## Some doubts concerning the mathematical bases of GR

Btw, there seems to be a mistake in the wikipedia entry on differentiable manifolds, they present topological manifolds as hausdorff, but if one reads Hawking and Ellis "The large scale structure of spacetime, they devote a few pages to discuss non-Hausdorff manifolds.

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 Quote by TrickyDicky You introduce something that could help clarify things a bit. Could you specify how exactly a pseudo-riemannian manifold (in thi case a Lorentzian one) is not a pseudometric space? Consider points on the light cone for instance, the distance betwen two different points can be zero, right? Wrt the comment about all differentiable manifolds being Hausdorff, that is true in a limited sense, that is, it is true locally, forgetting about the topology. But being a Hausdorff space is usually considered a global property of a space, and here we find the problem that a manifold with singularities is not differentiable globally. Other important limitations of pseudoriemannian manifolds are listed in the wikipedia entry " On the other hand, there are many theorems in Riemannian geometry which do not hold in the generalized case. For example, it is not true that every smooth manifold admits a pseudo-Riemannian metric of a given signature; there are certain topological obstructions. Furthermore, a submanifold does not always inherit the structure of a pseudo-Riemannian manifold; for example, the metric tensor become zero on any light-like curve".
I believe you are mixing up different fields of mathematics. That is also (I think) what George Jones is hinting at.

As I understand it:

- A metric space is just a set with a function on pairs of elements in it meeting certain properties. It is in the field of point-set topology, not differential geometry. Similarly for a pseudometric space. The key is that it's definition starts from set not topological space or manifold.

- Orthogonal to these concepts, there is a hiearchy of constructs with increase structure as follows:
topological space -> hausdorff space -> manifold -> riemannian or pseudo-reimannian manifold.

Topological space starts from imposing an arbitrary collection of subsets meeting certain properties such that we may call them 'the open sets'.

The quote you refer to above simply makes that point that any (smooth) manifold can be made riemannian, while some maniolds cannot be made pseudo-rieamannian.

 Quote by George Jones A metric space (X , d) is a set X together to together with a function d that maps pairs of elements of X to real numbers (the distances between the elements of the pairs). For a Riemannian manifold (M , g), g is function that maps pairs of tangent vectors to real numbers, i.e., g doesn't map pairs of elements of M to real numbers. Consequentl, a Riemannian manifold (M , g) is not a metric space in the sense in which you are using the term "metric space". This is an unfortunate and confusing clash of terminology: "metric" in "Riemannian metric" and "metric" in "metric space" mean different things.
George, I am aware of the distinction between a metric tensor g and a metric as referred to in "metric space"( I even mentioned the difference in a previous post).
Every time I wrote metric I referred to the distance function, not the metric tensor.
So I am not yet clear if you agree a Riemannian manifold is a metric space, and a Lorentzian manifold is not.

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 Quote by TrickyDicky Btw, there seems to be a mistake in the wikipedia entry on differentiable manifolds, they present topological manifolds as hausdorff, but if one reads Hawking and Ellis "The large scale structure of spacetime, they devote a few pages to discuss non-Hausdorff manifolds.
This is covered here:

http://en.wikipedia.org/wiki/Non-Hausdorff_manifold

 Mentor Okay, I am finding this thread to be very confusing. Why should a pseudo-Riemannian manifold be a metric space? As Matterwave and PAllen have pointed out, pseudo-Riemannian manifolds are topological spaces via the manifold topology, even if they don't have natural metrics. What do you want to do with a metric? Why does this introduce doubts about the mathematical basis of GR?

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 Quote by TrickyDicky George, I am aware of the distinction between a metric tensor g and a metric as referred to in "metric space"( I even mentioned the difference in a previous post). Every time I wrote metric I referred to the distance function, not the metric tensor. So I am not yet clear if you agree a Riemannian manifold is a metric space, and a Lorentzian manifold is not.
Seems to me you could have metric space that cannot even be a manifold (e.g. it is defined on a finite set). Similarly, a disconnected Riemannian manifold would not allow introduction of a global distance function based on integrated metric distance. Maybe every complete Riemannian manifold can be treated as a metric space.

 Quote by George Jones Okay, I am finding this thread to be very confusing. Why should a pseudo-Riemannian manifold be a metric space? ?
Sorry, I thought my questions were clear, my fault.
But I'm not saying a pseudo-Riemannian manifold is a metric space, my claim is that it is not.
I can see that a metric space is not a Riemannian manifold but I had the notion a Riemannian manifold had the property of being a metric space, am I wrong?

Similarly a pseudometric space is not a pseudo-Riemannian manifold but my belief is that pseudo-Riemannian manifolds have as defining property their pseudometricity(that is the key difference wrt Riemannian manifolds) , is this also not correct?

 Maybe I should add that I am only considering connected manifolds.

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 Quote by TrickyDicky Sorry, I thought my questions were clear, my fault. But I'm not saying a pseudo-Riemannian manifold is a metric space, my claim is that it is not. I can see that a metric space is not a Riemannian manifold but I had the notion a Riemannian manifold had the property of being a metric space, am I wrong?
See my earlier post for an example of metric space that is not a manifold and Riemannian manifold that cannot be treated as a metric space (at least not by building up from the metric tensor).
 Quote by TrickyDicky Similarly a pseudometric space is not a pseudo-Riemannian manifold but my belief is that pseudo-Riemannian manifolds have as defining property their pseudometricity(that is the key difference wrt Riemannian manifolds) , is this also not correct? Thanks for your help and sorry again about the misunderstanding?
This has the same confusion of categories as the prior case.

[edit: and there may be additional problems. The pseudo-metric tensor does not, in general, lead to any unique mechanism for constructing a global pseudometric function of points. For example, for some topologies, you could have both spacelike path and a timelike path between two points. Then what pseudo-metric global function value do you assign? The minimum among spacelike paths or the minimum among timelike paths? ]

 Pallen, do you agree that a Lorentzian manifold has a pseudometric space structure? Furthermore it also has a semimetric space structure due to their triangle inequality axiom being the reverse of the usual.

Mentor
As PAallen has said, given a connected (need not be complete, though) Riemannian manifold (M,g), g can be used to define a d (in a natural way) such that (M,d) is a metric space.

When this is done, the metric topology and the manifold topology are the same.

I have never worked with pseudometric spaces (that I remember), but, off the top of my head, I don't think a corresponding statement can be made about semi-Riemannian manifolds and pseudo metric spaces. I don't think the statement
 Given a connected semi-Riemannian manifold (M,g), g can be used to define a d (in a natural way) such that (M,d) is a pseudometric space.
can be made. I could very well be wrong.

 Quote by PAllen See my earlier post for an example of metric space that is not a manifold
Yes, that is why I wrote that a metric space is not the same as a manifold, I'm not sure if you are reading my answers.
 and Riemannian manifold that cannot be treated as a metric space (at least not by building up from the metric tensor).
I added the connectedness assumption for Riemannian manifolds.

 Quote by George Jones As PAallen has said, given a connected (need not be complete, though) Riemannian manifold (M,g), g can be used to define a d (in a natural way) such that (M,d) is a metric space. When this is done, the metric topology and the manifold topology are the same. I have never worked with pseudometric spaces (that I remember), but, off the top of my head, I don't think a corresponding statement can be made about semi-Riemannian manifolds and pseudo metric spaces. I don't think the statement can be made. I could very well be wrong.
Thanks George.

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 Quote by TrickyDicky Pallen, do you agree that a Lorentzian manifold has a pseudometric space structure? Furthermore it also has a semimetric space structure due to their triangle inequality axiom being the reverse of the usual.
No, because of the problem of no unique way to go from semi-riemannian metric tensor to global pseudometric function.

The concepts seem completely orthogonal in this case. A pseudometric space need not even be a topological space (let alone a manifold, or Hausdorff). A pseudo-Riemannian manifold is necessarily Hausdorff (by normal definitions of manifold), but there is no natural way to define a global pseudometric function (at minimum, several definitions would need to be added, as well - I am pretty sure - additional topological restrictions (beyond connectedness)).

To clarify this, I believe I can construct connected, pseudo-riemannian manifods such that there exist points connected both by a spacelike geodesic and a timelike geodesic. How then, do you define the global pseudometric function of points?

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 Quote by TrickyDicky Yes, that is why I wrote that a metric space is not the same as a manifold, I'm not sure if you are reading my answers. I added the connectedness assumption for Riemannian manifolds.
It's the classic question of timing - composing posts while you are composing, but mine post later.

 Blog Entries: 1 Recognitions: Gold Member Science Advisor I think there is an even more basic issue going pseudo-Riemannian to pseudometric. Pseudometric simply allows a global distance function that is zero. It has no concept of spacelike versus timelike (e.g. positive or negative interval squared). Thus, I think (despite the name similarity), there is no meaningful connection between pseudometric spaces and pseudoriemannian manifolds.

 Quote by PAllen The concepts seem completely orthogonal in this case. A pseudometric space need not even be a topological space (let alone a manifold, or Hausdorff).
I thought we had clarified this misunderstanding, I'm not saying anything about a pseudometric space being a manifold. It is the other way around, why do you keep bringing it up?

 A pseudo-Riemannian manifold is necessarily Hausdorff (by normal definitions of manifold),
It is explained in the wikipedia link you provided that the "normal" definition ignores the general topology.
 To clarify this, I believe I can construct connected, pseudo-riemannian manifods such that there exist points connected both by a spacelike geodesic and a timelike geodesic. How then, do you define the global pseudometric function of points?
This doesn't seem to be connected to what I have been talking about.
First you would have to address in what sense two different points in a null light cone in a Lorentzian manifold can have zero distance between them (the definition of pseudometric space according to wikipedia) and not have pseudometric space structure.