# Light Deflection by a Black Hole

I think I've come across a paradox while studying general relativity. Wikipedia states that the deflection angle of light by a point mass is $4GM/(c^2b)$.

http://en.wikipedia.org/wiki/Gravitational_lensing_formalism

Where b is the impact parameter, the limit of the perpendicular distance of the light ray from the point mass as the light gets far away. By shining a light that gets pulled close to the photon sphere of a black hole I can make a photon orbit the black hole an arbitrary number of times before escaping, leading to a deflection angle much greater than 2*pi. However, to make this angle go to infinity (at the photon sphere) according to the equation, I need to make b go to zero. However, if I shine a light very close to to center, it passes the event horizon and will go into the black hole without orbiting forever. What am I missing in this analysis?

• – Qmechanic Aug 2 '14 at 22:09
• benrg provides the right answer. The formula is only the first term of an expansion of deflection "far away" from a black hole. A different way to see this is to say $4GM/(c^2 b)$ is the Green's function for deflection by low mass densities. – Void Aug 2 '14 at 22:09