Assume that my friend and I decided to explore a black hole. I parked the spaceship in a circular orbit safely away from the horizon. He puts on his spacesuit with a jet pack and carefully travels towards the horizon. We communicate by electromagnetic waves. He reaches near the horizon and is hovering above it at a height that is safe considering the power of his jet pack. Suddenly his jet pack fails and he is in free fall. He sends the message to the ship. I receive the message, suitably red shifted, in some time. From the red shift, I can calculate where exactly he was when he sent the message. I then deduce how much time it would have taken in his frame (that is his proper time) for him to cross the horizon from the time when he sent the signal.
I decide that I do not want to live in this world without my friend and resolve to go after him and catch up with him (assume that the black hole is big enough so that there is plenty of time in his proper frame before he hits the singularity), even if I perish eventually. Can I do it? Note that my aim is not to rescue him but just to catch up with him. What sort of a trajectory should I choose? In other words, how should I fire my jet pack as I am going in and later when inside the hole?
To make the problem more precise, let us consider Schwarzschild coordinates (t,r,\theta,\phi) outside the black hole of mass M. Assume that my friend was at some radius r_0 and time t_0 when he sent me his farewell signal. My ship is orbiting at a radius r_1 and I receive the signal at time t_1. For simplicity, let us assume that I was at the same (\theta, \phi) coordinates as my friend when I received the signal. Finally, let the time when I start out from my ship to go after my friend be t_1+T. What are the conditions on the various quantities above such that I have a chance to go and catch my friend? In case the quantities are favorable, how should I go about catching up with my friend?
Note that I was originally not looking for a very mathematical answer and hence I had specified things rather vaguely. But I have modified the question so that it will be easier for users to discuss things more concretely if they wish.