I'll recap for others: This is all about a vacuum Kerr solution, so the black hole (BH) is eternal. We might have three parts to the spacetime: "A" where ther

I'll recap for others: This is all about a vacuum Kerr solution, so the black hole (BH) is eternal. We might have three parts to the spacetime: "A" where there's some gravitational radiation far from but heading towards the BH, "B" where they and the BH are very close together, "C" where the remnants of the inbound radiation are far from the BH. "Stability" means that at "C" the BH is still essentially Kerr, because it was at "A". Stability includes a bigger BH or a BH with different spin comparing "A" and "C". "Instability" means that at "C" we no longer have something that's essentially Kerr. Now for the theoretical physics sketch: One way to get there that comes to mind is letting gravitational waves that are "just so" enter the ergoregion in a way which causes them concentrate into a caustic that does not run off to infinity as a new, free, black hole (which would just be a flavour of "C"). In that case we have a black hole binary, which sheds (arbitrarily weak) gravitational radiation, instead of e.g. the black hole we're interested in and the new one which is at infinity, neither of which sheds any gravitational radiation at all. So we might have a "C" configuration that is badly described by the Kerr metric, even though we had an "A" that was asymptotically Kerr. This paper (and several in its bibliography) show that despite questions about this that do arise in linearized gravity (and effective one body and numrel and self-force), the full theory says that for a << M (implying smaller and weaker ergoregion) we don't get two black holes (or anything that's badly described by the Kerr metric). Now for the astrophysics sketch: > in nature We don't have a vacuum spacetime, unlike in this paper. We also have a sky with a lot of black holes in binary systems of various types and closenesses, at least a couple triples, and probably many more orbital configurations somewhere in nature (especially the future if most baryons are in black holes). The paper doesn't deal with those. It doesn't want to help us figure out the "final-parsec problem". Also, natural black holes don't extend into the infinite past, unlike the Kerr family of solutions in this paper. Isolated ones might settle down into asymptotically Kerr, but only if they don't fully evaporate! Lastly, old gravitational waves are weak in recent epochs of the universe, so they aren't going to give much of a kick to even a low mass Kerr-like BH. Strong early GWs have been redshifted. A strong-like-this-paper-envisions GW kick from a binary black hole merger to a nearby third BH isn't something you'd describe in a slightly perturbed Kerr picture, and all three of those BHs could have started as primordial, direct collapse, or been ejected right the hell out of their parent galaxy clusters before meeting in deep space, so each can be effectively slowly-rotating Kerr for a long time before they get close to each other.