Topology of Black Holes: Decomposing the Manifold and the Role of Knots

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SUMMARY

The discussion centers on the topology of black holes, specifically examining whether a black hole can be represented as a Heegaard decomposition or as the complement of a knot. The topological manifolds relevant to black holes include ##S^2 \times R^2## for maximally extended Schwarzschild spacetime and ##R^4## for black holes formed by gravitational collapse. The conversation highlights the complexity of defining "cross sections" and emphasizes that the topology of black holes can vary based on the spacetime model used, such as the distinction between connected and disconnected regions in different models.

PREREQUISITES
  • Understanding of topological manifolds, specifically ##S^2 \times R^2## and ##R^4##.
  • Familiarity with Heegaard decompositions and knot theory.
  • Knowledge of general relativity (GR) and its implications for black hole topology.
  • Basic concepts of cobordism and its application in topology.
NEXT STEPS
  • Study the properties of the Schwarzschild spacetime and its topological implications.
  • Explore Heegaard decompositions in the context of knot theory.
  • Read "General Relativity" by Robert Wald to deepen understanding of black hole physics.
  • Investigate the Witten-Reshetikhin-Turaev invariant and its relevance to black hole topology.
USEFUL FOR

Mathematicians, physicists, and researchers interested in the topology of black holes, general relativity, and knot theory will benefit from this discussion.

nateHI
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Can a black hole be presented as a Heegaard decomposition or as the complement of a knot?

I'll try and elaborate: If I understand correctly, the cross section of spacetime near a black hole can be thought of topologically as a manifold. What manifold is it? Can the manifold be decomposed?
 
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nateHI said:
the cross section of spacetime near a black hole

What do you mean by "cross section"?

nateHI said:
What manifold is it?

The topological manifold that describes the maximally extended Schwarzschild spacetime is ##S^2 \times R^2##. However, that spacetime is not physically reasonable. A physically reasonable spacetime that describes a black hole formed by the gravitational collapse of an ordinary object has topology ##R^4##.

These manifolds are for the entire spacetime. As above, I don't know what you mean by "cross section".
 
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PeterDonis said:
What do you mean by "cross section"?

I'm not sure about the physics term so maybe I should have stuck with the math. By cross section, I mean one of the boundaries of a cobordism between two 3-manifolds.
 
nateHI said:
By cross section, I mean the boundary of a cobordism between two 3-manifolds.

AFAIK you can only have a cobordism between compact manifolds. Neither the 4-manifolds I described, nor any 3-manifold "cross sections" you could take from them, would be compact.

Also, the spacetime describing a black hole is a single manifold, not two.

It might help if you would take a step back and explain why you are interested in this.
 
PeterDonis said:
It might help if you would take a step back and explain why you are interested in this.

I want to understand the topology of a black hole so that I can think about how (or if it's even possible) to compute its Witten-Reshetikhin-Turaev invariant.
 
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nateHI said:
I want to understand the topology of a black hole so that I can think about how (or if it's even possible) to compute its Witten-Reshetikhin-Turaev invariant.

You're probably better off asking about this in the math forum. I've already given you the two relevant 4-manifolds, topologically speaking, that I know of that have anything to do with black holes: ##S^2 \times R^2## and ##R^4##. A 3-manifold cross section of these would have topology of either ##S^2 \times R## or ##R^3##. Anything beyond that would be better asked of the regulars in the math forum.
 
Well, add the point at infinity to the real line and you are in business. I have no idea if that makes any physical sense though. The space S^2xS^1 is the complement of the unknot with no framing. The associated invariants are easy to compute. Unfortunately nothing interesting falls out as I had hoped.
 
nateHI said:
add the point at infinity to the real line and you are in business. I have no idea if that makes any physical sense though.

It doesn't.
 
To me it is still unclear what you mean by the black hole topology. The topology of the whole space-time? The event horizon? The black hole region? The intersection of the event horizon with a space-like hypersurface? Also do you assume that the black hole region is connected (one black hole) or not?
 
  • #10
martinbn said:
To me it is still unclear what you mean by the black hole topology.

The OP would have to say what he meant. The topologies I gave in post #2 were for the whole spacetime. In post #6 I gave possible topologies for 3-surfaces "sliced" out of the whole spacetime.

martinbn said:
do you assume that the black hole region is connected (one black hole) or not?

It depends on which spacetime you're looking at. In the maximally extended Schwarzschild spacetime, there are two "hole" regions which are disconnected; one is called the "black hole" and one is called the "white hole". But each of those regions, taken individually, is connected.

In the more realistic spacetime that describes a black hole formed by gravitational collapse, there is only one black hole region and it is connected.
 
  • #11
May guess is that he didn't mean just a Schwarzschild black hole, but was asking about black holes in general.
 
  • #12
It seems that in order to make my question less muddy I would need to study GR a bit myself first.
 
  • #13
nateHI said:
It seems that in order to make my question less muddy I would need to study GR a bit myself first.
At your shown level of mathematical sophistication, try first Wald's book then Hawking and Ellis, just for the physics part.
 

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