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 Sun's 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.
