Schwarzschild light deflection

A null Schwarzschild geodesic with $b\gg r_s$ is deflected through $2r_s/b=4GM/(b{c}^2)$ to first order.

This is the leading spherical-mass deflection underlying a gravitational lens.

A Schwarzschild static observer at areal radius $R$ emits an outward null geodesic at a small angle $\psi$ to the radial direction. Its impact parameter is $b=R\sin \psi /\sqrt{f}$, and the accumulated azimuth to infinity is ${\int }_R^{\infty }bdr/[r^2\sqrt{1-f(r)b^2/r^2}]=b/R+O(b^3/R^3)$. Thus weak curvature magnifies a narrow outward beam angle by $1/\sqrt{f(R)}$, rather than producing the full infinity-to-infinity deflection $4GM/(b{c}^2)$.

The 1919 solar eclipse expeditions organized by Frank Watson Dyson and Arthur Eddington compared background-star positions near the obscured Sun with reference photographs. The optical train used a slowly rotating flat mirror to send the light into a horizontal photographic telescope. A smaller telescope at Sobral provided a separate set of plates. Focus and plate-scale changes had to be distinguished from the radial pattern predicted by Schwarzschild light deflection. The experiment therefore tested a spatial pattern and its amplitude, not merely whether a star image had moved.^{[ref] [ref]}

Writing $y=b^{-1}\sin \varphi +\Delta y$ in $y^{\prime \prime }+y=3M{y}^2$ and retaining first order in $M$ givesThe displacement of the outgoing zero is $4M/b$.