General relativity and curvilinear coordinates

  • #126
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 cant 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)?
 
  • #127
116
25
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.
 
  • #128
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 [itex]\mathbb{R}^{n}[/itex]. 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 [itex]\mathbb{R}^{n}[/itex], 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 [itex]\mathbb{R}^{n}[/itex]. 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)?
 

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