Some doubts concerning the mathematical bases of GR

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".

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.

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.

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.

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?
Thanks for your help and sorry again about the misunderstanding?

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.

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).

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.

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.

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.

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),

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?