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There is famous No-hair theorem:

The no-hair theorem states that all black hole solutions of the Einstein–Maxwell equations of gravitation and electromagnetism in general relativity can be completely characterized by only three externally observable classical parameters: mass, electric charge, and angular momentum.

But there is also Majumdar-Papapetrou solution, which describe few critical black holes. Repulsive Coulomb force between two black holes exactly cancels the attractive Newtonian gravitational force. It describes N black holes sitting at arbitrary positions.

Such solution have $(N-1)*3$ additional parameters. So it's look like no-hair theorem is not true for such solution.

What no-hair theorem say about such solutions?

Why such solutions is not prohibited by no-hair theorem?

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  • $\begingroup$ Since that solution contains multiple black holes, i guess the theorem just doesnt apply. Else you could say the same for any number of black holes in any weird configuration. $\endgroup$
    – NDewolf
    Commented Nov 17, 2020 at 18:43
  • $\begingroup$ @NDewolf, two Schwartzchild BH is not static solution. $\endgroup$
    – Nikita
    Commented Nov 17, 2020 at 19:22
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    $\begingroup$ Does the Majumdar-Papapetrou solution have an event horizon? If not, does it qualify as a black hole according to the conditions of the theorem (which are not stated in the excerpt)? $\endgroup$ Commented Nov 18, 2020 at 0:31

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First thing to note that the "no-hair theorem" is not a single theorem, but a collection of theorems and results each with their own conditions and limitations. For probably the most commonly quoted result - regarding the uniqueness of the Kerr-Newman family of electrovac black holes - one of those condititions is that the horizon has a single connected component, thereby explicitly excluding the possibility of a multicenter Majumdar-Papapetrou solution.

However, some of the other results do apply to multicenter solutions. For example, one can show (if your name is Stephen Hawking in the 1970s) that each connected component of the event horizon (in 4D) must have a spherical topology.

Ultimately, the various available results point towards the Majumdar-Papapetrou solution being the only possible stationary asymptotically-flat multi-component black hole solutions (in 4D Einstein-Maxwell theory). However, a full proof is still missing (according to this review by Robinson, the closest is a proof by Gibbons assuming all components had the same charge to mass ratio.)

Anyway, I highly recommend the review I mentioned above, which gives a nice overview of the various results and their historical context.

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