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A black hole exists. Eventually, it will completely evaporate away.

Alice falls into the hole. Imagine you are Alice. According to logical positivism, the interior of the black hole exists. But you will die at the singularity. You can't measure anything after you're dead, so any event outside after the black hole has vanished doesn't exist.

Bob stays outside the hole. Imagine you are Bob. The interior doesn't exist.

Enter Quentin. Right now, he holds a qubit in the state ${1\over\sqrt 2} (|0\rangle + |1\rangle)$. He will measure the value of the qubit shortly in the future. If 0, he will jump into the hole. If 1, he will stay out forever. Imagine you are Quentin. Right now, does the interior exist?

Curiouser and curiouser.

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If you believe in black hole evaporation, it's possible to escape the interior of the black hole--a shrinking horizon is transversible. – Jerry Schirmer May 13 '11 at 17:27
Define an observable for "does the interior exist". – dmckee May 13 '11 at 17:31
Why doesn't the interior exist for an outside observer? I don't see how that follows from logical positivism + GR? – Raskolnikov May 16 '11 at 8:14

"Alice falls into the hole. Imagine you are Alice. According to logical positivism, the interior of the black hole exists."

No. Logical positivism requires you to look at all possible observers (or rather all possible experiments) to determine if something exists. So statements like "for observer A, logical positivism implies..." are meaningless.

Given that understanding, black hole interiors definitely do exist. Plenty of experiments can deduce the effect of the interior like measuring tidal forces, or detecting Hawking radiation, or hitting a singularity or sending a probe towards the hole and seeing it get more and more redshifted.

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Black hole interiors are dual to exteriors under what is called by Susskind "Black hole complementarity". The logical positivism is there--- the interior and exterior observers disagree on the description of their space time, but both descriptions are correct. The question can be asked in a de-Sitter space too, where Alice and Bob start out talking, but then fall into each other's cosmological horizon. There it is more difficult to understand.

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The moment Alice passes through the event horizon, she is doomed to death, and nothing she observes will come out. How can the interior of the black hole exist if the only observer is dead?

What Quentin is doing is committing quantum suicide. Subjectively, he will have quantum immortality, and will never die. He will always measure his qubit to have value 1, and be spared a visit to the nonexistent interior. It's a miracle! On the other hand, other observers will see him die with a 50% probability.

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