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So I know for an observer outside the event horizon, objects become infinitely time dilated at the event horizon, making them look as if time stopped for them. If I then throw an electron into a black hole with known momentum, and it "freezes" at the event horizon, don't I now know both its momentum and position, violating the Uncertainty Principle?

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  • $\begingroup$ Why bring black holes into it? According to any classical theory (like GR) I can know the position and momentum of a particle in any situation. $\endgroup$ – Rob Jeffries May 3 '17 at 22:45
  • $\begingroup$ I think the comment here, and the question it addresses, are particularly relevant. An excerpt: "I should also say that trying to apply non-relativistic quantum mechanics in the neighborhood of a black hole is never going to work successfully. " $\endgroup$ – Hal Hollis May 4 '17 at 1:19
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In GR the location of an event horizon is precisely defined, but GR is a classical theory so has no uncertainty.

We have no theory of quantum gravity, so we don't know what an event horizon looks like when we include quantum effects. There are no end of suggestions, e.g. the firewall theory, but the truth is that we simply don't know. However it is generally accepted that when quantum effects are included the event horizon will become fuzzy in some sense so its position is subject to some uncertainty.

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