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When looking at a (eternal) Schwarzschild Black Hole, we may identify two worlds. The region $R_1$ (right) - our world -, and the region $R_2$ (left) - an other world.

The "black hole interior" corresponds to the upper region, that we named $B_+$.

$B_+$ contains the future singularity $S_+$.

The regions $R_1$ and $R_2$ are "separated" from $B_+$ by their respective future horizons $H_1^+$ and $H_2^+$

We consider here a very large black hole.

Considering the point of view of free falling observers $FF_1$, $FF_2$:

1) They don't encounter something special as they "cross" the future horizon. (Tidal forces are considered as neglectible because the black hole is big)

2) There is a limit, a very sad end, at the singularity (practically, near the singularity), where these free falling observers $FF_1$, $FF_2$ are supposed to die.

Now, we want to understand black hole complementary, and see the point of view of outside fixed observers $O_1, O_2$.

For these observers $O_1, O_2$. the limit is their respective future horizon $H_1^+$ and $H_2^+$.

The degrees of freedom at the singularity (or near singularity), as seen by the free falling observers $FF_1$, $FF_2$, should have an equivalence from the point of view of the outside fixed observers $O_1, O_2$.

The question is : What are the equivalent degrees of freedom of the singularity, from the point of view of the outside fixed observers $O_1, O_2$ ?

Note : The simpler solution would be that the equivalent degrees of freedom would be the degrees of freedom of the future horizons $H_1^+$ and $H_2^+$, because it is a "limit" for the outside fixed observers, in the same way as the future singularity is a "limit" for the free falling observers. But of corse, the equivalent could be more complex.

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What does "the singularity freedom degrees should have an equivalence from the point of view of the outside observers" means ? –  Bru Jun 18 '13 at 16:12
For the outside fixed observers, the "limit" it their (future) horizon. These observers never see something cross the horizon, they see only things approach asymptotically the horizon. However, from the point of view of free falling observers, the singularity is a (very hard) reality, so the singularity, should corresponds to effective freedom degrees of fields (The free falling observers are "roasted" at the singularity). Now, the question is, from the point of view of the fixed outside observers, what is the equivalence, that is, where are these freedom degrees of fields. –  Trimok Jun 18 '13 at 16:18
My problem with this question is that I have the feeling that two different times are identified: the one for the falling observer and the one for the outside, fixed observers. The times for these two observers are different. For example the falling observer cross the horizon in a finite amount of proper time, while it takes an infinite time for the outside obs. to see the falling one cross the horizon, as you mentioned. Since there is nothing beyond infinity, it is meaningless to ask what does the outside horizon see when the falling obs. reach the singularity. –  Bru Jun 18 '13 at 17:37
You are perfectly right, but "it is meaningless to ask what does the outside horizon see when the falling obs. reach the singularity." is not at all my question. My question is that freedom degrees of singularity are seen by free falling observers, but, for fixed outside observers, these freedom degrees are felt differently, my hypothesis is that these freedom degrees lie on the (future) horizon, from the point of view of fixed outside observers. –  Trimok Jun 18 '13 at 17:44
I stopped understanding the question at the point where you started talking about black hole complementarity. I also don't understand why it matters that you've used the maximally extended Schwarzschild spacetime; I don't see how this ties in to anything else in the question. The part about "degrees of freedom of the singularity" also doesn't make sense to me. The singularity isn't even a subset of the manifold, so I don't see how it could have degrees of freedom. –  Ben Crowell Jun 19 '13 at 2:38
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