Exploring Topology: How Can Local Theory Affect Global Structure?

[JustinLevy]

Many interesting proofs in GR (regarding black holes and singularities, etc.) involve topological methods.

However, I don't understand how a theory embodied in Einstein's equations, which appear to me to be local rules of evolution, can ever change the topology since a patch of spacetime around any event is locally equivalent to minkowski / flat-spacetime.

Said another way, how can a local theory _ever_ change the global topology?
I'm really having trouble picturing this.

 

[George Jones]

Many interesting proofs in GR (regarding black holes and singularities, etc.) involve topological methods.

However, I don't understand how a theory embodied in Einstein's equations, which appear to me to be local rules of evolution, can ever change the topology since a patch of spacetime around any event is locally equivalent to minkowski / flat-spacetime.

Said another way, how can a local theory _ever_ change the global topology?
I'm really having trouble picturing this.

The use of topological methods does not necessarily mean that topology change is involved. For example, see the proof that all compact spacetimes contain closed timelike curves,

https://www.physicsforums.com/showthread.php?p=1254758#post1254758.

A local theory might be consistent with more than one global topology.

Also, what physicists and mathematicians mean by "a patch of spacetime around any event is locally equivalent to minkowski / flat-spacetime" are two different things.

 

[tiny-tim]

Hi Justin!

… a patch of spacetime around any event is locally equivalent to minkowski / flat-spacetime.

In the limit …

in the limit, you can treat a local patch of space as if it was the tangent space …

so you can map a small patch of the Earth's surface onto flat paper, and the smaller the patch the more accurate it is …

but you can't argue that since the Earth is locally equivalent to flat space, that means it is flat!

 

[JustinLevy]

In the limit …

in the limit, you can treat a local patch of space as if it was the tangent space …

so you can map a small patch of the Earth's surface onto flat paper, and the smaller the patch the more accurate it is …

but you can't argue that since the Earth is locally equivalent to flat space, that means it is flat!

I'm not claiming it is flat. But I DO think it is topologically locally equivalent to flat spacetime (maybe that is wrong?). Sure, a donut and a coffee mug have different curvature, but locally a patch on either is topologically the same (and in this case they also have the same global topology).

Basically, it looks to me like a local theory like Einstein's field equations can never change the global topology. But clearly this is wrong since Penrose's singularity thoerem (the first big application of topology to GR that I know of) essentially discusses the change in global topology since it shows that a star, which if it didn't collapse would leave spacetime where worldlines have no boundary, when allowed to collapse below a certain radius the global topology would change such that some worldlines now have a boundary.

I'm really having trouble picturing how something that essentially describes local stretching / curving can lead to global topological changes. Can you suggest a better way to visualize this?

 

[George Jones]

But clearly this is wrong since Penrose's singularity thoerem (the first big application of topology to GR that I know of) essentially discusses the change in global topology since it shows that a star, which if it didn't collapse would leave spacetime where worldlines have no boundary, when allowed to collapse below a certain radius the global topology would change such that some worldlines now have a boundary.

No, the topology of spacetime doesn't change when a star collapses to form a black hole. Maybe you're thinking of space evolving, instead of thinking of spacetime as a whole.

 

[JustinLevy]

No, the topology of spacetime doesn't change when a star collapses to form a black hole. Maybe you're thinking of space evolving, instead of thinking of spacetime as a whole.

Would you agree that the topology of spacetime is different if the star is allowed to collapse as opposed to if the star never collapses? It seems we could choose to change the global topology.

Even in the most naive steps towards a quantum gravity, we would at least expect it to tell us the probabilities of spacetime evolving into a particular state. Despite GR being a local theory, it seems to allow the probability of states having different topology to be non-zero (as the black hole example seems to show).

If I'm still misunderstanding, hopefully I've said enough now that someone can spot what is throwing me off.

 

[javierR]

The topology of spacetime (i.e., the manifold) is fixed. Einstein's equations determine a metric *on* that manifold. This ascribes geometrical meaning to the space. In particular, a black hole is a geometric solution (that happens to have singularities). You can ask about the topology of a *solution* such as solitons in general relativity (or other field theories for that matter). But topological properties of a solution to field equations are not topological properties of the bare spacetime manifold (i.e., without the metric). In quantum gravity, can look to quantize the *geometry* of spacetime on some background topological space.

In contrast, string theory admits actual topological transformations of the space as was first studied in the case of an internal Calabi-Yau manifold.

 

[JustinLevy]

I would appreciate other responses / viewpoints regarding my previous post, as I believe I am still not understanding. Tiny-tim, GeorgeJones, CH?

The topology of spacetime (i.e., the manifold) is fixed.

So your answer to my question "Would you agree that the topology of spacetime is different if the star is allowed to collapse as opposed to if the star never collapses?" seems to be no. How can you justify that? That seems to me to be like saying a plane (R^2) and a plane with the origin removed (R^2 - {0,0}) have the same topology.

Let's simplify this. Say I have a device that if I flip a switch, a spacelike wormhole bridge will appear. Do you agree that the topology of spacetime is different if I flip the switch vs if I never flip the switch?

You seem to be saying that you can prove such a device violates GR because the topology of spacetime must be fixed. Is that really true?

 

[George Jones]

So your answer to my question "Would you agree that the topology of spacetime is different if the star is allowed to collapse as opposed to if the star never collapses?" seems to be no.

No, this isn't what javierR meant.

The spacetimes for a non-collapsed and collapsed stars come from different solutions to Einstein's equation, and these different solutions have different topologies.

Let's simplify this. Say I have a device that if I flip a switch, a spacelike wormhole bridge will appear. Do you agree that the topology of spacetime is different if I flip the switch vs if I never flip the switch?

Yes, but again, these are *different* solutions to Einstein's equation.

Are you really asking "Does free will play a role in the solution to Einstein's equation realized in nature?"

 

[JustinLevy]

No, this isn't what javierR meant.

The spacetimes for a non-collapsed and collapsed stars come from different solutions to Einstein's equation, and these different solutions have different topologies.

So you agree they have different topologies. But javierR is claiming that the topology of the manifold is given ahead of time, and on top of that, the topology is fixed. Did we have to know if the star was going to collapse or not in order to work out Einstein's equations in the first place? No. There is some set of initial conditions that can be supplied, and we can use Einstein's equations to work out the evolution.

It seems to me that the topology is NOT fixed.

Maybe the problem here is that I am not understanding what you two mean by 'fixed'. Is it being used in some specific mathematical sense where it has a deeper meaning?

Yes, but again, these are *different* solutions to Einstein's equation.

Are you really asking "Does free will play a role in the solution to Einstein's equation realized in nature?"

No, that is not what I am asking.
Let us leave free will out of it if we can.

I'm clearly having difficulty communicating this question. So let me try approaching from a different angle. Given a spacelike slice with initial conditions, since GR is a local theory, how can this then be used to solve for a global topology (ie. worm hole forms or it doesn't)?