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Black holes are firstly a solution of classical general relativity, suggesting that when a sufficiently large mass is in a sufficiently small region, a black hole would appear. This is supported by the Hoop conjecture and the closed trapped surface theorem, and many papers have supported that these theories are well-established (like this). We see their applications in theorems like the TOV limit and models of gravitational collapse.

On the other hand, we have no evidence of non-contact black hole mergers(edit:a blackhole forming before its two precursor's horizons collide), and numerical relativity has already shown that binary black hole mergers will form a bridge between their black holes' event horizons (EH). Black hole thermodynamics also supports the idea since black hole mergers must contact to increase their surface area (entropy). The Weak Cosmic Censorship Hypothesis also rejects the idea of a black hole whose singularity can form out of nowhere during a non-contact merger.

So now we have two sides: one suggests that high mass density guarantees a black hole, even if you don't have an existing black hole; while the other argues that black holes must contact before mergers. Since a black hole's Schwarzschild radius is proportional to its mass, the closed surface area increases much faster than the enclosed volume. This scaling allows for the possibility of multiple black holes with a combined mass exceeding the Hoop conjecture's limit existing without making direct event horizon (EH) contact. If you push this idea to its extreme, you get black hole cosmology.

Another extreme scenario involves forming toroidal black holes or potentially hollow shell black holes (at a certain moment), whose center could be considered both inside and outside the EH simultaneously. However, they are mathematically impossible under the interpretations of the mass density theory. For example, a D=10 shell can only be covered by more than 400 D=1 black holes, but if combined into a single black hole, this mass would have a Schwarzschild radius of D=400(there's a thread about shell blackhole btw).

The question: Is black hole formation guaranteed by mass density, or does it require the direct merging at event horizons? Where did I get wrong about the theories I mentioned? Are there specific theories, simulations, or observations that already solved (or are planned to solve) the problem?

Clarification: The title means: Does meeting a certain mass density will inevitably create a blackhole?

(For this discussion, we focus on the conditions for black hole formation or merger involving at least 1 pre-existing black hole, rather than the initial formation process from matter or primordial black holes.)

ElfredaCyania
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I think the question is posing a false dichotomy. It is not a case of either/or but both/and. I'm not sure what is meant in the question by the phrase 'non-contact merger' since a merger surely involves contact, by definition. I think the question is asking about the case where there is a collection of massive bodies at some distance from one another, but such that the total mass $M$ within a circumference $C$ is $\ge C/4\pi$. One normally imagines these bodies to be ordinary, like stars, but the question proposes that they might themselves be black holes.

I think the answer is as follows, at least in the case where there is no electric charge.

When $M \ge C/4\pi$ then a horizon forms. This statement means simply that there is an enclosed spatial region from which no timelike or null geodesics extend to spatial infinity. It does not imply that a singularity has formed, but it does imply that a singularity will eventually form. So before the singularity forms we have a region correctly called a black hole (in that nothing can escape) but which is dynamic and not yet showing the characteristic final state of a black hole. For this reason one would not normally simply say "it is a black hole" without adding some further qualifications to say exactly what you are and are not saying by that.

Andrew Steane
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we have no evidence of non-contact black hole mergers

A non-contact merger is a contradiction in terms. If there is no contact then it is not a merger.

So now we have two sides: one suggests that high mass density guarantees a black hole, even if you don't have an existing black hole; while the other argues that black holes must contact before mergers.

These are not two opposing sides.

Black hole formation through gravitational collapse is a solution to the Einstein field equations.

Black hole merger is also a solution to the Einstein field equations.

Is black hole formation guaranteed by mass density, or does it require the direct merging at event horizons?

Actually, with enough mass a black hole can form from material at an arbitrarily low density. High density is only required for the formation of small black holes. But yes, black hole formation is a robust prediction of general relativity for suitable masses and/or densities.

Dale
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