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I have read several other questions about black holes merging and I am left with the question about the criterion that considers two black holes merged.

From what I have read on PSE, the singularity of a black hole can cross the horizon of another black hole. This implies that for some time, both horizons have already merged, but does this also mean that one has to wait until a singularity crossed an horizon to say that the black holes have merged? Or does one has to wait until the two horizons have "touched"? In such a case, can't we consider the black holes as a single non spherical black hole, with two singularities? So a black hole that results from the merging of two others would be highly non spherical, at least until one of the singularity crosses the closest horizon.

In short, when can one consider that two black holes have merged?

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Your question can't exactly be answered unambiguously because in General Relativity, the concept of time becomes irregular — not totally arbitrary, but even weirder than in Special Relativity. Specifically, we can choose how to define time in our own way, changing it from point to point. There are some restrictions: time has to vary smoothly from point to point, and it can't vary too quickly. But you can definitely push the value of time earlier or later in one region without changing it in another. The set of points that have the same time is called a "slice". Basically, we can wiggle that slice around and GR still works.

Now, typically we talk about the shape of the horizons on a slice and we say that they have merged when the topology of the horizons on that slice has changed from two separate spheres to a single sphere — or more precisely, two touching spheres as you suggest. And we usually think of them touching at a point. However, it is possible to re-slice a spacetime so that the spheres touch for the first time at two points — or possibly even more than two — so that the horizon has the topology of a torus. This was done numerically for the first time in this paper.

Of course, that's just talking about event horizons, and there are different types of horizons. In particular, there is also the concept of an apparent horizon. When you change the slicing of a spacetime, you can actually change the apparent horizon itself. In fact, it was shown in this paper that you can even reslice the plain old Schwarzschild spacetime so that there is no apparent horizon anywhere in the spacetime.

It so happens that, in numerical simulations, you don't normally see such crazy effects — not for any good reason, but just because of arbitrary choices of coordinates. And the apparent horizon is the easiest one to find, so they're normally the ones we talk about. Typically, each black hole is surrounded by its own apparent horizon. And then suddenly you can find another apparent horizon surrounding those two. The moment that "common" horizon forms is the moment you say they have merged. And this horizon is indeed spherical in the topological sense, but usually very distorted in whatever coordinates happened to have been used.

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  • $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$ – David Z Aug 24 '18 at 3:35

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