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:

  1. take a circle of radius $R$

  2. take $N$ equidistant points in the circle.

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

  4. pick an orientation (CW or CCW), and then throw black holes of radius $r \approx \frac{\pi R}{N - \pi}$ 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 $t_0$, their event horizons become connected. Even assuming that nature is abhorrent to this event horizon topology, it will take at least $t= \frac{R}{c}$ 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?

  • $\begingroup$ I can't answer your question, but +1 for the cool paper. Thanks. $\endgroup$ Aug 11, 2012 at 6:19
  • $\begingroup$ Just a clarification to understand the question: You have in mind a spacetime with a nontrivial spatial topology - maybe there's a spatial hole in the middle that lasts forever. Your idea is that by constructing this toroidal event horizon you can surround the hole, but because the EH is null you can construct a causal curve which links the topology but escapes to future null inf, thus invalidating the censorship theorem? (I don't know the answer - I just wanted to see if I understood what you were asking...) $\endgroup$
    – twistor59
    Aug 11, 2012 at 7:07
  • 1
    $\begingroup$ @twistor59, no, the spacetime is asymptotically flat minkowski, but once the toroidal EH forms at $t_0$, topology censorship says (or that is my reading at least) that any causal curves crossing inside will never reach infinity, regardless how big $R$ is. But since there is plenty of time before the information of the EH formation reaches the center and stuff to cross in the meantime, is not clear how that can happen $\endgroup$
    – lurscher
    Aug 11, 2012 at 7:12
  • $\begingroup$ Hmm, I was reading topology censorship as applying to the topology of the underlying manifold, i.e. it wasn't making a statement with respect to the topologies of subregions of spacetime defined by the positions of event horizons. When you say "crossing inside" in the last comment, you mean "crossing inside the region defined by the event horizons"? $\endgroup$
    – twistor59
    Aug 11, 2012 at 7:27
  • $\begingroup$ @twistor59, once the toroidal EH is formed, causal paths that cross inside the torus (and manage to reach infinity) cannot be deformed in trivial paths (those that lie far away from the circle and the affected region), so the spacetime is not simply connected anymore $\endgroup$
    – lurscher
    Aug 11, 2012 at 7:40

1 Answer 1


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.

  • $\begingroup$ damn! great answer, +1 $\endgroup$
    – lurscher
    Aug 11, 2012 at 12:55

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