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 $4GM/(c^2b)$.


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?

  • $\begingroup$ 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

1 Answer 1


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.)


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.