How Does the Topology of Spacetimes Influence the Structure of Curved Manifolds?

[micromass]

Ok, no problem with the coarsest topology in the topological manifold structure, but there is certainly problem with the finer topology required for the differentiable manifold (that must be Hausdorff and second-countable) that I'd like to understand how it can be compatible with singular points in a manifold.

I'm sorry, but this statement is making no sense to me.

First, there is no coarser and finer topology. The topology of a topological manifold is equal to the topology of a differentiable manifold. That is because a differentiable manifold is a topological manifold with some extra structure.

By making a topological manifold into a differentiable manifold, the topology is not changed in any way. We don't add or remove open sets.

Second, any topological manifold must already be Hausdorff and second countable by definition. The definition varies of course from author to author, but the standard condition seems to be Hausdorff and second countable.

Third, Singular points on a manifold are not a concept depending on the topology.

 

[dextercioby]

Micromass, is it true that second-countable, Hausdorff and paracompact topological manifolds are metrizable ? If so, do you know a reference for a proof ? I've been looking for this fact for about an hour or so.

Thanks!

 

[micromass]

Micromass, is it true that second-countable, Hausdorff and paracompact topological manifolds are metrizable ? If so, do you know a reference for a proof ? I've been looking for this fact for about an hour or so.

Thanks!

In fact, if M is a Hausdorff locally Euclidean space, then second countable actually implies paracompact. So there is no need to add the paracompact condition. The proof of this fact can be found in Munkres: Theorem 41.5, page 257 (note that every topological manifold is in fact regular and Lindelof).

Also, a note on terminology. A topological manifold is defined as a locally euclidean, Hausdorff and second countable space. So the term "second countable topological manifold" has some unnecessary words (as does "paracompact topological manifold")

A more general result is the Smirnov metrization theorem. This states that any paracompact, Hausdorff and locally metrizable space is actually metrizable. A proof can be found in Munkres: Theorem 42.1, page 261 This proves in particular that every topological manifold is metrizable.

 

[George Jones]

Having said this your definition of singular spacetime might clear up something for me, is it defining something like a metric space that is not complete, that has missing points? Can this missing points be considered singularities? In that case things would start to make sense to me.

Because the idea is so attractive, over the decades, there have been a number of attempts to define spacetime singularities as missing points or adjoined boundaries, but many (all?) have had various problems.

The work of Susan Scott and collaborators has possibly shown the most promise, but I know very little about this stuff. An interesting recent paper:

http://iopscience.iop.org/0264-9381/28/16/165003/

Unfortunately, at this link, the paper is behind a paywall, and I can't find it on the arXiv. I have access to it, but I haven't a chance to look at it yet.

 

[TrickyDicky]

micromass, this is just a technical note that I really don't think adds much to the main discussion and I have no interest whatsoever in arguing about it but just for the record and since you have insisted on it several times...
You have said repeatedly that the Hausdorff condition is already required for topological manifolds, well I'm not the only one here that has pointed out that the reference text for this stuff, the 1973 book by Ellis and Hawking cites a few examples of topological manifolds that are not Hausdorff.
Besides, I've consulted several standard texts on differential geometry and they also name the Hausdorff condition only when defining smooth/differentiable manifolds.

 

[TrickyDicky]

Third, Singular points on a manifold are not a concept depending on the topology.

Well, since there seems to be no commonly accepted definition of the singularity concept (only of singular spacetime), it is at the very least hard to say. It might not depend on the topology but it might be incompatible with it, just the same way a manifold won't admit certain metrics incompatible with its manifold topology.

 

[TrickyDicky]

Because the idea is so attractive, over the decades, there have been a number of attempts to define spacetime singularities as missing points or adjoined boundaries, but many (all?) have had various problems.

The work of Susan Scott and collaborators has possibly shown the most promise, but I know very little about this stuff. An interesting recent paper:

http://iopscience.iop.org/0264-9381/28/16/165003/

Unfortunately, at this link, the paper is behind a paywall, and I can't find it on the arXiv. I have access to it, but I haven't a chance to look at it yet.

Thanks, I'm trying to clarify things with a book called "The Analysis of Space-Time Singularities" by C. J. S. Clarke. So far the impression I get is that this is a bit of a mined field.

 

[WannabeNewton]

micromass, this is just a technical note that I really don't think adds much to the main discussion and I have no interest whatsoever in arguing about it but just for the record and since you have insisted on it several times...
You have said repeatedly that the Hausdorff condition is already required for topological manifolds...

I didn't look back to see if he actually did (I doubt he did though) but yes it depends on the author. Some authors take Hausdorff as one of the conditions for a topological space to be a manifold and others don't. In the context of space - times if you want physically relevant ones you would probably include the condition.

 

[micromass]

micromass, this is just a technical note that I really don't think adds much to the main discussion and I have no interest whatsoever in arguing about it but just for the record and since you have insisted on it several times...
You have said repeatedly that the Hausdorff condition is already required for topological manifolds, well I'm not the only one here that has pointed out that the reference text for this stuff, the 1973 book by Ellis and Hawking cites a few examples of topological manifolds that are not Hausdorff.
Besides, I've consulted several standard texts on differential geometry and they also name the Hausdorff condition only when defining smooth/differentiable manifolds.

It does depend on the author. I'm sure there are people who do things differently (although I would like to know which differential geometry texts you are talking about). But I think the standard definition is to require topological manifold to be Hausdorff. My posts try to reflect the standard position as much as possible. But yes, there are probably some authors who do things differently.

 

[kevinferreira]

My confusion comes from not seeing how an structure that is supposed to act only locally can have global effects.

This is my original cause of confusion that began the discussion, still in the other thread. This happens for the metric tensor, the energy-momentum tensor, the Riemann tensor, etc.

 

[atyy]

This is my original cause of confusion that began the discussion, still in the other thread. This happens for the metric tensor, the energy-momentum tensor, the Riemann tensor, etc.

I think this the "global" in the other thread is different from the "global" in this thread. There you were wondering about the locality in the EP, and the non-locality in things like the Riemann tensor for which the EP fails. Both of those are "local" relative to the issues in this thread. For example, one can allow non-Hausdorff manifolds on which to place solutions of the Einstein equations. Or decide before you solve the equations that the manifold has the topology of a torus. So there are now at least 3 "levels" of "locality" - local for the EP (roughly less than first derivatives), non-local for the EP (roughly spacetime curvature or second derivatives or higher), and manifold topology (global). You cannot choose the manifold completely arbitrarily - for example - Ben Niehoff some time ago commented on PF that pseudo-Riemannian metrics cannot be placed on even-dimensional spheres (I think, I can't quite remember what the result was).