Astronomy
PF Gold
P: 23,235
 Quote by Machinus I stated in my post that the object has a mass m. Photons do not have mass. Light is not an appropriate example for this orbit. The radial velocity reaches a maximum at some point between infinity and the minimum r. The fact that the velocity is all radial at infinity is irrelevant to finding the maximum $$\dot{r}$$.

the massive particle that is sent at the black hole at speed 0.999 c
will follow approximately the same trajectory as the photon

if you want to make the approximation better, send a new particle at
0.9999 c

and then one at 0.99999 c

the particles follow geodesics in spacetime---the spacetime created by the massive central body, the black hole. approximately the same paths as photons of light

if you think about it in an open-minded fashion, I believe you will come to the conclusion that the maximum radial velocity (what you asked about in your question) is c.