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Imagine two close, really big black holes rapidly spinning around each other. That setup would emit a terrible amount of gravitational waves. My question is, could those gravitational waves, if big enough, temporarily collapse space around them into a singularity?

That is, not the black holes themselves forming a singularity, but the ripples they form on space-time, constructively interfering to form a singularity in a place where there is no actual mass.

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As Ron states, gravitational waves can in fact collapse to form black holes. There are exact solutions representing this, such as John Archibald Wheeler's geon solutions and the "Brill Wave" solutions of Dieter Brill, both of which feature vacuum solutions that evolve into black hole solutions at late times.

The majority of the radiation produced isn't going to happen until the holes are orbiting very close to each other.${}^{1}$ The plunge process after the instability hits is also very rapid, creating a common horizon in much less time than, say, an orbit.

This means that it is highly unlikely that you will be able to fine-tune the radiation in such a way as to focus the gravitational radiation in such a way to create a third, separate horizon or a third singularity.

${}^{1}$For an extreme ratio binary pair of unspinning holes, the last stable orbit is at a radius of $\frac{6GM}{c^{2}}$, where $M$ is the mass of the larger hole. Radii of instability for extreme ratio spinning holes are typically closer than this.

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Gravitational waves act like any other matter, so they can form black holes like any other matter. But for two spinning orbiting black holes, the waves just go out to infinity. The solutions for any mass ratio of black hole collision is a recent benchmark for the progress of numerical relativity, and the gravitational wave emission is always regular.

The process would not be constructive interference, however, it would be collapse by concentrating waves in a small region.

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