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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?

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  • $\begingroup$ Related: physics.stackexchange.com/q/52824/2451 $\endgroup$
    – Qmechanic ♦
    Aug 2, 2014 at 22:09
  • $\begingroup$ 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. $\endgroup$
    – Void
    Aug 2, 2014 at 22:09

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That formula is an approximation derived in linearized GR; it's not correct near a black hole.

(The test particle is non-Newtonian, since it's traveling at c, and that's why the answer can differ so much from the Newtonian prediction even though the gravitational field is assumed to be weak.)

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