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Mathematician here with a speculative physical question -- feel free to boot me if the level isn't right.

Suppose one finds, or builds, a constellation of several black holes arranged in a circle. (To get a stable arrangement, presumably one could put a bunch of small positively charged black holes at equal angles around a central positively charged object, or alternatively suspend uncharged black holes gravitationally in a circle surrounded by a quickly rotating ring of heavy matter). Suppose moreover that the event horizons of neighboring holes overlap -- or if there's some reason this is theoretically impossible, suppose they're really, really close to being tangent, while leaving room for light and matter to pass through the middle of this ring. (Is this possible?)

My question is whether a trajectory from point A to point B that goes through the middle of the ring is causally related to one that does not. For example say a space explorer measures the spin of an electron, writes down the result and puts it in a box which he leaves at some fixed point in space (we've chosen a reference frame), then goes in a loop through the middle of the ring of black holes and comes back. When he comes back and opens the box again, will the recorded result be the same?

I got this question after someone told me about quantum decoherence, and I was curious whether the world can change significantly if you go around a loop that either cannot be contracted or cannot be contracted without at some point becoming ridiculously long (where it's up to you to decide the value of "ridiculously").

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"Suppose moreover that the event horizons of neighboring holes overlap" You're going to run into real trouble here ... I believe, that there is no such stable relationship, though I'd have to ask one of our theorists for a reference. –  dmckee Mar 15 '13 at 19:34
    
Yes, such a ring immediately forms one huge black hole larger than the size of the ring. If nonrotating, the radius of the big black hole will equal the sum of the radii of the smaller black holes. –  Jim Graber Mar 15 '13 at 19:42
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I don't see any reason why a ring of equally spaced black-holes, in circular orbit around a common center of mass, wouldn't be dynamically stable (of course it wouldn't be stable to perturbations, and for indefinite times). –  zhermes Mar 15 '13 at 20:27
    
I see. Thanks Jim! So then say the radius of the ring is very large and the holes are placed within epsilon of being as close as possible to each other without collapsing (so I'm approximating an infinite line of black holes). –  Dmitry Vaintrob Mar 15 '13 at 20:29
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Hi Dmitry, If your ring of uniform little black holes is spaced so that the inter black hole distance is more than pi times the diameter or two pi times the radius of the little black holes, it can be stable. Less than that and it forms one big black hole whose diameter is equal to the sum of the diameters of the little black holes. –  Jim Graber Mar 16 '13 at 2:40
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1 Answer

Two things would happen:

If the ring is sparse, meaning that there are large gaps between the holes, then you fly through normally and nothing interesting happens.

If you try to make the ring dense, so that the event horizon formes a torus, then the entire ring becomes a black-hole and you fall in and never come out. This is because the radius of a black hole scales linearly with it's mass. Since the sum of all the radii making up the ring has to be greater than the ring radius (if you want to avoid gaps), the entire thing becomes a black hole.

I am not sure what charge/rotation does but it probably doesn't change the picture much. There is a limit to how fast you can spin up and/or charge a hole for a given mass.

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