If you only look at the deflection angle, which is the best way to go about it, you can drop the mass. In classical theory it doesn't matter, and in GR, it means something completely different.
So method one. Take a particle of arbitrary mass m (you can take limit m->0 in the end, as it doesn't matter) and shoot it past, say, a star of mass M. You fire it originally distance b from radial, and infinitely far away the particle's initial velocity (hyperbolic excess velocity) is c. You compute classic trajectory, and observe the particle leave the star with the same distance b from radial, same velocity c, but heading in a slightly different direction, making angle θ with original.
\theta = sin^{-1}\left(2\sqrt{\frac{GM^2}{GM^2+b^2 v^4}}\right)
This is general formula that works for any v, so you can take v->c. It can be derived from formualae on
this page by keeping in mind that b*v
∞ = v(r)*r at closest approach r = -a(1-e) due to conservation of angular momentum. Note, also, that what I'm calling θ is the deflection angle. The angle in the article is angle between asymptotes. Hence the inverse sine instead of inverse cosine and the factor of 2. In the limit bv² >> GM and v->c, the deflection angle is small, and the above simplifies to the following.
\theta = \frac{2GM}{bc^2}
Method two. You find a null-geodesic that corresponds to the parameter b above. It also yields you an angle, some θ'. I am not going to point to derivation of that, because it involves Christoffel Symbols of Schwarzschild Metric. But the result is given in
this article on Gravitational Lensing. Note that what I call b they call r.
\theta = \frac{4GM}{bc^2}
Both of these describe a "photon". First method describes it in classical approximation, second under GR. The difference in the deflection angle is exactly factor of 2.
This lead to one of the original tests of GR described
on this page. The deflection of the light from the stars due to gravity of the Sun can be measured, and does, in fact agree with GR rather than the classical result.
However, because the difference in deflection between GR and classical result is by a constant factor, it can be said that influence on the gravity on light does drop as 1/r as it does in classical theory. In fact, both formulae give deflection as 1/r from distance to the star.
And that was my original point. I just didn't expect to have to explain it in that much detail in the GR section of this forum.
