Shadows and deflection angle of charged and slowly rotating black holes in Einstein-{\AE}ther theory

In this paper, we study the shadow cast by two types of charged and slowly rotating black holes in Eisntein-aether theory of gravity. This two types of black holes are corresponding to two specific combinations of the coupling constants of the aether field, i.e., $c_{14} = 0$ but $c_{123}\neq 0$ for the first type and $c_{123} = 0$ for the second type, respectively. For both types of black holes, in addition to the mass and charge of the black holes, we show that the presence of the aether field can also affect the size of the shadow. For the first type black hole, it is shown that the shadow size increases with the parameter $c_{13}$, while for the second type black hole, the shadow size still increases with c13 but decreases with the parameter $c_{14}$. With these properties of these aether parameters, we also discuss the observational constraints on these parameters by using the data of the first black hole image by the Event Horizon Telescope. In addition, we also explore the effect of the aether field on the the deflection angle of light and the time delay by using the Gauss-Bonnet theorem. It is shown that, for a specific combination $c_{123} = 0$, the deflection angle/time delay is slightly affected by the aether parameter $c_{13}$ at the leading order.