General relativity and curvilinear coordinates

[PeterDonis]

This induces a second topology on the spacetime which is not the same as the a priori topology.

Is what would be induced by the Minkowski inner product even a valid topology? It gives zero distance between distinct points.

 

[micromass]

Yes, it's a valid topology. But it will be non-Hausdorff and very ugly.

 

[George Jones]

Well, there is the Hawking-King-McCarthy topology,

http://authors.library.caltech.edu/11027/

Fifteen years ago, I wrote "If anybody knows of applications of the Hawking-King-McCarthy topology to anything, I would be most interested."

I am still interested.

 

["Don't panic!"]

induces the topology. Many other distance functions also induces that same topology.

So by defining the topology this way we don't determine the geometry a priori then, as we haven't specified the form of the distance function?
Is this why we are able to map patches of a manifold, in which the geometry is non-Euclidean, into open sets of $\mathbb{R}^{n}$ (and thus have to use non-Cartesian coordinates)?

 

[Cruz Martinez]

So by defining the topology this way we don't determine the geometry a priori then, as we haven't specified the form of the distance function?
Is this why we are able to map patches of a manifold, in which the geometry is non-Euclidean, into open sets of $\mathbb{R}^{n}$ (and thus have to use non-Cartesian coordinates)?

Without the metric tensor and the Levi-Civita connection the notion of cartesian or non-cartesian isnt'ty even meaninful.
Think about it as layers of structure. First we have a set which has no structure, on that set define a locally euclidean (locally homeomorphicto R^n) topology, then a smooth structure. Up to this point you can have coordinate charts, they are just the homeomorphisms from open sets of the manifold to R^n, but these coordinates are not cartesian or non-cartesian, it doesn't even make sense to ask that question yet.
Now once you define the metric tensor and the Levi-Civita connection you can ask wether a particular choice of coordinates is cartesian or not. If the Levi-Civita connection induced by the metric is curved then you can't have cartesian coordinates except for a very small patch on the manifold.

 

["Don't panic!"]

Now once you define the metric tensor and the Levi-Civita connection you can ask wether a particular choice of coordinates is cartesian or not. If the Levi-Civita connection induced by the metric is curved then you can't have cartesian coordinates except for a very small patch on the manifold.

Ok, I think it's all starting to make sense a bit more now.
Just to clarify though (and then I'll stop bugging everyone), if we consider a manifold with a metric that induces a non-Euclidean geometry, then if we consider a patch on such a manifold that is large enough that the local geometry cannot be considered as Euclidean, the coordinate maps that we use to map points in such a patch will be non-Cartesian as it will not be possible to construct such coordinate maps (unless we consider smaller patches around each point in the patch)?

 

[Cruz Martinez]

Ok, I think it's all starting to make sense a bit more now.
Just to clarify though (and then I'll stop bugging everyone), if we consider a manifold with a metric that induces a non-Euclidean geometry, then if we consider a patch on such a manifold that is large enough that the local geometry cannot be considered as Euclidean, the coordinate maps that we use to map points in such a patch will be non-Cartesian as it will not be possible to construct such coordinate maps (unless we consider smaller patches around each point in the patch)?

Correct.

 

["Don't panic!"]

So am i correct in saying that the Cartesian coordinate system is a special kind of mapping which directly relates the intrinsic distance between two points on a manifold to the 'numerical' distance between their coordinates in $\mathbb{R}^{n}$. As, in general, a coordinate patch on a manifold will have a non-Euclidean geometry, although it will be possible to construct a one-to-one mapping such that these points can be labeled by coordinates in $\mathbb{R}^{n}$, it will not be possible to construct a map that preserves the intrinsic distance between two points in this patch such that it corresponds to the 'coordinate distance' between their corresponding coordinates in $\mathbb{R}^{n}$. In other words, although we will be able to construct a coordinate map, it will be impossible to construct a Cartesian coordinate map for this patch (apart from within a small neighbourhood around each point in this patch)?