Confusion in General Relativity

[martinbn]

Not in the same way. That's an important misconception you should look into.
-Being locally euclidean has nothing to do with the Euclidean metrics or signature types, it is a topological or differentiable structure equivalence with ℝn locally depending on the kind of manifold(topological or smooth) one is talking about. It has nothing to do with having a Euclidean metric no matter how small a region you want to consider.

That is true, but in the context of this discussion, it is clear what PeterDonis means.

 

[PeterDonis]

Being locally euclidean has nothing to do with the Euclidean metrics or signature types

It depends on how we interpret the word "Euclidean"--it can be a purely topological term, as you claim here, or it can be a term describing the metric. I was using it in the latter sense. The former sense is irrelevant to the discussion because we are talking about equations with covariant derivatives in them, and you can't take covariant derivatives unless you have a metric.

I think that is what is behind the split between the "spacetime tells matter how to move" and "matter tells spacetime how to curve" separate parts put forward by Wheeler and also determines the peculiar implementation of "general covariance" in GR that I refer to in my last post.

I disagree. All of these things require a metric. The purely topological sense of "Euclidean" is irrelevant.