Compactness of V in Minkowski Space: A Question

In summary, the closed forward light cone V in Minkowski space is not a compact set as it is not bounded. This can be proven using the criterion that all compact sets in metric spaces are bounded. However, this does not apply to subsets of Minkowski space, so the compactness of V cannot be determined with certainty without using the general definition of compactness.
  • #1
parton
83
1
Consider the closed forward light cone

[tex] V = \left \lbrace x \in M \mid x^{2} \geq 0, x^{0} \geq 0 \right \rbrace [/tex]
and M denotes Minkowski space.

My question is whether V is a compact set or not. If it is a compact set, how do I show it?

Intuitively I would say it is compact, but I don't know how to proof it.

I hope someone can help me.
 
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  • #2
parton said:
Consider the closed forward light cone

[tex] V = \left \lbrace x \in M \mid x^{2} \geq 0, x^{0} \geq 0 \right \rbrace [/tex]
and M denotes Minkowski space.

My question is whether V is a compact set or not. If it is a compact set, how do I show it?

Intuitively I would say it is compact, but I don't know how to proof it.

I hope someone can help me.

Why does your intuition tell you that a light cone is compact?
 
  • #3
A simple criterion for deciding if a set is not compact is boundedness: if a set is not bounded, then it's not compact!
 
  • #4
I'm sorry, I did a mistake and the problem was no cone, but a certain subset of a cone.

Of course the cone defined above is not bounded and therefore not compact. Thanks.
 
  • #5
That is only true for subsets of Rn. Are you working with subsets of Rn here?
 
  • #6
HallsofIvy said:
That is only true for subsets of Rn. Are you working with subsets of Rn here?

hmmm, no, I am working with subsets of Minkowski space M. So the Heine Borel theorem doesn't apply to M. On the other side we could equip M with an Euclidean topology. And this space would be homeomorphic to R^4, so the basic topologic properties should be the same in both spaces. But I am not sure whether this holds for compactness, too.

But to be sure, maybe one should use just the general definition of compactness to prove the compactness of a subset of M, i.e. if each of its open covers has a finite subcover. But I think it is rather difficult to prove it in this way, but I don't know another possibility.
 
  • #7
Sorry, I misread. It is only in Rn or variations that "closed and bounded" sets must be compact. It is true in any metric space that all compact sets are bounded. Since this set is not bounded it cannot be compact.
 

1. What is compactness in Minkowski space?

Compactness in Minkowski space refers to the property of a set of points in this space to be bounded and closed. In other words, a compact set in Minkowski space is a set that contains all its limit points and has a finite size.

2. Why is compactness important in Minkowski space?

Compactness is important in Minkowski space because it allows us to define and study important geometric concepts such as continuity and convergence. It also helps us to understand the structure and behavior of spacetime in the context of special and general relativity.

3. How is compactness related to the topology of Minkowski space?

Compactness is closely related to the topology of Minkowski space as it is a topological property. It is used to define the concept of compactness in topological spaces, including Minkowski space. In fact, compactness is one of the fundamental properties of a topological space.

4. Can a set be compact in one topology but not in another in Minkowski space?

Yes, a set can be compact in one topology but not in another in Minkowski space. This is because different topologies can define different notions of compactness. For example, a set may be compact in the Euclidean topology but not in the discrete topology.

5. How is compactness of a set in Minkowski space determined?

The compactness of a set in Minkowski space is determined by checking if the set satisfies the definition of compactness. This involves checking if the set is bounded and closed. Additionally, in some cases, the compactness of a set can also be determined using topological tools such as the Heine-Borel theorem.

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