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

[micromass]

But result for both functions is distance, right?

The pseudo-Riemannian metric induces a distance on the tangent space T_p(M), not on M. I don't think it induces a well-defined distance on the manifold. And even if it does, nobody says that the topology of this distance should with the topology of the manifold.

 

[zonde]

Check http://www.damtp.cam.ac.uk/research/gr/members/gibbons/dgnotes3.pdf
Page 6:

So all spaces are taken to be Hausdorff.

The reason why they specify Hausdorff in this case is because they don't define manifold to be Hausdorff (which is not standard practice). But in any case, this shows that all spacetimes are taken to be Hausdorff.

Fine there is someone else who thinks like you.
Now can you provide arguments? In that link there is only definition (belief) and no arguments.

Why do you believe that all spacetimes are Hausdorff?

 

[micromass]

Fine there is someone else who thinks like you.
Now can you provide arguments? In that link there is only definition (belief) and no arguments.

Why do you believe that all spacetimes are Hausdorff?

Because it is the definition...
How do you define a spacetime??

Since you like wikipedia, here is another reference: http://en.wikipedia.org/wiki/Spacetime_topology

 

[WannabeNewton]

Why do you believe that all spacetimes are Hausdorff?

It is a definition. For example see pg 12 of Wald.

 

[zonde]

Because it is the definition...
How do you define a spacetime??

Since you like wikipedia, here is another reference: http://en.wikipedia.org/wiki/Spacetime_topology

Let me try such question:
How do you define neighborhood if you have distance function that can give zero for two distinct points?

 

[micromass]

Let me try such question:
How do you define neighborhood if you have distance function that can give zero for two distinct points?

Let (X,d) be a pseudometric space. A set V is a neighborhood of x\in X if there exists an \varepsilon>0 such that B(x,\varepsilon)\subseteq V.

 

[micromass]

Another good reference is "Fundamental of differential geometry" by Serge Lang. He covers pseudo-Riemannian metrics on page 175. It's a fun book to read, so I recommend it.

 

[zonde]

Let (X,d) be a pseudometric space. A set V is a neighborhood of x\in X if there exists an \varepsilon>0 such that B(x,\varepsilon)\subseteq V.

What is \varepsilon - a point or a set or an open set? And B()?

According to wikipedia http://en.wikipedia.org/wiki/Neighbourhood_(mathematics) neighborhood should contain an open set containing the point. Given spacetime properties neighborhood of any event in spcetime should include it's lightcones. But for any two distinct points there will be some place where their lightcones (future or past or future with past) will intersect. So they can't have disjoint neighbourhoods which is required to say they belong to Hausdorff space.

 

[WannabeNewton]

What is \varepsilon - a point or a set or an open set? And B()?

According to wikipedia http://en.wikipedia.org/wiki/Neighbourhood_(mathematics) neighborhood should contain an open set containing the point. Given spacetime properties neighborhood of any event in spcetime should include it's lightcones. But for any two distinct points there will be some place where their lightcones (future or past or future with past) will intersect. So they can't have disjoint neighbourhoods which is required to say they belong to Hausdorff space.

A good book on topology would probably clear up much of the confusion here. Hausdorff property states there exists a pair of neighborhoods for two distinct points that are themselves disjoint; a point on a manifold will have multiple neighborhoods. As for your comment on the causal structure of space - time, please take a look at chapter 8 of Wald which should clear up confusion or something else if anyone else has another reference. In particular note that a light cone emanating from a point p on a space - time M is a subset of Tp(M) not M itself.

 

[atyy]

All the usual spacetimes are of course Hausdorff. But just for interest, Hawking and Ellis mention one example of a non-Hausdorff spacetime, and mention a paper by Hajicek.

 

[TrickyDicky]

A pseudo-Riemannian metric is a function

g:T_pM\times T_pM\rightarrow \mathbb{R}

for each p.

A pseudometric is a function

d:M\times M\rightarrow \mathbb{R}

So how can they be the same thing??

This is a key distinction IMO. The tangent space at a point and the manifold itself are two very different objects, and this is manifested even more clearly when the manifold is curved.
One shouldn't be able to draw conclusions about the global spacetime features from the purely local effect of the pseudoriemannian metric at a point, more so when the distance metric function that acts on the manifold doesn't coincide with the one that would be derived from the pseudoriemannian metric tensor, due to the smooth structure of the manifold.
When I mention the global structure of the manifold I refer to things like its maximal extended form, its singularities or its Killing vector fields nature(timelike, spacelike,lightlike).

 

[zonde]

If we say that spacetime is Hausdorff then we can't include complete lightcones in the neighborhood of an event. But then we should relay on some concept of nearness that is positive-definite and rather unrelated to spacetime distances.

It seems like a kind of double standard.

 

[micromass]

If we say that spacetime is Hausdorff then we can't include complete lightcones in the neighborhood of an event. But then we should relay on some concept of nearness that is positive-definite and rather unrelated to spacetime distances.

It seems like a kind of double standard.

Please read this again:

In particular note that a light cone emanating from a point p on a space - time M is a subset of Tp(M) not M itself.

 

[TrickyDicky]

Please read this again:

In particular note that a light cone emanating from a point p on a space - time M is a subset of Tp(M) not M itself.

Exactly, but a complete light cone structure is usually attributed in GR not only to the point p and its neighbourhood, but to the whole manifold. This is the double standard IMO.

 

[WannabeNewton]

Exactly, but a complete light cone structure is usually attributed in GR not only to the point p and its neighbourhood (TpM), but to the manifold. This is the double standard IMO.

Tp(M) is not a neighborhood it is the tangent space to M at p.

 

[zonde]

Please read this again:

In particular note that a light cone emanating from a point p on a space - time M is a subset of Tp(M) not M itself.

Done.

So what was the point? There is no analog of light cone on spacetime itself? And all spacetime distances are positive-definite? Or what?

 

[WannabeNewton]

Done.

So what was the point? There is no analog of light cone on spacetime itself? And all spacetime distances are positive-definite? Or what?

That is the subset of M generated by null geodesics emanating from p but you are talking about light cones as they relate to causal structure. Also, I'm not sure how you are concluding that the metric tensor must suddenly be positive - definite.

 

[micromass]

Done.

So what was the point? There is no analog of light cone on spacetime itself? And all spacetime distances are positive-definite? Or what?

I'm saying that the topology of T_pM determined by the pseudo-Riemannian metric is of course non-Hausdorff. But this is a topology on T_pM and not on M. The topology on M is Hausdorff and has nothing to do with the metric tensor.

 

[TrickyDicky]

Tp(M) is not a neighborhood it is the tangent space to M at p.

Yes, strictly you are right, but note that the whole justification of the concept of manifold depends upon the possibility of making the neighbourhood of a point and its tangent space "equivalent" in the sense of homeomorphic to R^n.

 

[zonde]

The topology on M is Hausdorff and has nothing to do with the metric tensor.

So we do relay on some positive-definite concept of nearness when we speak about topology of M, right?

 

[micromass]

So we do relay on some positive-definite concept of nearness when we speak about topology of M, right?

A topology has nothing to do with "positive-definiteness". Positive-definite is a property about inner products.

 

[TrickyDicky]

That is the subset of M generated by null geodesics emanating from p but you are talking about light cones as they relate to causal structure.

That is the point, it is hard to find (at least for me) mathematical justification for deriving a causal structure for the whole manifold only from the local action of the pseudoriemannian metric at the tangent space, when the distance function that prevails in smooth manifolds is not even the same as the one that integrates from the pseudoriemannian metric tensor.

 

[George Jones]

If we say that spacetime is Hausdorff then we can't include complete lightcones in the neighborhood of an event.

As has been indicated, some care is needed with respect to the meaning of "lightcone". Depending on the context and reference, "lighcone at p" can either mean a subset of T_p \left(M\right), or it can mean a subset of M. I think that you mean the latter. In this case, M is a neighbourhood of p that contains its light cone.

 

[George Jones]

Fine there is someone else who thinks like you.
Now can you provide arguments? In that link there is only definition (belief) and no arguments.

Why do you believe that all spacetimes are Hausdorff?

All the usual spacetimes are of course Hausdorff. But just for interest, Hawking and Ellis mention one example of a non-Hausdorff spacetime, and mention a paper by Hajicek.

We want to model physics. For most situations, spacetime Hausdorffness seems to be a reasonable, physical separation axiom. Two distinct physical events always admit distinct neighbouhoods.

Having said this, we have strayed far off-topic with respect to the original post. Physics Forums rules advises that, instead of posts that are off-topic, new threads should be started.

 

[dextercioby]

IMHO, off-topic or not, this is by far the best thread in the Relativity section for quite some time. :)

 

[micromass]

I have moved the off-topic posts to a new thread so we can keep discussing this.

 

[George Jones]

OK, I guess they must just use the length of the shortest path, regardless of whether or not there are multiple geodesics.

Almost. Consider \mathbb{R}^2 with its standard positive-definite norm. Now obtain a new Riemannian manifold M by removing the origin. In this new manifold M, what is the distance between the points (-1 , 0) and (1 , 0)? There is no geodesic in M that joins these points. There isn't even a shortest path in M that joins these points points, i.e., if someone gives me a path in M between (-1 , 0) and (1 , 0), I can always find a shorter path in M.

This leads to a slightly subtle definition of distance in a Riemannian manifold. The distance between points p and q is the greatest bound on the lengths of all "nice" paths between p and q.

In my example, 2 is greatest lower bound of the lengths of paths between, even though there is no path of length 2, and 2 is the distance between (-1 , 0) and (1 , 0).

 

[Dale]

Indeed even though the two topological spaces mentioned are homeomorphic, they need not have same distance functions. Metrizable implies there exists some metric for the set but it doesn't state there is a single, unique metric. By the way, I think there is some confusion arising here in the terminology.

OK, from my understanding a metric space must have a unique distance between any two points in the space. A metrizable space seems to be one that can be given a metric, not necessarily one that has a metric. So a differentiable manifold is metrizable, but by itself that doesn't make it a metric space. You know that you can equip it with a metric, and once you do so then it is a metric space, not before. Does that agree with your understanding?

Almost. Consider \mathbb{R}^2 with its standard positive-definite norm. Now obtain a new Riemannian manifold M by removing the origin. In this new manifold M, what is the distance between the points (-1 , 0) and (1 , 0)? There is no geodesic in M that joins these points. There isn't even a shortest path in M that joins these points points, i.e., if someone gives me a path in M between (-1 , 0) and (1 , 0), I can always find a shorter path in M.

This leads to a slightly subtle definition of distance in a Riemannian manifold. The distance between points p and q is the greatest bound on the lengths of all "nice" paths between p and q.

In my example, 2 is greatest lower bound of the lengths of paths between, even though there is no path of length 2, and 2 is the distance between (-1 , 0) and (1 , 0).

Thanks, that helps my understanding. So what happens for spacelike paths? Also, you should be able to connect any pair of events with a null path, how are those avoided?

 

[micromass]

Almost. Consider \mathbb{R}^2 with its standard positive-definite norm. Now obtain a new Riemannian manifold M by removing the origin. In this new manifold M, what is the distance between the points (-1 , 0) and (1 , 0)? There is no geodesic in M that joins these points. There isn't even a shortest path in M that joins these points points, i.e., if someone gives me a path in M between (-1 , 0) and (1 , 0), I can always find a shorter path in M.

This leads to a slightly subtle definition of distance in a Riemannian manifold. The distance between points p and q is the greatest bound on the lengths of all "nice" paths between p and q.

In my example, 2 is greatest lower bound of the lengths of paths between, even though there is no path of length 2, and 2 is the distance between (-1 , 0) and (1 , 0).

I wonder what exactly the problem is in this example. Intuitively, the problem is of course the hole at the origin. But is there a condition that we can place on our manifold such that this situation doesn't arise? I guess I'm asking for a condition where there always exists a shortest path.

 

[WannabeNewton]

Does that agree with your understanding?

Yessir.