L.B. Castro, Eur. Phys. J. C
**76**, 61 (2016)ADSCrossRefGoogle Scholar - 26.L.C.N. Santos, C.C. Barros Jr., Eur. Phys. J. C
**78**, 13 (2018)ADSCrossRefGoogle Scholar - 27.M. Hosseinpour, H. Hassanabadi, Eur. Phys. J. Plus
**130**, 236 (2015)CrossRefGoogle Scholar - 28.H.F. Mota, K. Bakke, Gen. Relativ. Gravit.
**49**, 104 (2017)ADSCrossRefGoogle Scholar - 29.Y. Aharonov, D. Bohm, Phys. Rev.
**115**, 485 (1959)ADSMathSciNetCrossRefGoogle Scholar - 30.M. Peshkin, A. Tonomura,
*The Aharonov–Bohm Effect in Lecture Notes in Physics*, vol. 340 (Springer, Berlin, 1989)CrossRefGoogle Scholar - 31.S. Bruce, P. Minning, Nuovo Cimento A
**106**, 711 (1993)ADSCrossRefGoogle Scholar - 32.E.R. Bezerra de Mello, Braz. J. Phys.
**31**, 211 (2001)ADSCrossRefGoogle Scholar - 33.A.L. Cavalcanti de Oliveira, E.R. Bezerra de Mello, Class. Quantum Gravity
**23**, 5249 (2006)ADSCrossRefGoogle Scholar - 34.A. Boumali, H. Aounallah, Adv. High Energy Phys.
**2018**, 1031763 (2018)CrossRefGoogle Scholar - 35.R. Jackiw, A.I. Milstein, S.-Y. Pi, I.S. Terekhov, Phys. Rev. B
**80**, 033413 (2009)ADSCrossRefGoogle Scholar - 36.M.A. Anacleto, I.G. Salako, F.A. Brito, E. Passos, Phys. Rev. D
**92**, 125010 (2015)ADSMathSciNetCrossRefGoogle Scholar - 37.V.R. Khalilov, Eur. Phys. J. C
**74**, 2708 (2014)ADSCrossRefGoogle Scholar - 38.C. Coste, F. Lund, M. Umeki, Phys. Rev. E
**60**, 4908 (1999)ADSMathSciNetCrossRefGoogle Scholar - 39.C. Furtado, F. Moraes, V. Bezerra, Phys. Rev. D
**59**, 107504 (1999)ADSMathSciNetCrossRefGoogle Scholar - 40.E.V.B. Leite, K. Bakke, H. Belich, Adv. High Energy Phys.
**2015**, 925846 (2015)CrossRefGoogle Scholar - 41.H. Belich, E.O. Silva, M.M. Ferreira Jr., M.T.D. Orlando, Phys. Rev. D
**83**, 125025 (2011)ADSCrossRefGoogle Scholar - 42.K. Bakke, Int. J. Theor. Phys.
**54**, 2119 (2015)MathSciNetCrossRefGoogle Scholar - 43.K. Bakke, H. Belich, J. Phys. G Nucl. Part. Phys.
**42**, 095001 (2015)ADSCrossRefGoogle Scholar - 44.K. Bakke, C. Furtado, Eur. Phys. J. B
**87**, 222 (2014)ADSCrossRefGoogle Scholar - 45.K. Bakke, Eur. Phys. J. B
**85**, 354 (2012)ADSCrossRefGoogle Scholar - 46.G.B. Arfken, H.J. Weber,
*Mathematical Methods for Physicists*, 6th edn. (Elsevier Academic Press, New York, 2005)zbMATHGoogle Scholar - 47.M. Abramowitz, I.A. Stegum,
*Handbook of Mathematical Functions*(Dover Publications Inc., New York, 1965)Google Scholar - 48.V.B. Bezerra, J. Math. Phys.
**38**, 2553 (1997)ADSMathSciNetCrossRefGoogle Scholar - 49.M.G. Sagnac, C. R. Acad. Sci. (Paris)
**157**, 708 (1913)Google Scholar - 50.M.G. Sagnac, C. R. Acad. Sci. (Paris)
**157**, 1410 (1913)Google Scholar - 51.W. Greiner,
*Relativistic Quantum Mechanics: Wave Equations*, 3rd edn. (Springer, Berlin, 2000)zbMATHCrossRefGoogle Scholar - 52.G. Soff, B. Müller, J. Rafelski, W. Greiner, Z. Naturforsch.
**28a**, 1389 (1973)ADSCrossRefGoogle Scholar - 53.M.S. Cunha, C.R. Muniz, H.R. Christiansen, V.B. Bezerra, Eur. Phys. J. C
**76**, 512 (2016)ADSCrossRefGoogle Scholar - 54.Z. Wang, Z. Long, C. Long, B. Wang, Can. J. Phys.
**95**(4), 331 (2017)ADSCrossRefGoogle Scholar - 55.S.M. Ikhdaira, M. Hamzavi, Chin. Phys. B
**21**, 110302 (2012)CrossRefGoogle Scholar - 56.M.K. Bahar, F. Yasuk, Adv. High Energy Phys.
**2013**, 814985 (2013)CrossRefGoogle Scholar - 57.R.L.L. Vitória, C. Furtado, K. Bakke, Eur. Phys. J. C
**78**, 44 (2018)ADSCrossRefGoogle Scholar - 58.H. Asada, T. Futamase, Phys. Rev. D
**56**, R6062 (1997)ADSCrossRefGoogle Scholar - 59.I.C. Fonseca, K. Bakke, J. Math. Phys.
**56**, 062107 (2014)ADSCrossRefGoogle Scholar - 60.P.M.T. Barboza, K. Bakke, Ann. Phys.
**361**, 259 (2015)CrossRefGoogle Scholar - 61.K. Bakke, H. Belich, Ann. Phys.
**360**, 596 (2015)CrossRefGoogle Scholar - 62.R.L.L. Vitória, H. Belich, K. Bakke, Adv. High Energy Phys.
**2017**, 6893084 (2017)CrossRefGoogle Scholar - 63.M. Moshinsky, A. Szczepaniak, J. Phys. A Math. Gen.
**22**, L817 (1989)ADSCrossRefGoogle Scholar - 64.Y. Nogami, F.M. Toyama, Can. J. Phys.
**74**, 114 (1996)ADSCrossRefGoogle Scholar - 65.W. Moreau, R. Easther, R. Neutze, Am. J. Phys.
**62**, 531 (1994)ADSCrossRefGoogle Scholar - 66.V.M. Villalba, Eur. J. Phys.
**15**, 191 (1994)CrossRefGoogle Scholar - 67.N.A. Rao, B.A. Kagali, Mod. Phys. Lett. A
**19**, 2147 (2004)ADSCrossRefGoogle Scholar - 68.Victor M. Villalba, Phys. Rev. A
**49**, 1 (1994)CrossRefGoogle Scholar - 69.J. Cravalho, C. Furtado, F. Moraes, Phys. Rev. A
**84**, 032109 (2011)ADSCrossRefGoogle Scholar - 70.K. Bakke, C. Furtado, Ann. Phys.
**336**, 489 (2013)ADSCrossRefGoogle Scholar - 71.K. Bakke, H.F. Mota, Eur. Phys. J. Plus
**133**, 409 (2018)CrossRefGoogle Scholar - 72.L. Deng, C. Long, Z. Long, Ting Xu, Adv. High Energy Phys.
**2018**, 2741694 (2018)CrossRefGoogle Scholar - 73.A. Boumali, H. Hassanabadi, Eur. Phys. J. Plus
**128**, 124 (2013)CrossRefGoogle Scholar - 74.H. Hassanabadi, S. Sargolzaeipor, B.H. Yazarloo, Few-Body Syst.
**56**, 115 (2015)ADSCrossRefGoogle Scholar - 75.N.A. Rao, B.A. Kagali, Phys. Scr.
**77**, 015003 (2008)CrossRefGoogle Scholar - 76.J.-Y. Cheng, Int. J. Theor. Phys.
**50**, 228 (2011)CrossRefGoogle Scholar - 77.B. Mirza, R. Narimani, S. Zare, Commun. Theor. Phys.
**55**, 405 (2011)CrossRefGoogle Scholar - 78.M.-L. Liang, R.-L. Yang, Int. J. Mod. Phys. A
**27**, 1250047 (2012)ADSCrossRefGoogle Scholar - 79.A. Boumali, N. Messai, Can. J. Phys.
**92**, 11 (2014)CrossRefGoogle Scholar - 80.K. Bakke, C. Furtado, Ann. Phys. (NY)
**355**, 48 (2015)ADSCrossRefGoogle Scholar - 81.R.L.L. Vitória, K. Bakke, Eur. Phys. J. Plus
**131**, 36 (2016)CrossRefGoogle Scholar - 82.R.L.L. Vitória, C. Furtado, K. Bakke, Ann. Phys. (NY)
**370**, 128 (2016)ADSCrossRefGoogle Scholar - 83.R.L.L. Vitória, H. Belich, K. Bakke, Eur. Phys. J. Plus
**132**, 25 (2017)CrossRefGoogle Scholar - 84.R.L.L. Vitória, H. Belich, Eur. Phys. J. C
**78**, 999 (2018)ADSCrossRefGoogle Scholar - 85.B. Khosropour, Indian J. Phys.
**92**(1), 43 (2018)ADSCrossRefGoogle Scholar - 86.B. Hamil, M. Merad, Eur. Phys. J. Plus
**133**, 174 (2018)ADSCrossRefGoogle Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Funded by SCOAP^{3}