# Topology of Black Holes: Possible Topologies & Examples

• martinbn
In summary, the result of Hawking's theorem says that the topology of a black hole has to be that of a 2-sphere.
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?

##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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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.

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.

## 1. What is a black hole topology?

A black hole topology refers to the geometric structure of a black hole. It describes the shape, curvature, and connectivity of spacetime within and around the black hole.

## 2. Are there different types of black hole topologies?

Yes, there are several possible topologies for black holes, including spherical, toroidal, and non-orientable topologies. Some theories also suggest the existence of higher-dimensional topologies.

## 3. What determines the topology of a black hole?

The topology of a black hole is determined by its mass, spin, and charge. These properties influence the curvature and structure of spacetime in and around the black hole.

## 4. Can we observe the topology of a black hole?

Currently, we cannot directly observe the topology of a black hole. However, scientists have been able to infer the topology of black holes through mathematical models and observations of their effects on surrounding matter and light.

## 5. What are some examples of black hole topologies?

Some examples of black hole topologies include the Schwarzschild topology for non-rotating black holes, the Kerr topology for rotating black holes, and the Reissner-Nordström topology for charged black holes. Other possible topologies include Kerr-Newman and Kerr-Taub-NUT topologies.

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