How can I make sense of black hole complementarity if the universe consist of one manifold and observers are not married to coordinates?

Question

I'm reaching way over my head here, so bare in mind my knowledge base is at best upper undergraduate. This is, unfortunately, yet another byproduct of discussions in this page that is itself a byproduct of another question.

User benrg says,

Judging from this and some other comments, safesphere believes that different observers occupy different "private universes," and it can be true in one observer's coordinate system that an object crosses the horizon and in another's that it never does. That simply isn't true.

Then commenter Accumulation replies,

If it is untrue, it being untrue certainly isn't simple. According to black hole complementarianism, to the outside observer, nothing ever crosses the event horizon. To an infalling observer, they do cross the event horizon.

I know black hole complementarity is only a rough postulate, but I'm not sure how to make sense of it. If the universe, including those having black holes, consists of one spacetime manifold, does that not mean there is only one possible account of what happens?

Yes, the event horizon allows us to have events that are causally disconnected, but they lie on the same manifold so they can't occupy different universes.

Suppose I throw a rock radially into the black hole. There are at least two different descriptions of what happens according to different coordinates.

• According to the Schwarzschild coordinates, the rock goes in slower and slower inward (according to the Schwarzschild $r$ and $t$) and it reaches the event horizon only at $t\rightarrow\infty$.
• According to the Kruskal–Szekeres coordinates, the rock passes the event horizon like nothing happened with finite Kruskal–Szekeres coordinates.

Now both descriptions are true, because the difference is a matter of coordinates. So far so good.

Here is my problem: observers are not coordinate systems. An outside observer does not occupy Schwarzschild coordinates and an observer is not married to Schwarzschild coordinates. An observer occupies a coordinate-independent universe modeled by a spacetime manifold.

Moreover, an observer does not experience Schwarzschild coordinates. The only thing an observer experiences is his worldline and all the photons that reach him. That's it. So in that case, it makes no sense to identify a person with any coordinate system.

Question: Are my statements correct so far?

Now, this leads me to the follow conundrum: black hole complementarity says that to an outside distant observer, an infalling object never gets past the event horizon. What I don't understand is what does it even mean to say, "to an outside distant observer [this thing] happens?" Like I said above, the only thing an observer experiences is his worldline and all the photons that reach him.

Moreover, can't a distant observer decide to use Kruskal–Szekeres coordinates in his or her calculations to conclude that the object does fall past the event horizon with no issues whatsoever.

Question: How can black hole complementarity make sense if a distant outside observer is not married to any particular coordinate system AND he or she can simply choose coordinates that extend past the EV?

Question: Is there a more precise phrasing of black hole complementarity that would clear up my confusion?

Answer 1

Black hole complementarity is supposed to be a duality like AdS/CFT. That is, the physics on the boundary (event horizon) and the physics in the bulk (black hole interior) are supposed to be the same physics described in different coordinate systems, though the coordinate systems are bases for something like a quantum Hilbert space rather than charts on a manifold.

I don't think it's impossible that a theory could be subjective in the sense that whether a black hole interior exists or not depends on whether or not you fall in – i.e., there are different worlds like those of many-worlds/relative-state QM, but you choose which one you end up in by your spacetime motion instead of being duplicated in all of them – but black hole complementarity isn't an idea of that kind.

(And my other answer was about classical general relativity, where there isn't any horizon complementarity, much less private universes.)

How can black hole complementarity make sense if a distant outside observer is not married to any particular coordinate system AND he or she can simply choose coordinates that extend past the EH?

If complementarity is correct, then anyone, even if they've fallen through the horizon, is free to pick a boundary or bulk description of the interior, since they're equivalent.