# If black holes almost or barely collide

Consider black holes approaching each other at some relative velocity and varying impact parameters. For impact parameters above some value they will sail past each other. For impact parameters below it they will merge. What happens if the impact parameter is exactly that value? It seems physically and mathematically unintuitive that an infinitesimal change in the impact parameter could make the difference between between one black hole and two.

Edit: There are some possibilities I did not consider:

"there could be a range of impact parameters where the black holes do not merge during the initial encounter, but have lost sufficient energy that they now form a bound system"

"It has been suggested that there may even be a regime where a third or more black holes are formed during the interaction of the two black holes before they scatter, essentially due to the strong focusing of gravitational waves by the shock-fronts representing the boosted black holes"

"approaching a critical value ... of the impact parameter, the two black holes settle into the analogue of an unstable circular geodesic orbit, whirl around for an amount of time ... then either fly apart or plunge together"

• When you get right down to it, the difference between a collision and a near miss is always infinitesimal. Black holes aren't special. Aug 27, 2015 at 5:07
• @Asher For material bodies such as planets, there is a continuous range of intermediate outcomes between merger and miss. They can splatter into multiple bodies, which is impossible for black holes. Black holes are special. Aug 27, 2015 at 5:22
• but for black holes the only options are miss or merge, whether the second body is another black hole or a mere photon. In that regard it's fitting to model them in the same manner as, say, marbles: either they hit or they don't, with no in between. Aug 27, 2015 at 5:52
• The "miss" as I understand it, has to be outside the photon-sphere, 1.5 times the Schwarzchild radius. Inside the photon-sphere all orbits degrade and escape is impossible without an outside force. Also, there's really 3 options for 2 objects sailing towards each other. 1) Collision, 2) they sail past each other, and 3) they remain in orbit around each other. Aug 27, 2015 at 7:05
• I'm not going to argue over the meaning of "collide" . The point of my question is that for material bodies, the final state of an interaction varies continuously in some sense with the impact parameter, whereas for black holes it is discontinuous. Aug 27, 2015 at 22:40

Who says they merge?

And I don't mean, give a citation, not that I object to citations. I mean who observers it. Even head on, the observers far away do not see them merge.

This is key. If it is someone on the outside the answer is that it never forms a merged black hole. In the sense that after any finite amount of observer time you can still wait and see a test particle emerge from the nonzero space between them. So the problem is that test particles between them need an infinite amount of far away observer time to escape even in the case when they "merge."

So to people far away it remains two black holes that swirl around each other with the event horizons getting closer and closer but never touching. And to the people far away the universe just doesn't last long enough to see them merge.

If they don't merge then there remains space between them (in some sense) where the observers right there can wait forever and not see themselves get swallowed up.

So the cutoff is more like the critical density or a parabolic orbit. The critical density still had the universe expand forever and the parabolic orbit is still unbound. It's just the cutoff. And cutoffs can be bound. The event horizon is the cutoff between events that can escape and events that cannot and the cutoff is itself bound.

So when you are at the critical impact parameter for a given relative velocity (which isn't very precise, the exact metric between the two objects is relevant) you are going to have some space in between and it is going to get squished and the things inside will start to take an infinite amount of time to get out.

So you can think of it like this. For a first impact parameter something in between can see a million years older universe when it gets out and up to a reference sphere in the asymptotic region. Then a slightly smaller impact parameter makes it see a billion year older universe when its gets to safety. And then an even smaller impact parameter makes it see a trillion year older universe. The critical impact parameter means the observer in between won't see the older universe because they aren't getting out. Even closer impact parameters just start to have a bigger region of the "in between" get swept up in it all the sooner.

Plus, there is actually lots of gravitational radiation going on if they merge, that is how over time it starts to approach a curvature outside that is closer and closer to the Kerr solution.

As for splattering. If the black holes are astrophysical instead of primordial then they can still be broken apart. By definition the events we see are events on our side of the horizon which from the infalling material's point of view is before it crosses. So we see the material always from before any horizon forms and so it is actually extremely red shifted time dilated regular matter we see moving towards each other if they are astrophysical objects and so by definition they can still be torn away from each other. It is hard. But that is exactly what those near misses do, they allow delicate interactions that take trillions of years of outside time to peel off bits of the black holes.

"So to people far away it remains two black holes that swirl around each other with the event horizons getting closer and closer but never touching. And to the people far away the universe just doesn't last long enough to see them merge." My definition of "not merge" is that astronomers observe two bodies moving away after the interaction. What will astronomers observe if they don't "not merge"?

The regions outside the event horizon press closer and and closer together. As they swirl around each other they emit gravitational radiation. Whether or not they slow down their joint spin depends on how they interact with stuff outside the horizons. But they definitely get closer and closer together. They themselves don't act like they are spin, it is frame dragging, they experience very little time, not enough time to orbit around something an infinite number of times.

If they are astrophysical then you see very redshifted images of the material that formed each object and you see them get closer. And they still have the theoretical ability to splatter and break apart like how you can leave the solar system by a series of gravitational slingshots. There are lots of bodies making up each object so there isn't one object with a fixed photo sphere. And the one that don't escape can still come close so you can watch the parts climb up a bit and come back down. So the frame dragging is important there too, they aren't going to feel like they are orbiting.

The real long term dynamics become determined by what happens to in falling matter at short time scales. If in short time scales you interact with your neighbors on ways that normally cancel out then there is room for interesting physics to happen.

You might lose the ability to have time to go influence them into leaving, but you are seeing them from back when they could escape. Over time you see little bits escape by being lucky and getting assistance from their neighbors. But you see most looking redder and you see them being frame dragged around in a circular type orbit. Whether they just have s finite emission of photons and hence you eventually see a last one from the ones that don't come out or whether you see interesting physics about things that happen on super short time scales that we normally only see the average remains to be seen.

• "So to people far away it remains two black holes that swirl around each other with the event horizons getting closer and closer but never touching. And to the people far away the universe just doesn't last long enough to see them merge." My definition of "not merge" is that astronomers observe two bodies moving away after the interaction. What will astronomers observe if they don't "not merge"? Sep 7, 2015 at 14:28
• "Over time you see little bits escape by being lucky and getting assistance from their neighbors." You seem to be suggesting that astronomers would observe some kind of oscillating blob emitting matter,(and small black holes?), photons, and gravitational waves. This might resolve the unintuitive discontinuous behaviour at the critical impact parameter, since there would appear to be a continuous range of outcomes below it. Has anyone else predicted that black holes can behave in this way (perhaps this should be another Question)? Sep 7, 2015 at 17:33
• @KeithMcClary It isn't uncommon. The whole idea of black home complementarity is based on the fact that to outside observers nothing ever crosses a horizon. Sep 7, 2015 at 17:55