On relativistic corrections to microlensing effects: Applications to the Galactic black hole
The standard treatment of gravitational lensing by a point mass lens M is based on a weak-field deflection angle α̂ = 2/xo, where X0 = r0c/2GM with r0 being the distance of closest approach to the mass of a lensed light ray. It was shown that for a point mass lens, the total magnification and image centroid shift of a point source remain unchanged by relativistic corrections of second order in 1/X 0. This paper considers these issues analytically, taking into account the relativistic images, under three assumptions A1-A3, for a Schwarzschild black hole lens with a background point and extended sources having arbitrary surface brightness profiles. The assumptions are A1, the source is close to the line of sight and lies in the asymptotically flat region outside the black hole lens; A2, the observer-lens and lens-source distances are significantly greater than the impact parameters of the lensed light rays; and A3, the distance of closest approach of any light ray that does not wind around the black hole on its travel from the source to the observer, lies in the weak-field regime outside the black hole. We apply our results to the Galactic black hole for lensing scenarios where A1-A3 hold. We show that a single factor characterizes the full relativistic correction to the weak-field image centroid and magnification. As the lens-source distance increases, the relativistic correction factor strictly decreases. In particular, we find that for point and extended sources approximately 10 pc behind the black hole, which is a distance significantly outside the tidal disruption radius of a Sun-like source, the relativistic correction factor is minuscule, of the order of 10 -14. Therefore, for standard lensing configurations, any detectable relativistic corrections to microlensing by the Galactic black hole will most likely have to come from sources significantly closer to the black hole.
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