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Suppose Alice and Bob are in train carriage at each end and tossing a ball to each other as their carriage crosses the event horizon of a super massive black hole at 0.001 m/s (i.e., very slowly, much slower than the speed of the ball relative to the carriage). The carriage is aligned radially with the singularity.

Just as the carriage falls in with Bob's end first, he throws the ball back to Alice, who has not yet crossed the horizon.

Will the ball make it to Alice before she crosses the horizon?

If not, then will that not be a clear indication to Alice that "something is wrong"?

Jus12
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The key to all of these "cross the event horizon slowly" scenarios is that all of the statements about not being able to notice crossing the horizon involve making LOCAL measurements. But local means "in a sufficiently small spacetime interval". Note that it does NOT mean "in a sufficiently small spatial interval". If the time of flight of the ball from Bob to Alice is greater than the freefall time from Alice's location to Bob's location, then it is decidedly not a local measurement, and you will be able to detect the event horizon.

Other things to worry about in a scenario like this:

  1. The conditions required to maintain structural integrity of the train
  2. The inward motion of Alice as Bob throws the ball to her
  3. the differential of speed between Alice's throw to bob and the reverse

You can get around a lot of these by making the train small, and throwing the ball really hard, but the important thing is the relative size of the train to freefall time. There is probably some limit you can work out that shows the obvious thing from a kruskal diagram -- that this cannot work. My first guess is that this would be resolved in a way similar to the way that the "lorentz contract a long pole inside a small barn" paradoxes are -- the formulation is not properly considering simultaneity, but I don't see the exact solution right now. I will edit this answer if I do.

Zo the Relativist
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Rephrase the question:

Can something escape from a (standard general relativistic) black hole to infinity? Answer: No.

Can something escape from a (standard general relativistic) black hole to a finite distance from the horizon? Answer: No.

So the situation will be the following: 1)Alice is too far away (the train is very long), the ball doesn't reach Alice. In this option is the setup that it's impossible, Alice and Bob cannot pass a ball to each other if Bob is arbitrary close to a black hole horizon (at least without noticing).

2)Alice gets the ball and the train is short. Alice is now inside the black hole, by definition.

3)Alice gets the ball and the train is so short that it can be seen as a point. Boundary situation in which both Bob and Alice are on the horizon, but this makes little sense since we want a train with finite length.

The black hole horizon is a teleological objecs, you should really know the entire future history of the space time in order to know where it is. Of course this cannot be done locally. What happens when Bob gives away the ball (I guess) is that the horizon get deformed in some odd way to include Alice when (if) she gets the ball. This nice article https://arxiv.org/abs/gr-qc/0508107 describe for instance how the horizon is deformed if some mass is falling into the black hole. An odd effect is that infalling matter can decrease the rate of expansion, while the horizon expand more if something it's NOT falling!

Rexcirus
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As soon as Bob crosses the event horizon, he's gone. No light from him can reach beyond it to Alice. The carriage itself is (somehow) slowly drifting into the event horizon at 1mm per second and Alice would be in the same frame of time relativity as the ball so would catch it before her part of the carriage entered the event horizon. This would happen seemingly instantly to an observer a safe distance away.

Considering the effects of gravity this close to a black hole, I think Alice would already have guessed something was wrong, especially as Bob suddenly redshifts and vanishes.

Strictly speaking, an event horizon is the point where the escape velocity is greater than the speed of light so very slowly entering a black hole in that carriage, Alice would see the carriage in front of her, in the direction of the black hole, vanishing as light from it can no longer reach her.

Michael Hill
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