Deflection angle with electromagnetic interaction and gravitational-electromagnetic dual lensing

The trajectory deflection and gravitational-electromagnetic dual lensing (GEL) of charged signal in general charged static and spherically symmetric spacetimes are considered in this work. We showed that the perturbative approach previously developed for neutral particles can be extended to the electromagnetic interaction case. The deflection angle still takes a (quasi-)series form and the finite distance effect of both the source and observer can be taken into account. Comparing to pure gravitational case, the apparent angles of the images in the GEL, their magnifications and time delay all receive the electromagnetic corrections starting from the first non-trivial order. The sign and relative size of the leading corrections are determined by $\sim \frac{Q}{M}\frac{q}{E}$ where $M,~Q,~q,~E$ are the spacetime mass and charge, and signal particle charge and energy respectively. It is found that for $qQ>0$ (or $<0$), the electromagnetic interaction will decrease (or increase) the deflection angle, and in GEL the impact parameters, apparent angles, magnifications and total travel time for each image. The time delay is increased for small $\beta$ and $qQ>0$, and otherwise always increased regardless the sign of $qQ$. The results are then applied to the deflection and GEL of charged protons in cosmic rays in Reissner-Nordstrom, charged dilaton and charged Horndeski spacetimes.