Chasing someone who has fallen into a black hole 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.   
 A: ANSWER WITHDRAWN
I am withdrawing my answer because I am persuaded by Henning and others that I am mistaken about the impossibility of catching up with someone who has crossed the Event Horizon.
I have also withdrawn my Vote-to-Close.
Original Answer
What do you mean that your friend has "fallen into a black hole"?  If you mean that he has crossed the Event Horizon, whatever is left of him will be hurtling towards the Singularity at the speed of light.  There is no catching up with him however much you fire the thrusters on your jet pack.  Both he and you and the jet pack will be ripped apart* long before you reach the Event Horizon.
[* Apologies : As you point out, if the black hole is massive enough the tidal force just above the Event Horizon might be survivable. But the g-force on you as you "hover" there will not be : you will be flattened like a pancake.]  
If there is still time for you to reach your friend long before he reaches the Event Horizon, and before you are both shredded by the intense gravitational field, what part does the black hole and black hole physics play in this scenario?  Indeed, what part do general relativity and space-time play in this case?  
In any environment in which you and he have a realistic chance of survival, this is simply a question about manoeuvring in a (possibly varying) gravitational field.  You must reach him before he reaches a field of about $30g$, and in reaching him you will have to avoid a deceleration of more than $30g$.  But even this question cannot be answered because of the sparsity of detail in your question about the starting conditions. The only condition which you do state is that you set out after your friend the following day.  
If you do supply further detail and are asking for a calculation of optimum trajectory, this avoids the "black hole" setting which interests you.  Moreover, it becomes a "homework-like" problem which requires that you demonstrate some effort to work it out yourself.  Somehow I doubt that is what you want.
Of course your question might be re-interpreted by Heather or ACuriousMind as a question about black hole physics and space-time, by removing all reference to people and jet packs.  But having done that it would be their question, not yours.
Response to your Revised Question :
If your friend's remains have crossed the Event Horizon, there is no way of catching up with them. He will be travelling at the speed of light.  Your jet pack will not make the slightest difference.  
You seem to misunderstand "proper time".  Even if your friend has not yet crossed the EH and survives, the fact that time slows down for him does not mean that it will buy you extra time to reach him before he does so.  Likewise the fact that your friend appears to slow down as you observe him approaching the EH does not buy you extra time either.  In your time frame you are still 1 day away from following him into the black hole.  (Or is it 1 hour now?  Or the vague $t_1+T$ seconds?)  
The questions of "choosing a trajectory" and "firing the jet pack" - and indeed of existence inside the EH - are still meaningless.    
A: Assuming that the black hole is large enough that one can cross the event horizon without being spaghettified by tidal forces, and the that when the accident happened, the both of you were hovering in place above the black hole with your jetpacks, rather than orbiting it:
You can still see your friend (no matter how long you dally, the part of his worldline that is inside your past light-cone will not have crossed the horizon yet). Accelerate in that direction, i.e. straight down.
If the black hole is large enough, it's possible that you may catch the friend before he reaches the horizon -- in the limit of an infinitely massive black hole, if you were able to hover above it at a comfortable 1 G, then you will be about a light-year from the horizon (give or take some factors of 2 and/or $\pi$ that I don't care to derive right now), and there'll be plenty of time to catch a free-falling friend even if he has a day's head start.
Otherwise you will see your friend's crossing of the horizon exactly at the instant when you cross it yourself. Keep going in his direction and prepare to match velocities when you begin coming closer (for a large enough hole, space both immediately inside and outside the horizon will be flat enough that you can plan the rendezvous just as you would in Minkowski space). It you're lucky you may be able to catch up and exchange a few last words before both of you snuff it.
On the other hand, if the black hole was too small, or you waited too long, it may be that the friend's impact with the singularity is already outside your futureward light-cone, in which case you'll inevitably hit the singlarity yourself before you see your friend doing so, and nothing will be gained.
(In any case, if you're chasing your friend into an ideal Kruskal black hole, then don't miss paying some attention to what you see behind your friend after crossing the horizon. At that instant the backdrop will stop being the past singularity and instead you should be able to see stars and galaxies of the other outer Schwarzschild region of Kruskal space, potentially a completely inacessible separate universe. Too bad you won't be able to send observations back home by then. Bring goggles; the blueshift may be overwhelming at first).
(On the other hand, if it's an ordinary black hole created by collapsing matter at some time in the past, then never mind; there won't be any universe at the other end of the hole).
Calculating all this more precisely, based on the size of the hole and your initial positions, is strongly recommended before you set out on a rescue mission.
