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

[kevinferreira]

For instance, the topology of a smooth manifold (that by definition has no discontinuities) is altered by introducing singularities based precisely on the peculuarities of the pseudoriemannian metric tensor.

As micromass pointed out, the singularities are metric tensor's singularities, not topological! The topology is only used in GR in order to be able to define open sets and local coordinates on them. And this you may always do, I think, the singularities that arise in GR do not affect this in no way.

 

[TrickyDicky]

I don't get this. A manifold has a topology. Only then do we introduce a metric tensor. So the metric tensor is an extra structure.
How can an extra structure possibly change the topology of a manifold??

That IS my point. I'm saying that it shouldn't change it.

 

[TrickyDicky]

As micromass pointed out, the singularities are metric tensor's singularities, not topological! The topology is only used in GR in order to be able to define open sets and local coordinates on them. And this you may always do, I think, the singularities that arise in GR do not affect this in no way.

Correct me if I'm wrong but singularities may be viewed as. discontinuities, at least in the wikipdia page about singularity theory they are defined as failures of the manifold structure, in which case they would affect the topology. But even in the case one decides this is not the case, it is clear that at the very least it affects the differential structure, and we wouldn't be dealing with smooth manifold in its presence.

The things you mention about GR wouldn't be affected, but all the physical assertions in GR either for cosmological or asymptotically flat cases that deal with the manifold globally would.

 

[zonde]

I'm not sure if you are understanding what it means for a topological space to be Hausdorff. Sure two events connected by a null geodesic represent a light pulse being able to get from one to the other but what does that have to do with Hausdorff? Hausdorff simply states there exist a pair of neighborhoods, for the two (distinct) events, that are disjoint but you seem to be thinking that this implies we could not anymore connect the two events with the aforementioned null geodesic. If the null geodesic connects the two events then that is that; the Hausdorff property won't break anything.

The way you say it null geodesic is just line with some idea of length along the line. Well, the idea about spacetime is that the length along null geodesic is always zero.

Hmm, maybe I would convey my view on this if I would speak about spacetime as a space whose elements are null geodesics not events. If you think about it it makes sense from physical perspective. Events have no relevance if there is no null or timelike worldline extending from it (and toward it).

 

[atyy]

Do you really think the spacetime invariant interval between events is physically irrelevant? That's odd.

Let's consider flat spacetime for simplicity. In flat spacetime the invariant interval determined by the pseudo-Riemannian metric tensor is physically relevant. It's just that this isn't the metric of a metric space. I guess there are 3 notions: pseudo-Riemannian metric, Riemannian metric, metric of a metric space. The first two are related (both are defined by their action on vectors in the tangent space at each point), the last two are related (the Riemannian metric tensor can be used to define the metric of a metric space), but the first and the last are not (the pseudo-Riemannian metric tensor cannot be used to define the metric of a metric space), and it's the first that is physically relevant in spacetime.

 

[TrickyDicky]

Let's consider flat spacetime for simplicity. In flat spacetime the invariant interval determined by the pseudo-Riemannian metric tensor is physically relevant.

Why is the spacetime interval relevant in SR and not in GR?

 

[atyy]

Why is the spacetime interval relevant in SR and not in GR?

The pseudo-Riemannian metric acts on tangent vectors at a point. To get a "distance" (which is not the distance of a metric space:) between events at different spacetime points, we have to specify a path to integrate over. In flat spacetime, there is a unique extremal path between every pair of points and we use that path to define the spacetime interval from the pseudo-Riemannian metric. I think this idea can be generalized to curved spacetime in some circumstances, but the generalization wasn't immediately obvious to me (how to choose the path?), so I restricted my discussion to flat spacetime in the earlier post to focus on the 3 different quantities (pseudo-Riemannian metric tensor, Riemannian metric tensor, metric of metric space).

 

[strangerep]

I don't get this. A manifold has a topology. [...]

This is the crucial point. A manifold is a mathematical abstraction which we use to construct models of physics. So of course, we physicists tend to assume too easily that this mathematical abstraction is homeomorphic, isomorphic, etc, to something out there in the real world.

Also, many physicists don't understand that a given set can be equipped with various inequivalent topologies -- which is strange since the distinction between strong and weak topology on a Hilbert space is something of which any self-respecting physicist ought to be at least vaguely aware. Both can be useful -- in different contexts.

What's most important for physics are the observables -- fields on the manifold. One may think of these as mappings from an abstract state space (the points on the manifold) to some more convenient linear space that's closely relatable to measurement data. Thus, they may be regarded as generalized functionals, and hence define a weak topology on the manifold state space. IMHO, such weak topologies are more important for physics because they come from physically meaningful observables.

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For the benefit [or perhaps confusion?] of other readers: an ordinary "weak topology" is constructed essentially by demanding that a certain function be continuous on the underlying set. One typically does not actually construct the open sets explicitly in this weak topology, since what is important is that the function is deemed continuous -- meaning that every open set in the range of the function comes from an open set in the domain, the latter being thereby defined by the function rather than by some independent lower level construction.

 

[George Jones]

Correct me if I'm wrong but singularities may be viewed as. discontinuities, at least in the wikipdia page about singularity theory they are defined as failures of the manifold structure, in which case they would affect the topology. But even in the case one decides this is not the case, it is clear that at the very least it affects the differential structure, and we wouldn't be dealing with smooth manifold in its presence.

The things you mention about GR wouldn't be affected, but all the physical assertions in GR either for cosmological or asymptotically flat cases that deal with the manifold globally would.

I am going to give examples to try and illustrate what micromass and kevinferreira have written.

I don't get this. A manifold has a topology. Only then do we introduce a metric tensor. So the metric tensor is an extra structure.

How can an extra structure possibly change the topology of a manifold??

As micromass pointed out, the singularities are metric tensor's singularities, not topological! The topology is only used in GR in order to be able to define open sets and local coordinates on them. And this you may always do, I think, the singularities that arise in GR do not affect this in no way.

As a differentiable manifold, what is the spacetime of an open Friedmann-Lemaitre-Robertson-Walker universe? This differentiable manifold is \mathbb{R}^4. There is no problem with topological or manifold structure, yet this spacetime is singular.

As far as I know, there is no reasonably generic, accepted definition of "spacetime singularity". There is, however, a reasonably generic definition of "singular spacetime". A rough, sufficient condition: spacetime is singular if there is a timelike curve having bounded acceleration that ends in the past or the future after a finite amount of proper time. For example, and speaking very loosely, a spacetime is singular if a person can get in a rocket, and, after using a finite amount of fuel wristwatch time, can fall "off of spacetime" at a "singularity".

The example of an open FLRW universe shows that "singular" is due to the extra structure of a pseudo-Riemannian metric tensor field.

My guess is that the distance is just not a very useful one as it won't agree with the pseudo-Riemannian metric. ... In the same way, we can endow a distance on a spacetime. But nothing tells us that this distance actually has a physical significance or that it agrees with some metric tensor.

As \mathbb{R}^4, clearly, we can introduce a positive-definite distance function on open FLRW universes, but, as micromass notes, this wouln't have physical significance. The differentiable manifold together with the added structure of a particular pseudo-Riemannian metric nicely models some physics. Particular pseudo-Riemannian metric, because different pseudo-Riemannian metrics can be added to the same differentiable manifold, with very different results! For example, adding the Minkowski metric to \mathbb{R}^4 results in Minkowski spacetime. The same underlying differentiable manifold, yet one spacetime is singular and the other spacetime is non-singular!

I'm not sure what you mean here, but I'd say the metric(distance function) is never physically irrelevant in GR. One of the pillars of the theory is the invariance of length across arbitrarily long distances, think of cosmological redshifts. If we didn't care about metrics (distances) in GR there would be no need for a curvature concept or unique connections.

As the example of open FLRW universes demonstrates, the added pseudo-Riemannian metric is the physically significant structure that is added.

Why is the spacetime interval relevant in SR and not in GR?

The spacetime interval is very relevant physically. For example, consider an observer's worldline that joins events p and q. The worldline doesn't have to be a geodesic, as the observer could be in a rocket. How much observer wristwatch time elapses between p and q? Appropriately integrate the spacetime metric along the worldline to find out.

In cosmology, 4D spacetime is cut into 3D spatial slices that change with time. On the 4D spacetime, there is a pseudo-Riemannian metric tensor, and no physically relevant metric space metric. On each 3D spatial slice there is a Riemannian metric tensor, which can be used to define a metric space metric.

As another example, again consider FLRW universes. What is the present proper spatial distance between galaxies A and B? Appropriately integrate the spacetime metric along a path in the "now" spatial hypersurface to find out.

I think that this is a beautiful interplay between physics and mathematics.

 

[TrickyDicky]

As a differentiable manifold, what is the spacetime of an open Friedmann-Lemaitre-Robertson-Walker universe? This differentiable manifold is \mathbb{R}^4. There is no problem with topological or manifold structure, yet this spacetime is singular.

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. Singularities seem to be incompatible also with a global(not just local) differentiable structure.

As far as I know, there is no reasonably generic, accepted definition of "spacetime singularity". There is, however, a reasonably generic definition of "singular spacetime". A rough, sufficient condition: spacetime is singular if there is a timelike curve having bounded acceleration that ends in the past or the future after a finite amount of proper time. For example, and speaking very loosely, a spacetime is singular if a person can get in a rocket, and, after using a finite amount of fuel wristwatch time, can fall "off of spacetime" at a "singularity".

I would find really upsetting if there is no accepted definition of spacetime singularity when such an important part of GR theory deals with singularities (BHs, BBT) and so much physics literature is devoted to them.
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.

The example of an open FLRW universe shows that "singular" is due to the extra structure of a pseudo-Riemannian metric tensor field.

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

The spacetime interval is very relevant physically. For example, consider an observer's worldline that joins events p and q. The worldline doesn't have to be a geodesic, as the observer could be in a rocket. How much observer wristwatch time elapses between p and q? Appropriately integrate the spacetime metric along the worldline to find out.

As another example, again consider FLRW universes. What is the present proper spatial distance between galaxies A and B? Appropriately integrate the spacetime metric along a path in the "now" spatial hypersurface to find out.

I think that this is a beautiful interplay between physics and mathematics.

Agreed, that's why I found atyy's statement odd.

 

[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).