I'm having a hard time understanding what the arguments against stable toroidal black holes are saying. For many of these, I can't figure out if they're talking about:

  1. A non-rotating toroidal event horizon
  2. A toroidal event horizon, for which it's matter is moving in the tangential direction (of its center ring) such that the gravitational force is balanced by its acceleration

I am unimpressed and uninterested in #1, since this is predicted by basic Newtonian physics, and the matter starts out violently accelerating toward the center of mass.

A number of academic arguments about the viability of toroidal black holes in the book Black Hole Physics, page 164-165. I think that the term "non-stationary" might be referring to my #2 above, but this isn't explicitly established in that page or two. The most detailed mechanistic logic I found was this quote:

Jacobson and Venkataramani (1995) pointed out that a black hole with toroidal surface topology provides a potential mechanism for violating topological censorship. Specifically, a light ray sent from past null infinity to future null infinity and passing through the hole in the torus would not be deformable to a ray which did not come close to the black hole. Thus, topological censorship implies that a toroidal horizon (if it exists) must close up quickly, before a light ray can pass through.

The 1995 Jacobson and Venkataramani paper does seem to make very strong statements along this thread. But nothing I read about the Cosmic censorship theorem itself sounds even remotely convincing. It's more-or-less saying you can't have naked singularities or causality violations. Neither of those are obvious to me from the above quote's argument.

In layman's terms, it sounds like the arguments say that a stable toroidal black hole would allow you to time travel by going through the donut hole. The notation is probably just beyond my grasp. Does the argument really apply to #2 as I've defined it? And, if so, how can I convince myself of the proposition?

  • $\begingroup$ Is this somehow connected with black rings solutions in higher-dimensional gravity? $\endgroup$
    – user23660
    Sep 27, 2013 at 20:00
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    $\begingroup$ Possible duplicate? physics.stackexchange.com/q/33963 Not that there is terribly much detail over there. $\endgroup$
    – user10851
    Sep 27, 2013 at 20:24
  • $\begingroup$ You shouldn't say "cosmic censorship theorem." The result isn't proved. And in fact, over a precisely fine-tuned subset of normal matter distributions, cosmic censorship will be violated. $\endgroup$ Sep 27, 2013 at 20:43
  • $\begingroup$ And I should say that it is very definitely the case that you can build a time machine out of a Kerr naked singularity by travelling a closed path that goes through the ring. I don't necessarily see why this should generically be true for all toroidal black hole horizons, but I remember my advisor saying that known numerical solutions that included toroidal horizons quickly evolved to spherical black holes. $\endgroup$ Sep 27, 2013 at 20:46
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    $\begingroup$ @AlanSE I just wrote up a somewhat lengthy review of toroidal horizons related to this question: physics.stackexchange.com/q/92224. I don't know if you've already read those papers, but they show many ways you can't have tori, including topological (not cosmic) censorship. $\endgroup$
    – user10851
    May 20, 2014 at 7:37

1 Answer 1


It is not the cosmic censorship theorem/conjecture that rules out stable toroidal black holes but no-hair theorem/conjecture. If such black hole existed its field expansion would have contained additional terms (anapole moment ?) which would have constituted 'hair' in terms of the theorem.

For instance, in higher dimensions there is 'cosmic censorship' conjecture but no 'no-hair' theorem. And there are 'black ring' solutions: in 5D it hash $S_2\times S_1$ horizon topology (which would be the analogue of toroidal black hole).


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