A small black hole asymptotically approaches a big black hole's event horizon. Will it seem to be frozen there, or will it seem to merge? On this site, there are currently two scenarios described:

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*two black holes merge in a finite time

in any sensible meaning of the term merge the two black holes do indeed merge in a finite, and very short, time.

So Black Holes Actually Merge! In 1/5th of a Second - How?


*anything falling to a black hole's event horizon will seem to be frozen there

Indeed, nothing can get under the horizon. The stuff close to the event horizon does move outwards as the black hole radius increases. Even more with any black hole deformations such as waves on its surface, the tidal deformations or the change of the rotation speed, all the objects close enough to the horizon remain "stuck" to it and follow all the changes of the black hole form.

How can anything ever fall into a black hole as seen from an outside observer?
So basically, the first one says, that two black holes will merge in a finite time from an outside observer's view (in fact we have witnessed black hole mergers and the gravitational waves coming from them), and the second one says that anything approaching the horizon will remain stuck to it (seem to be frozen there).
Now I cannot resolve this, because if a small black hole approaches a bigger black hole, then they should seem to merge in a finite time, but the small black hole should seem to be frozen at the event horizon of the bigger one. These two scenarios cannot seem to happen at the same time. We should either see the holes merge, and a common horizon to be created in a finite time, or we should see the small black hole to be frozen at the event horizon of the bigger one. I do understand that black holes are black, but they do have photon spheres and accretion disks, so the merger or the frozen state should be clearly distinguishable.
Which one will we observe? A merger, or a smaller black hole frozen at the event horizon of the bigger one?
A small black hole asymptotically approaches a big black hole's event horizon. Will it seem to be frozen there, or will it seem to merge?
 A: These are two different situations.
When an ordinary object falls into a black hole, The black hole curves spacetime. The object is assumed to not curve spacetime. A typical solution is given by the Schwarzschild metric. In that solution, as seen by a distant observer, the object takes an infinite time to reach the event horizon.
When a small black hole is dropped in, both objects curve spacetime. The small black hole is not going to get arbitrarily close to the big event horizon without disturbing it. The metric between the black holes is not well described by the Schwarzschild metric. They will merge.
A: You shouldn't take "frozen at the horizon" to mean that an extended body will appear to freeze as soon as the front end of it reaches the horizon, and retain its shape. Rather, the front appears to freeze as it nears the horizon, and the back also appears to freeze as it nears the horizon, so the object ends up seeming to be flattened against the horizon (Lorentz contracted, if you like; that's a fairly apt analogy since the object also appears redshifted as though it was receding rapidly). Meanwhile, the event horizon expands by an amount corresponding to the mass of the infalling object. This starts as a bulge at the location of the infalling object but it also flattens quickly.
When two black holes merge, essentially the same thing happens. If the smaller hole glowed brightly enough to be seen, you would see it pancake on the horizon and redshift to invisibility much like an ordinary object of the same size and temperature. The event horizon of the smaller hole becomes a bulge in the horizon of the combined hole which then flattens out.
A: I would like to add another answer that gets at this another way. I am not totally sure of this answer. If anyone has corrections, I would like to hear about it.
Suppose two black holes approach each other and freeze in place without actually merging, as described in Nathaniel's answer to How can anything ever fall into a black hole as seen from an outside observer?.
This is not a spherically symmetric distribution of matter. We should see an asymmetric gravitational field. If the pair are rotating, we should see gravitational waves continue to be emitted. We should see oscillations in the positions of background stars. We should see interesting effects in any accretion disk.
