Topology of Black Holes: Possible Topologies & Examples

[martinbn]

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 spatial 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?

 

[Dale]

The topology of Schwarzschild spacetime is $S^2 \times R^2$.

http://arxiv.org/abs/1111.5790

 

[martinbn]

That is different, I am not asking about the topology of the space-time, just a cross section of the event horizon.

I found this [PLAIN]http://arxiv.org/abs/gr-qc/9410004[/PLAIN]
http://arxiv.org/abs/gr-qc/9410004
which is relevant, but my questions remain.

 

[Dale]

$S^2 \times R$ 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"

 

[Ben Niehoff]

He is asking about the topology of (a spatial slice of) the horizon, not the topology of the whole spacetime.

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.

 

[martinbn]

That's interesting, I saw some papers about generalizations but dismissed them, may be i should go back and take another look. About the 4d case my curiosity is what happens if the energy conditions are not satisfied. Is the topology still spherical or could it be different or is the question not good because without that assumption one cannot guarantee that the event horizon will not disappear and so on anything along those lines. But the proof in Hawking's paper uses the dominant energy condition at one step.