Gravitational lensing of massive particles in Reissner-Nordstr\"om spacetime

In this work we study the deflection angle $\Delta \varphi$ and gravitational lensing of both lightlike and timelike neutral rays in Reissner-Nordstr\"om (RN) spacetimes. The exact deflection angle is found as an elliptical function of the impact parameter $b$ and velocity $v$ of the ray, and the charge $Q$ of the spacetime. In obtaining this angle, we found the critical impact parameter $b_c$ and radius of particle sphere $r_c$ that are also dependent on $v$ and $Q$. In general, both the increase of velocity and charge reduces the $b_c$ as well as $r_c$. To study the effect of $v$ and $Q$ on the deflection angle $\Delta\varphi$, its weak and strong deflection limits, relativistic and non-relativistic limits, and small charge and extremal RN limits are analyzed carefully. It is found that both the increase of velocity and charge reduces the deflection angle. For weak deflection, the velocity and charge corrections appear respectively in the $\mathcal{O}(1/b)$ and $\mathcal{O}(1/b^2)$ orders. For strong deflections, these two corrections appear in the same order. The apparent angles and magnifications of weak and strong regular lensing, and retro-lensing are studied for both lightlike and timelike rays. In general, in all cases the increase of velocity or charge will decrease the apparent angle of any order. We show that velocity correction is much larger than that of charge in the weak lensing case, while their effects in the strong regular lensing and retro-lensing are comparable. It is further shown that the apparent angle and magnification in strong regular lensing and retro-lensing can be effectively unified. Finally, we argue that the correction of $v$ and $Q$ on the apparent angle can be correlated to mass or mass hierarchy of timelike particles with certain energy. In addition, the effects of $v$ and $Q$ on shadow size of black holes are discussed.