Relativistic Aharonov–Casher effect in 1+2-dimensional Gürses spacetime

Abstract

In this work, we investigate the topological effects of spacetime on the Dirac oscillator in the presence of the Aharonov–Casher effect. We establish the Dirac equation that defines the model. The energy spectrum and wave functions are derived as a function of various physical parameters involved in the problem. The energy spectrum of the oscillator is studied in detail. An important conclusion is that the combined influence of parameters changes the allowed energy states of the oscillator.

Appendix A. (1+2) dimensions Gürses metric

The spacetime metric considered by Gürses in three dimensions [34] (see Eq. (7) , notations are same) is given in the form

$$\begin{aligned} ds^{2}=\phi dt^{2}-2 qdt d\varphi -\frac{-q+h^{2}\psi }{\phi }d\varphi ^{2}-\frac{1}{\psi }dr^{2} \end{aligned}$$

(A.1)

where

$$\begin{aligned} \phi =a_{0},~~~~~~~~~~\psi =b_{0}+\frac{b_{1}}{r^{2}}+\frac{3\lambda _{0}}{4}r^{2}, ~~~~~~~~~~\nonumber \\ q=c_{0}+\frac{e_{0}\mu }{3}r^{2},~~~~~~~~~~~h=e_{0}r,~~~~~~~~~~~~~~~~~~~~~ \lambda _{0}=\lambda +\frac{\mu ^{2}}{27} \end{aligned}$$

(A.2)

and $a_{0}, b_{0}, b_{1}, c_{0}, e_{0}$ are arbitrary constants.

After the constants in (A.2), [35] are chosen as

$$\begin{aligned} b_{1}=0,~~~~~~~~~~~~~~~~ c_{0}=0,~~~~~~~~~~~~~~~~\lambda _{0}=0 \end{aligned}$$

(A.3)

substituting the found forms of $\phi ,\psi ,h, q,$ in (A.1), the spacetime metric takes on the following new form

$$\begin{aligned} ds^{2}=a_{0} dt^{2}-\frac{2 e_{0}\mu }{3}r^{2}dt d\varphi -\frac{1}{a_{0}}(e_{0}^{2}b_{0}r^{2}-\frac{e_{0}^{2}\mu ^{2}}{9}r^{4})d\varphi ^{2}- \frac{1}{b_{0}}dr^{2} \end{aligned}$$

(A.4)

Finally, choosing the constants in (A4) as

$$\begin{aligned} a_{0}=1,~~~~~~~~~~~~~~~~ b_{0}=1,~~~~~~~~~~~~~~~~e_{0}=1,~~~~~~~~~~~~~~~~\varOmega =-\frac{\mu }{3} \end{aligned}$$

(A.5)

(1+2) dimensions Gürses metric (7) presented in this work became

$$\begin{aligned} ds^{2}= dt^{2} -dr^{2}+2\varOmega r^{2}dt d\varphi -r^{2}(1-\varOmega ^{2} r^{2})d\varphi ^{2} \end{aligned}$$

(A.6)