How does nature prevent transient toroidal event horizons?.. and does it really

Question

How does nature prevent transient toroidal event horizons?.. and does it really need to?

Steps to construct a (transient) toroidal event horizon in a asymptotically flat Minkowski spacetime:

take a circle of radius R

take N equidistant points in the circle.

consider tangent lines on each equidistant point, label the infinite directions on each tangent as their clockwise or counterclockwise direction relative to the circle

pick an orientation (CW or CCW), and then throw black holes of radius r??RN?? from each tangent line from the asymptotic infinite. Choose the tangential momenta which they are sent to be p

When all the black holes arrive at the circle in time t0, their event horizons become connected. Even assuming that nature is abhorrent to this event horizon topology, it will take at least t=Rc for the event horizon to reach the center of the circle. So there is "plenty" of time for causal curves to pass through the inner region and reach infinity

How does the topology censorship theorem avoid this to happen?

Explanation / Answer

The problem with this argument is that in 4d, the horizon of a black hole scales linearly with the mass. If you divide a circle into N segments, and have black holes whose radius is order R/N, where R is the radius of the big circle, their total mass is order R, so that the light rays passing through the center can be trapped by the total gravitational field of all the black holes inside.

This argument is specific to 4d, where the mass/radius relationship is linear. In 4d, you probably can't form a toroidal horizon even transiently. But in 5d and above, you can have spinning black holes with a toroidal horizon topology, and this argument is what shows that this is possible. The exact stable spinning toroidal black hole solutions were found in the last decade, and are now a major focus of research.

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