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Does the apparent radius of the event horizon of a black hole get smaller and smaller as an observer approaches a black hole? If this were true, it would seem to allow communications to be relayed from observer to observer from within the classical Schwarzschild radius to the outside world.

Roger Wood
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Alice, Bob, and Charlie are sitting on a stationary platform a long way from a black hole. Alice tells Bob and Charlie to go jump in the black hole. Bob grabs a clock off the mantelpiece and jumps. Exactly one minute after Bob jumps, Charlie grabs another clock and jumps. They follow the same trajectory into the black hole, so as they each reach a given position, Charlie’s clock will show a time exactly one minute later than Bob’s clock showed at that position. Bob calculates that he will cross the event horizon exactly at 12:00 when his clock chimes midnight. Charlie’s clock will show 12:01 when he crosses the event horizon himself.

As Bob and Charlie fall, they see the apparent event horizon receding in front of them. Bob turns round and sees both Charlie behind him and Alice in the far distance. He feels a sudden pang of regret. But just then his clock chimes midnight. He has just crossed inside the event horizon and can no longer send a message to Alice. Fortunately, he can still see Charlie who is falling in one minute behind him. So he shouts to Charlie: “Tell Alice I love her”. Bob has calculated that it will take less than one minute for his message to reach Charlie, so Charlie should still be outside the event horizon and able to relay the message to Alice.

Unfortunately, Bob’s clock is running quite a bit slower than Charlie’s because Bob is at a deeper gravitational potential. According to Bob’s clock, Charlie received the message just before 12:01. But according to Charlie’s clock it is already just after 12:01. It's too late, Charlie has already crossed the event horizon. Alice will never know.

Roger Wood
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    "Bob turns and looks back at Alice. He sees her rapidly get older and die and disappear just as his clock chimes midnight. " This is incorrect for a free falling observer. – ProfRob Sep 13 '20 at 20:45
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    @Rob Jefferies As he falls towards the event horizon, doesn't Bob's clock run slower and slower compared with Alice's clock - to the extent that he appears to never actually reach the horizon. Conversely, doesn't Alice's clock run faster and faster when viewed by Bob? Ultimately doesn't Bob look back and see the entire future of the universe pass by at an increasingly fast rate through a vanishingly small window? – Roger Wood Sep 14 '20 at 05:58
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    "Ultimately doesn't Bob look back and see the entire future of the universe pass by at an increasingly fast rate through a vanishingly small window?." No. https://physics.stackexchange.com/questions/82678/does-someone-falling-into-a-black-hole-see-the-end-of-the-universe – ProfRob Sep 14 '20 at 06:39
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    @Rob Jeffries really excellent reference! I gather Bob will continue to see Alice even after he passes the event horizon. She will appear to age more quickly than he will, but not indefintely. The exact amount will depend on the relative position of the starting stationary platform from which he jumped. If this is correct, I will try to edit my 'answer' accordingly. – Roger Wood Sep 14 '20 at 19:07
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    For any realistic scenario she will age imperceptibly quicker. Try plugging some numbers into the equations. You are confusing a falling observer with one that is hovering/orbiting just outside the event horizon. – ProfRob Sep 14 '20 at 20:21
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    @Rob Jeffries Yes, that was my initial confusion. That reference you pointed me to is very helpful. I'll go ahead and edit my 'answer'. I'll ignore Alice's age - she'll probably appreciate that anyway. – Roger Wood Sep 15 '20 at 05:37
  • Roger, I don't think this is right. Sorry. – John Duffield Sep 20 '20 at 13:27