This is some kind of a follow up of this nicely to the point answer to a provocative (but nevertheless upvoted!) question, about the legitimacy of black hole physics. The answer mentions, that the interior and exterior of a black hole are not completely decoupled, and about this exact relationship I'd like to learn some more not just hand waving details.
Of course I know the complementarity principle, which says that the perspective of what happens near the event horizon of a black hole an infalling observer who does not note anything special when crossing the event horizon and the description of the physics near the horizon by a staying outside observer who sees the infalling observer getting frozen on the horizon, are both equally legitimate descriptions of the same physics.
Is there some kind of a map, or exact dictionary by which one can transform back and forth between these two descriptions of the same physics, that describes the complementary principle for black holes mathematically?
An aside: I always thought that the holographic principle is soemething different from the complementarity principle in this context, so this should not be the trivial answer to my question (?) ... Maybe I have managed to confuse myself now :-)
Clarification after some comments
I am not just up to a coordinate transformation between spacetime inside and outside the even horizon, but I'd like to learn about how the degrees of freedom used by an infalling observer and the ones used by a staying outside observer to describe the same physics transform into each other.