Basically, my question is the title. If a black hole crosses the Roche limit of another black hole, what happens?
For a hypothetical example, let’s say you have a two black holes: one at 5 solar masses and one at 300 solar masses. If the smaller black hole crosses the Roche limit of the larger what happens? Does they simply merge? Would the event horizon of one or both black hole’s be geometrically distorted in some way or retain their spherical shape?
https://en.m.wikipedia.org/wiki/Binary_black_hole
The two event horizons stretch out toward each other, form some interesting shapes, connect into a cylindrical bridge shape, and then the combined horizon smooths itself out while emitting large amounts of gravitational radiation.
For “realistic” scenarios where these black holes start out in a binary elliptical orbit, the final ringdown phase concludes very rapidly. Gravitational waves are emitted continuously during the inspiral phase, in a manner analogous to how an electron in a circular orbit emits electromagnetic radiation.
The event horizon itself is a mathematical boundary of neither matter nor energy, so it does not appear to slow down or stop from time dilation. (From the perspective of a distant outside observer).
The no-hair theorem applies to stationary black hole solutions. That is, after event horizon ring down is completed.
Editing myself to directly answer the information question:
If you observe an event horizon on a complex distorted bridge shape, you can deduce information about the original merger partners. This is not a violation of any principle, because the famous no hair theorem does not apply in this situation.
The complex shape condition is not stable, and it relaxes to a “simple” shape that provides no information about the individual merger partners. This process completes in finite time, and is usually quite fast.
During this process, undulations and ripples in the shape of the event horizon result in emitted gravitational waves. Presumably, these gravitational waves contain the last information you can possibly get about the original merger partners.
Second edit: I am not a physicist, but I can read Wikipedia. Feel free to correct me.
No, not just presumably. LIGO picks up and measures black hole mergers semi-regularly. They’re just loud (in gravitational waves) and very easy to interpret.