# Black holes: Is merger inevitable when horizons touch?

I watched a simulation of the binary black hole merger of 2019 April 12
https://www.ligo.caltech.edu/video/ligo20200420v1

When the "apparent horizons" (their terminology; are those different from event horizons?) touched, the two combined and the merger was complete. So I'm wondering: would it be possible for two black holes to quickly pass by each other, so that their horizons briefly touched, but then they continued on their ways separately.

PS: The video was nice, but I would have preferred it stay on a 2D plane. I don't know if the changing points of view was done by the animators, or if the two objects really did move around in three dimensions.

• What do you mean by "touched"? The event horizon is not a physical place but a mathematical boundary. Commented Nov 10, 2021 at 0:45
• Here is a much better video: youtube.com/watch?app=desktop&v=Y1M-AbWIlVQ Commented Nov 10, 2021 at 4:07
• Once two horizons touch, it is already a single black hole. Commented Nov 10, 2021 at 4:15

## 2 Answers

I'm not sure what you mean by "touch" since the event horizon is not a physical location$$^1$$.

If you mean overlap, and if you are asking if it's possible that merging can be avoided even if the event horizons of both black holes were to overlap, then the answer is still no. They will merge (regardless of how fast the black holes are moving). The escape velocity at an event horizon is equal to the speed of light, and nothing (with mass$$^2$$) can have this speed, so even if two black holes approach with a great speed, they will merge if their event horizons overlap.

$$^1$$ The event horizon is a mathematical boundary, where for example, in a Schwarzschild black hole, the event horizon is defined by the Schwarzschild radius $$r_S=\frac{2GM}{c^2}$$ where $$r_s$$ describes a mathematical sphere (meaning non-physical) surrounding the gravitating mass $$M$$.

$$^2$$ Even something massless, like light, still will not be able to escape since the future of all events (within the EH) lie within the event horizon.

Event horizons are defined as trapped surfaces, closed and orientable surfaces that act as one-way boundaries. The apparent horizon is the union of all trapped surfaces, essentially building up a 4-dimensional boundary in spacetime no particle can get out of. Note that they are not physical things, and can move about faster than light. The topology censorship theorem implies that they cannot have holes like doughnuts.

The second law of black hole thermodynamics gives a very strong reason to think the horizon area cannot decrease, which it would do if a $$2M$$ mass hole (area $$4M^2$$) could turn into two mass $$M$$ holes (total area $$2M^2$$). There is a somewhat devious possibility of mergers followed by splitting into three or more holes that obeys this, but considering the entropy carefully this is disfavoured.

What I have not seen is a purely classical argument for the non-separation of a black hole merger. One can obviously take the time reversed spacetime manifold of a merger and get a valid manifold that splits - but this is a black hole disturbed by a complex ingoing set of strong gravitational waves that splits, a bit like how a loud crash sound converging on some shards can in principle make them reassemble into a vase jumping back up on a shelf. Possible, but thermodynamically impossible.

But I have a feeling there should be topological arguments about the trapped surfaces that could also be used.