Gravitomagnetic bending angle of light with finite-distance corrections in stationary axisymmetric spacetimes

By using the Gauss-Bonnet theorem, the bending angle of light in a static, spherically symmetric and asymptotically flat spacetime has been recently discussed, especially by taking account of the finite distance from a lens object to a light source and a receiver [Ishihara, Suzuki, Ono, Asada, Phys. Rev. D 95, 044017 (2017)]. We discuss a possible extension of the method of calculating the bending angle of light to stationary, axisymmetric and asymptotically flat spacetimes. For this purpose, we consider the light rays on the equatorial plane in the axisymmetric spacetime. We introduce a spatial metric to define the bending angle of light in the finite-distance situation. We show that the proposed bending angle of light is coordinate-invariant by using the Gauss-Bonnet theorem. The non-vanishing geodesic curvature of the photon orbit with the spatial metric is caused in gravitomagnetism, even though the light ray in the four-dimensional spacetime follows the null geodesic. Finally, we consider Kerr spacetime as an example in order to examine how the bending angle of light is computed by the present method. The finite-distance correction to the gravitomagnetic deflection angle due to the Suns spin is around a pico-arcsecond level. The finite-distance corrections for Sgr A$^{ast}$ also are estimated to be very small. Therefore, the gravitomagnetic finite-distance corrections for these objects are unlikely to be observed with present technology.