Returning Light

Question

What happen if someone is just close enough to a black hole, and he emits light tangentially out, and the light got bent and circles around, what will he see, can he see the light source from his back?

             +---------->
            /            \
   light   /              \
          ^                \
observer  +   O black hole /
sees light\               /
           \             /
             <----------- light path

Answer 1

It seems that you want a position where light can get emitted and then return to where it started and then continue on forever.

Such places do exist; what you're looking for is a circular null orbit. Uh, "null" means having no change in proper time, that is, the sort of path (or orbit or geodesic) that is suitable for light. The radius for this orbit is $3r_s/2$ where $r_s$ is the Schwarzschild radius. See wikipedia, Schwarzschild metric / orbital motion.

The spacecraft emitting the light would have to use thrusters (or something else, maybe a rope or what have you) to keep itself stationary, but the value is outside the event horizon at the Schwarzschild radius (where $r_s=1$), so this is possible.

Answer 2

I found “The mathematical theory of black holes” by S. Chandrasekhar (1992) quite useful for such a purpose. The already mentioned $3 r_s/2 $ orbit for light really exists, but it is not stable, i.e. any small fluctuation or error in initial direction make impossible for light to return in the initial point and walk around black hole forever. It is because the last stable circular orbit is $3 r_s$ (see already cited wikipedia article). Yet in the mentioned book is described null geodesic (i.e. path of a light beam) asymptotically spiralling around the circle $3 r_s/2 $.