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My question is if we have two black hole and they will merge into each other, then where will be the singularity?

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The issue of the interior of merged black holes (or for that matter any black holes if one considers quantum gravity and not GR) is not settled. However, I would like to point out that the question needs to be phrased carefully. First the starting singularities are spacelike and the final singularity would also be spacelike. In between could it be timelike? I doubt it since then one could avoid it. Second during the merger the manifold would change drastically and so the original coordinate systems would become largely useless except close to infinity. Thus answering "where" the singularity is would be very difficult.

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  • $\begingroup$ Is it possible that if we had an exact interior solution for merging black holes, there would really just be a single spacelike singularity inside, in the same way that merging black holes really only have a single continuous event horizon (one with two 'trouser legs') from the perspective of a spacetime diagram? $\endgroup$ – Hypnosifl Jan 13 '15 at 22:25
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There is of course the hope that quantum gravity will resolve the singularity within a black hole, but if we restrict to the classical case for the present purpose, then yes singularities can merge. Imagine throwing two separate black holes into each other, the final state, after sufficient time has elapsed, will be a Kerr black hole. At intermediate times there will be radiation and complicated physics associated with the merger, but the late time behavior will be Kerr (because of the angular momentum of the starting configuration). Kerr has a timelike singularity. It also has a inner Cauchy horizon which is known to be unstable, and it is actually a subject of current research what the endpoint of this instability is. I think the general outcome that is widely believed to be the case is that generic black holes will have singularities which are a combination of space like and null.

If you're interested in this last aspect concerning the nature of the singularity in general black holes, I would recommend the recent work of Dafermos.

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