# Are black holes bound to merge once their event horizons touch? [duplicate]

Even if they move toward each other at relativistic velocities and not head-on?

I looked at questions Black hole collision and the event horizon and Dynamics of Event Horizons between head-on colliding non-rotating black holes, but did not find the answer.

• What do think it means to "merge" other than for their horizons to touch? – Brick Sep 20 '19 at 14:33
• – G. Smith Sep 20 '19 at 16:20

I'll start by quoting Stephen Hawking, and then I'll explain how it relates to the question:

As time increases, black holes may merge together and new black holes may be created by further bodies collapsing but a black hole can never bifurcate.

This is from page 156 in

By definition, an event horizon is the boundary of a region of spacetime from which no light signal can reach "future null infinity." That's a technically more careful way of saying that light cannot escape. If two black holes come close enough together that their event horizons touch, then we now have a single connected event horizon — a single black hole by definition. The question seems to be asking whether or not such an event horizon can become disconnected again.

The quote by Hawking shown above says that a black hole cannot bifurcate, meaning that a connected event horizon cannot become disconnected. Note that Hawking is using the term "black hole" to mean a region of spacetime behind an event horizon, regardless of the structure of whatever singularities may be lurking inside.

Why can't an event horizon bifurcate? The basic idea is that the set of points at any time in the future of a given connected region of spacetime is again connected: the future can't "split." Since the future of the interior of a black hole is always inside a black hole (by definition, otherwise it wouldn't be a black hole), that means that if the black hole was connected originally, it will remain connected forever.

That's the basic idea, but of course that's not a watertight argument, and we could probably contrive unrealistic situations where it doesn't hold. Maybe somebody with more experience in general relativity can post an answer/comment saying what the necessary and sufficient conditions are.

## A more careful argument

For people who aren't satisfied with the fast-and-loose argument that I gave above, here's a more careful version from section 6.2 on page 62 of the excellent review article "Light rays, singularities, and all that" by Edward Witten (https://arxiv.org/abs/1901.03928):

Black holes can merge, but a black hole cannot split, in the sense that if $$Z$$ is a connected component of the black hole region on a given Cauchy hypersurface $$S$$, then the future of $$Z$$ intersects any Cauchy hypersurface $$S'$$ to the future of $$S$$ in a connected set. To prove this, suppose that the black hole region intersects $$S'$$ in disconnected components $$Z_i'$$ ... Consider first future-going causal geodesics from $$Z$$. Any such causal geodesic (like any causal path from $$Z$$) remains in the black hole region, so it intersects $$S'$$ in one of the $$Z_i'$$. But the space of future-going causal geodesics starting at $$Z$$ is connected, and cannot be continuously divided into two or more disjoint subsets that would intersect $$S'$$ in different components of the black hole region. So all causal geodesics from $$Z$$ arrive at $$S'$$ in the same component $$Z_i'$$ of the black hole region. Now consider any future-going causal path from $$Z$$. If there is a causal path $$\gamma$$ from $$Z$$ to a given component $$Z_i'$$ of the black hole region, then by maximizing the elapsed proper time of such a path, we learn that there is a causal geodesic (lightlike or timelike) from $$Z$$ to $$Z_i'$$. So in fact, precisely one component $$Z_i'$$ of the black hole region on $$S'$$ is in the future of $$Z$$.

Notice the logic: first Witten considers causal geodesics, and then he extends the conclusion from causal geodesics to all causal paths. At the beginning of the section, Witten assumes that cosmic censorship holds and that the spacetime is globally hyperbolic and asymptotically flat.

• This is all closely related to the 2nd law of black hole thermodynamics, and is discussed in the same part of Hawking and Ellis. I would think that the necessary conditions to forbid bifurcation would therefore be the same as the conditions for the 2nd law. I think those are an energy condition (NEC or WEC?), cosmic censorship, and 3+1 dimensions, because apparently bifurcation is possible in higher dimensions. See A.V.S.'s comment on your related answer: physics.stackexchange.com/a/470429/4552 – Ben Crowell Sep 20 '19 at 21:07
• @BenCrowell Thank you for the comment and for reminding me about A.V.S.'s comment about exceptions in higher-dimensional spacetime. According to arxiv.org/abs/1106.5184, the Gregory-Laflamme instability of a black string in 5d spacetime can lead to a naked singularity, violating cosmic censorship -- thus violating one of the conditions that Witten's argument assumes (in the excerpt that I added to the answer a few minutes ago.) – Chiral Anomaly Sep 21 '19 at 0:24