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Originally Posted by Blob
However, if we remember that Hawking basically used our normal space-time in his `no-hair theorem` and worked out that the BH is a sphere;
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Only if it is non spinning. A spinning black hole is not a sphere, the exact shape depends on how fast the hole is spinning. The no hair theorem has to include normal spacetime. After all, the no hair theorem describes what an observer outside of the event horizon sees.
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Originally Posted by Blob
but we add extra dimensions to his theory (I assume that they even exist of course) then the topology of the BH changes.
The sphere could be `hollow`, have `holes` in it, it could even be `inside out`.
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That would depend on the size of the extra dimensions. If they are the size of dimensions postulated for superstring theory, there would be no change in the topology of a black hole larger than the plank length.
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Originally Posted by Blob
All that would be possible as you cross the event horizon (basically our mathematical tools fail us at this point, though).
So as you pass the horizon you may strangely find yourself at every point inside the sphere.
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Actually, the math works quite well right down to r=0. It's only at the central singularity (r=0) where the math fails. There appears to be a failure of math (singularity) at the event horizon when using spherical coordinates. But that singularity disappears if you change coordinate systems, which means the singularity at the horizon in spherical coordinates is not a physical singularity. The central singularity is physical.
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Originally Posted by Blob
I would at this point stress that I don’t have a clue what I’m talking about.
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Well, I hope this helped.