My googling (as well as a textbook from college, Hartle's "Gravity") states that the deflection angle = 4GM/(b * c^2) = 2 R/b

where b is the impact parameter of the undeflected light ray and R is the Schwarzschild radius of the object in question.

-However, in Hartle (p. 259), it states that this is true for b >> M.

-Furthermore, this gives a maximum deflection angle of 2 radians (when b = R, as a lesser value of b would mean the light ray enters the object's event horizon). This doesn't jive with my conceptual understanding of light rays being deflected by gravity (it seems as if maximum deflection should equal 2*pi. If the impact parameter is greater, the deflection is less. But if the impact parameter is less than R, than the light ray should spiral inwards - essentially, a "deflection" or more than 2*pi, as the light ray spirals inwards.) Otherwise, why would 2 radians "magically" be the critical angle, which seems a random value?

These two facts lead me to believe that the simple formula above is an approximation and is very wrong in the vicinity of a black hole. Thus, I wonder if anyone has the formula for the deflection angle of light where b ~ R.

Secondly, I'm also interested in the "lensing" effect a black hole can have with an object, particularly when the black hole lies directly between the object and the viewer. I wonder if there is a formula for the magnification an apparent image has, given:

R (Schwarzschild radius, and thus M of the black hole)

dO (distance between object and black hole)

dV (distance between black hole and viewer)

w (width of object being viewed, not sure if this is important).

d (distance between the

The simpler the formulae, the better, but accuracy is more important.

The application is for real-time calculation of optical effects for a game I'm proto-typing. If I need to, I can simplify the formula myself (for quicker processing), but would rather do so given the accurate original formula, rather than an approximation (as above).

Thank you.