Non-existence of multiple-black-hole solutions that look locally like subextremal Kerr-Newman
by Willie Wong
Pin Yu and I just posted a paper on arXiv based on some joint work we started doing when we were both graduate students (and parts of which appeared in his dissertation). I think I did a decent job explaining the motivation in a mathematical way in the introduction to the paper, so I’ll motivate the problem here a little differently.
Ever since the discovery of the black hole solutions, there had been interest in whether static or stationary black hole solutions with more than one hole can exists in equilibrium. One of the earliest considerations was by Bach and Weyl in 1922, fairly soon after the discovery of the Schwarzschild solutions. In general it was concluded that a static configuration is impossible due to singularities forming between two holes. Roughly speaking, a black hole in vacuum spacetime is a strongly gravitating object, and gravity attracts. If two black holes were to form and were to be kept apart, an external force will be needed to hold on to them. This manifests in a singularity between the two black holes.
On the other hand, if we were to add electrical charge to the black holes, then two black holes with charges of the same sign stands a chance of being in equilibrium: the electromagnetic interaction between two like charges is repulsive, while gravity is attractive, so maybe they can cancel out! And indeed such a configuration is possible. Under what are now called Majumdar-Papapetrou solutions are precisely these types of balanced multiple black hole solutions. But the balance has to be precise! The electric charge has to be large enough to equal the gravitational mass. And this in turn requires that the black holes represent what are called “extremal black holes”.
For the stationary, as opposed to static, case, the situation is less clear. In the static case, the black holes must remain fixed in place. In the stationary case, we can allow the black holes to orbit each other. As we know well from our own solar system, orbiting systems can happen where static systems are disallowed. This is because of the “fictitious” centrifugal force which can balance out the gravitational attraction (or, more correctly stated in accordance to American high school physics curricula, the gravitational force provides the centripetal force for the bound orbit).
In our paper we show that orbiting systems (as opposed to inspiraling ones where the multiple black holes eventually crash into one another) cannot exist provided that outside the black hole things looks more or less like the situation with only one black hole. A posteriori, knowing that these systems cannot exist, we are justified in our inability to provide examples. But even a priori it is difficult to imagine a system which our assumptions do not immediately rule out.
- If we have two black holes with large mass placed close to each other, we’d expect there to be lots of gravitational interaction and the local geometry will be distorted heavily away from the single black hole Kerr Newman solution.
- If we have two black holes with significant mass place far from each other, we’d naively expect that the gravitational field from infinity sees a single body with the combined masses of the two black hole, but near one individual black hole we only see the gravitational effects from that one black hole. So intuitively a situation like this cannot have an exterior that looks everywhere just like that with one single black hole.
The situation that intuitively we cannot rule out here is the case where we start with one gigantic black hole and one tiny black hole. The effect of the tiny black hole on the gravitational field is but a mere blip compared to the big black hole. So we can say expect that the space-time metric looks like that of when there is just the one big black hole. This is the one to which our theorem applies.
Our theorem also says nothing about the extremal case. We only consider the case where the charge is insufficient to balance out the mass. As we know from the Majumdar-Papapetrou solutions we do have a need to ruling out that class of solutions. Where this restriction enters the proof, however, is in a cute way. First, let me describe the general strategy of the proof. Our method is roughly inspired by Morse theory. We construct a real valued function on the space-time such that its level surfaces foliate the space-time with homeomorphic leaves. We then show that the boundaries of the black holes (the event horizons) necessary are almost level surfaces for this real valued function. This will give rise to a contradiction: on the one hand near infinity, the level surfaces look like spheres and is a connected surface; the boundary of the black holes, together form a level surface that has two components. This contradicts the homeomorphism between level surfaces. In the actual argument, however, we do not prove homeomorphism. Instead, we show that the real valued function has no critical points. The contradiction then is provided by a mountain pass lemma. It is here we need to use subextremality.
We can only show that the real valued function has no critical points where its values are at least as large as the values on the event horizons (there is a sign issue). So to actually get a contradiction by the mountain pass lemma, we need that as we go out from the event horizon, the value of the real-valued function increases. For subextremal black holes, the event horizon is non-degenerate, and the near horizon geometry forces this to be true. For extremal black holes, the event horizon is degenerate, and the near horizon geometry allows the value of the function to remain the same or decrease.