I Black hole event horizon confusion

[Yukterez]

Are you perhaps not aware that after the point where a horizon crossing changes from causal future to possibly now, for a given external observer, then any signal sent by the the external observer to the horizon crosser will only be received by the horizon crosser inside the horizon? This is an aspect of the change in causal relation that is wholly independent of coordinates.

I am well aware that any signal sent right at the horizon can only be received when the receiver crosses the horizon himself, that is because an outgoing lightray at the horizon has a constant radial coordinate. So the photon does not come to you, you have to come to the photon, so to speak:

In a spherical black hole, the river of space falls into the black hole at the Newtonian escape velocity, hitting the speed of light at the horizon. Inside the horizon, the river flows inward faster than light, carrying everything with it.

This quote is of course coordinate dependend.

 

[PAllen]

I am well aware that any signal sent right at the horizon can only be received when the receiver crosses the horizon himself, that is because an outgoing lightray at the horizon has a constant radial coordinate.

That is not at all what I said. Consider an observer staying one light year away from a BH. They fire a test vehicle toward the BH. They send a light signal to it every second per their clock. There are then two signals they send such that the test vehicle receives one just outside the horizon and the next just inside the horizon. This is a coordinate independent fact. That transition for the external observer is when the crossing changes from being in the causal future to the possibly now = spacelike separation. Do you get this? Your answer above strongly suggests you do not.

 

[Yukterez]

Consider an observer staying one light year away from a BH. They fire a test vehicle toward the BH. They send a light signal to it every second per their clock. There are then two signals they send such that the test vehicle receives one just outside the horizon and the next just inside the horizon. This is a coordinate independent fact.

That is correct, I calculated that a lot. It is by the way also a coordinate independend fact what time the stationary observer reads off his clock when he receives a given timestamp from the freefaller, so that statement goes both ways, with the only difference that all the signals the external observer receives come from outside of the horizon.

That transition for the external observer is when the crossing changes from being in the causal future to the possibly now = spacelike separation. Do you get this? Your answer above strongly suggests you do not.

You are right, I really don't get how that helps your case, but I will sleep over it. Maybe I get it tomorrow (or the other way round).

 

[PAllen]

That is correct.You are right, I really don't get how that helps your case, but I will sleep over it. Maybe I get it tomorrow (or the other way round).

There is nothing for me to sleep on since I don’t see you as having any case at all, sorry. Do you understand the distinction between the future light cone, the past light cone, and what is in between? And that after the event described for the external observer, the horizon crossing event has moved out of the future light cone and can no longer be considered causal future irrespective of coordinates? And that for events between future and past light cones, what is now is entirely a coordinate convention? This is really basic to SR, let alone GR.

 

[Yukterez]

Do you understand...

I understand all of this, but what you don't seem to understand is that in the future light cone of the external observer, the freefaller is always outside of the horizon, so it would make no sense to say the crossing of the horizon was "possibly now". But whatever, better having you against me (of course not on a personal level, but regarding the subject) and Susskind on my side, than the other way round!

 

[PAllen]

I understand all of this, but what you don't seem to understand is that in the future light cone of the external observer, the freefaller is always outside of the horizon, so it would make no sense to say the crossing of the horizon was "possibly now".

This is just false. We have hit on your key misunderstanding. For any external observer, independent of coordinates, there is a precise moment when the infaller’s horizon crossing is no longer in the future light cone of the external observer. This should be easier to address now since you are making a false statement. The future light cone of an event is the set of all events that can receive a signal from that event. Once the horizon crossing event can no longer receive a light signal from the external observer, it is no longer in its future light cone.

 

[Yukterez]

Once the horizon crossing event can no longer receive a light signal from the external observer, it is no longer in its future light cone.

Correct, I didn't mean the freefaller is outside the external observer's future light cone, I meant that in his future you are always outside of the event horizon. The external observer is outside of the freefaller's future light cone once he passed the horizon, I made a mistake in my last posting but replace either the two observers or the future light cone with future and the statement is correct.

 

[PeterDonis]

in the future light cone of the external observer, the freefaller is always outside of the horizon

No, this is not correct, as @PAllen points out.

didn't mean the freefaller is outside the external observer's future light cone

You should have. He is.

I meant that in his future you are always outside of the event horizon

Sorry, this is still wrong. More precisely, there are two possible referents for "his future" in this statement, and the statement is wrong for both of them.

Possibility #1: You mean that in the future of the free-faller, the external observer is always outside of the horizon. This is false; the external observer's worldline exits the future light cone of the free-faller when the free-faller crosses the horizon. After that point all events on the external observer's worldline are spacelike separated from the free-faller, so anyone of them could possibly be "now".

Possibility #2: You mean that in the future of the external observer, the free-faller is always outside the horizon. This is also false, as @PAllen has already explained.

At this point I am closing the thread since this particular subthread is going nowhere, and you are not the OP in any case; the OP's questions have been addressed.