# Topology of black holes

## Main Question or Discussion Point

This might be well known or even discussed here, though I couldn't find a thread about it, but the questions is what are the possible topologies of a black hole i.e. the topology of a spacial slice of the event horizon. I know there is a result of Hawking that says the topology has to be that of a 2-sphere. I am looking at his paper "Black holes in general relativity". In the proof the dominant energy condition is used, so my question is if the energy condition is violated, enough to make the integral change sign, is it possible to have a black hole with a different topology? Or is the condition needed only for this proof, but the result holds under weaker assumptions? In case different topologies are possible, are there any examples and which assumptions have to be violated? As a side, is there an exposition of the proof of Hawking's theorem written in a more textbook like style, with more details and more self contained?

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Dale
Mentor
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Dale
Mentor
$S^2 \times R$ or just $S^2$ depending on what you mean.

If you mean some other spacetime than a Schwarzschild spacetime then I don't know what you mean by "black hole"

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Ben Niehoff
In 4d, one can prove (as you have found) that the horizon is spherical. However, this proof works only in 4d. In higher dimensions, one can find other topologies, the simplest example being "black rings" in 5d, which have horizon topology $S^2 \times S^1$ (as opposed to $S^3$ for spherical black holes). I seem to remember that the general proof shows that the horizon is a manifold of positive Yamabe invariant, or something like that.