Leading-order deflection of particles by a moving Schwarzschild lens with a two-dimensional velocity

The gravitational deflection effect of relativistic massive and massless particles up to the first post-Minkowskian order caused by a moving Schwarzschild black hole with a two-dimensional equatorial velocity, which contains the radial and transversal components, is studied analytically, and a new unified formula for the deflection angle is achieved. The expression of the angle matches well with the results of the weak deflection of relativistic particles induced by a radially moving Schwarzschild source given in the literature, when the transversal component of the lens velocity vanishes. The joint velocity effect, which consists of the influences of the transversal and radial motions of the lens on the leading-order Schwarzschild deflection of the massive particles and light, is then discussed in the context of general relativity. We analyze the order of magnitude of this kinematical effect and evaluate the possibility of its astronomical detection subsequently.