Shadows and deflection angle of charged and slowly rotating black holes in Einstein-Æther theory

Abstract

In this paper, we study the shadow cast by two types of charged and slowly rotating black holes in the Einstein-\AE{}ther theory of gravity. These two types of black holes correspond to two specific combinations of the coupling constants of the \ae{}ther field, i.e., ${c}_{14}=0$ but ${c}_{123}\ensuremath{\ne}0$ for the first type and ${c}_{123}=0$ for the second type. For both types of black holes, in addition to the mass and charge of the black holes, we show that the presence of the \ae{}ther field can also affect the size of the shadow. For the first type of black hole, it is shown that the shadow size increases with the parameter ${c}_{13}$, while for the second type of black hole, the shadow size still increases with ${c}_{13}$ but decreases with the parameter ${c}_{14}$. With these properties of the \ae{}ther 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 \ae{}ther field on 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 \ae{}ther parameter ${c}_{13}$ at the leading order.