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Lets say there's two astronauts, Alice and Bob, going on a space trip to a super-massive black hole. So large that they wouldn't notice any significant spaghettification forces at the event horizon. Alice is sitting on the left, and Bob on the right.

They go so close to the event horizon, that Bob enters the event horizon, but Alice does not. The event horizon is splitting the spaceship. Alice's side of the spaceship is travelling at a velocity greater than the escape velocity of the black hole, so presumably will escape it's fate.

What do Alice and Bob see? Bob is apparently doomed, but Alice should be able to survive, but it seems like she has to tear the spaceship apart. And what do they see? Does Alice just see Bob moving closer and closer to the event horizon but never crossing it, whereas Bob sees Alice moving further and further away from him until he crosses the even horizon and she disappears?

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Don't know what a "specification force" is. Guess you mean tidal forces, or (equivalently) spaghettification forces? :) – Michael Brown May 3 '13 at 10:01

In order to escape Alice's section of the space ship has to accelerate. The required acceleration increases rapidly as you approach the horizon (to infinity right at the horizon, meaning escape from there is impossible). So what they both see initially is Alice's section of spaceship accelerating, placing the spacecraft under increasing tension until is splits in two. Then Alice accelerates rapidly away from Bob. Bob sees nothing special for the few seconds before he dies, except for Alice (and everything else outside the black hole) getting blue shifted. Alice sees Bob descend into the black hole, squishing, slowing down and red-shifting until invisibility just above the horizon. Here is a nice set of visualisations of falling into a black hole.

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They see nothing special. From the point of view of an infalling observer, the event horizon is just an invisible "point of no return". If Alice fails to start accelerating before her half of the ship crosses the line then she will be doomed along with Bob, but there is no physical measurement that either of them can make that would tell them where this invisible line actually is.

When put like that it sounds as if they'd be able to wave to each other across the horizon, allowing Bob to transmit information to Alice from inside the black hole. However, in order to escape after Bob has crossed the horizon, Alice will have to accelerate so rapidly that the light reflected from Bob's waving hand after he crosses the horizon will never reach her. So she never gets a chance to see Bob after the point he crosses the horizon, and when she reaches a safe distance he will indeed appear to be frozen in time just before the moment he crossed. (Though the image of Bob will be red-shifted into gamma rays, and will hence be impossible to detect in practice.)

Bob, however, can still see Alice up until the moment he hits the singularity. The event horizon is the point beyond which light cannot escape, but it can still pass into the black hole without obstruction. So although Alice can't see Bob after he crosses the horizon, Bob will be able to watch Alice fire up her thrusters and make her escape.

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On a pedantic note, the types of forces being applied to the ship itself would likely destroy anything (spaghettify) that enters the event horizon, so unless Alice can escape with half a ship, she is just as doomed as Bob. It is hard to imagine anything that could withstand the turmoil within 100 thousand kilometers of a black hole, much less be half-in half-out. – Neil Mar 29 at 8:59
@Neil the magnitude of tidal forces depends on the mass of the black hole. For smaller ones, yes, you'd be spaghettified long before you reached the event horizons, but for big ones like the one at the galactic centre the tidal forces at the event horizon are negligible - you would experience it as a region of space like any other. – Nathaniel Mar 29 at 11:17
That seems counterintuitive. If part of the ship is within the event horizon, then in theory, at the very least, that part of the ship is not getting out again or else it wouldn't be an event horizon. Why would this be more possible with smbh's? – Neil Mar 29 at 12:23
@Neil there is no contradiction. The event horizon is a point of no return, and if you pass it there is no way you'll ever make it back again. Eventually, no matter what you do, you will fall into the singularity. It's just that there isn't a sign to tell you that you've gone past it. Spaghettification and the event horizon are quite different issues - it's the spaghettification that disappears for a big enough black hole, not the point-of-no-return nature of the horizon. – Nathaniel Mar 29 at 13:27

You question is as same as asking what happens after a spaceship accelerates to faster than the speed of light. First I would say you have to reach c and the same goes for Rs. It may only take your one second to reach Rs in your own time but in that second the whole universe would change infinitely. Why? Because according to the scharzschild metrics time-dilation the moment your reach the scharzschild radius infite time has passed in the outside universe! Now what makes your think the BH would still exist while the rest of the universe is gone? Most likely as per thermodynamic even BHs must radiate the Rs becomes smaller and smaller and is gone before you could ever catch it! So if you not roasted by incoming and outgoing radiation Bob and Alice would be fine after all. They would have spend some billion years in one second and hardly meet any of you again but well that is for another episode.

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