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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.

sammy gerbil
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Curiosa
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    I'm voting to close this question as off-topic because it is pure science fiction. Do you imagine that you are going to rescue your friend?? Even as a thought experiment this scenario does not pose any interesting or useful problem in physics. – sammy gerbil Jul 17 '16 at 13:41
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    @sammygerbil, it may not be possible, but it is physics. If you wanted, you can answer by saying why this is totally impossible...I think this is an interesting question. – auden Jul 17 '16 at 13:51
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    It don't think this belongs closed as "science fiction", but it's not particularly clear to me what is being asked - you will never see your friend fall into the hole because that takes infinite time for an outside observer, so this is just asking how fast one needs to accelerate to catch up with something in free fall, I'm not sure why the black hole would be relevant. (If you are saying that your friend is already inside the blackhole, then that will never happen from your viewpoint - things take infinite time to outside observers to fall into a black hole) – ACuriousMind Jul 17 '16 at 14:00
  • Your edit is contradictory : para 1 says he has already crossed the "singularity" (do you mean Even Horizon?) but in para 2 (1 day later) you want to catch up with him before he hits the "singularity." – sammy gerbil Jul 17 '16 at 16:07
  • The new information that the friend only has one hour of proper time left limits the possible size of the black hole. It is not clear that you can then spend 24 hours on the outside without sweeping out of the past light cone of his meeting the singularity. – hmakholm left over Monica Jul 17 '16 at 17:07
  • Or did you mean, "crossed the horizon in one hour within an hour of his proper time"? – hmakholm left over Monica Jul 17 '16 at 17:32
  • @sammy gerbil - sorry for that. Replaced "crossed the singularity" by "crossed the horizon." – Curiosa Jul 17 '16 at 17:49
  • @HenningMakholm - I did not understand what you meant by "crossed the horizon in one hour within an hour of his proper time". But I have modified the question now to leave the different time periods as arbitrary. – Curiosa Jul 17 '16 at 18:14
  • @Curiosa: I accidentally garbled the quote -- what I meant to suggest was merely to replace the word "singularity" with "horizon". – hmakholm left over Monica Jul 17 '16 at 18:34
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    Closing this would make no sense. The "science fiction" is just a way to make the question clear. It could equally be phrased in terms like "given a trajectory for particle A such that ..., does there exist a trajectory for particle B such that...", but this would likely make it less clear, not more. – N. Virgo Jul 18 '16 at 00:00
  • @Nathaniel : I disagree about clarity. The sci-fi setting introduces numerous practical difficulties which could prevent reunion, thereby making the question far too broad and vague, leaving assumptions up to respondents. eg Avoiding spaghettification; having enough time to accelerate/decelerate to meet without losing consciousness; losing sight of the friend as he crosses the EH; being unable to "see" him inside the EH because of an expanded field of view. Far simpler to ask the abstract pursuit question - B chases A, setting off after time T with speed V - but in a black-hole setting. – sammy gerbil Jul 20 '16 at 02:15
  • Related : Falling into a black hole, http://physics.stackexchange.com/q/32693 – sammy gerbil Jul 21 '16 at 20:46

2 Answers2

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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.

hmakholm left over Monica
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  • +1 with the added caveats that depending on the size of the black hole and the resulting gravity gradient, you could both die from tidal forces long before crossing the event horizon, and the blueshift could be so overwhelming it's fatal. – Todd Wilcox Jul 17 '16 at 16:05
  • "in the limit ... a comfortable 1 G ... about a light-year" -- so just to confirm: is there a lower bound on the gravitational field strength at the event horizon of a black hole, and that bound happens to be somewhat more than 1G? Does that mean humans cannot comfortably hover using jet packs near the event horizon of any black hole, no matter how big? – Steve Jessop Jul 17 '16 at 16:39
  • @ToddWilcox: Hmm, I thought I had a caveat about the black hole being large enough, but it must have disappeared in editing. – hmakholm left over Monica Jul 17 '16 at 16:42
  • @SteveJessop: The gravitational field strength at the event horizon itself is always infinite ("somewhat more than 1 G" indeed), in the sense that if you stand on a platform above the black hole that remains in place according to a far-away observer with Schwarzschild coordinates, then the acceleration of gravity on the platform diverges to infinity the closer it is to the horizon. This is true for any size of black hole. – hmakholm left over Monica Jul 17 '16 at 16:50
  • Ah, gotcha. So the questioner's scenario, hovering on the jet pack a mere subjective hour away from the event horizon, basically can't happen. There's no black hole small enough that you can survive hovering that close, and large enough that you can fall in while retaining even approximately your original shape and size? – Steve Jessop Jul 17 '16 at 17:09
  • @Steve: I'm not sure about that. The "infinte mass limit" I'm describing gives an upper bound for the distance you need to go from the horizon to put an 1G platform, and that would be for black holes of absolutely stupendous mass, well above that of our entire galaxy. Smaller black holes allow you to come closer, and I think a merely supermassive one might actually allow you to fall in within an hour and still not be spaghettified until after you pass the horizon. But I haven't done the calculations. – hmakholm left over Monica Jul 17 '16 at 17:17
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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.

sammy gerbil
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  • Would me and my friend be ripped apart before reaching the event horizon? That will certainly happen near the singularity. But my understanding is that tidal forces can be very small near very huge black holes and hence one can cross the horizon without getting turned into sphagetti. – Curiosa Jul 17 '16 at 18:20
  • Ah yes, you are correct! Thank you for enlightening me! You can fall into a super-massive black hole without being aware of it. However, the g-forces on you while you "hover" at the edge of the black hole are going to squash you, aren't they? You may avoid becoming spaghetti - but you'll become a pancake instead! And your friend will still be travelling at the speed of light (relative to you and your ship) as he crosses the Event Horizon, giving you no chance of catching up 1 day later. – sammy gerbil Jul 17 '16 at 19:01
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    @sammygerbil: If we consider larger and larger black holes, spacetime near the horizon will approximate flat Minkowski space for larger and larger coordinate neighborhoods of an event on the horizon. With a large enough hole that will give room enough to maneuver to catch up with someone who crossed the horizon a day earlier. The first infalling observer is in free fall -- and in his inertial frame the other astronaut simply accelerated away from him at 1 G for a day. If he accelerates the other way for two days and then away again for one day, he will end up where he started -- – hmakholm left over Monica Jul 18 '16 at 01:15
  • -- that is, right next to the free-falling astronaut. It doesn't matter that an event horizon swept across the scene halfway through the maneuver, because the event horizon is not locally special, just a light front like any else. – hmakholm left over Monica Jul 18 '16 at 01:17
  • @HenningMakholm : Yes, the 1st observer is in free fall, but with what speed? Isn't he moving at the speed of light away from the event horizon and towards the singularity? If so, can anything catch up with him? – sammy gerbil Jul 18 '16 at 02:04
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    @sammygerbil: His speed is $0$ in his own inertial frame. This is relativity; there's no such thing as "with what speed" unless you specify relative to what, and nothing/nobody moves at or above the speed of light. The horizon sweeps past both observers at the speed of light. – hmakholm left over Monica Jul 18 '16 at 02:07
  • This is nonsense. No mass can accelerate (especially not instantaneously) to the speed of light. – OrangeDog Jul 18 '16 at 08:50
  • @OrangeDog : I was not suggesting instantaneous acceleration to light speed, since it is not possible to hover arbitrarily close to the Event Horizon. As Henning says, the gravitational field strength diverges to infinity at that point, requiring infinite energy to hover, and crushing anything which attempted to do so. However, I agree that I am mistaken in thinking that, because light cannot escape, the infalling obect must have reached light speed. – sammy gerbil Jul 20 '16 at 00:49
  • @HenningMakholm : I was referring to speed relative to the parked space-craft and/or the centre of the black hole. Of course an observer is at rest in his own frame of reference, regardless of whether that is inertial or not. What I am confused about though is how the EH can sweep past both observers at the speed of light but not be getting any further from the singularity. – sammy gerbil Jul 20 '16 at 01:03
  • @sammygerbil: There's not really any such thing as "distance from the singularity". The Schwarzschild metric is stationary outside the horizon, so it makes some sense to measure distances along spacelike geodesics whose time coordinate is constant -- but inside the horizon there is no stationary chart one can make a canonical speed or distance measurement against, and the $r$ coordinate is timelike. Try looking up "Kruskal coordinates". – hmakholm left over Monica Jul 20 '16 at 09:00
  • @HenningMakholm : Thanks. That looks horrendous. So it seems that even travelling along the same geodesic into the black hole, the calculation of whether B can catch up before A hits the singularity is going to be very difficult? That - or the even more difficult skew trajectory - is what the OP is asking for. – sammy gerbil Jul 20 '16 at 09:56
  • @sammygerbil: I suppose you mean along the same spatial geodesic in the stationary outer part of the spacetime? Yes, that gives only partial information -- the Kruskal diagram shows the history of one straight radius from the black hole, but the two observers can travel along it both at different times and different speeds, with different implications for whether they can catch up or not. (And I'm not good enough to do those actual calculations). – hmakholm left over Monica Jul 20 '16 at 10:41