- #1

Frank Castle

- 580

- 22

Since the space is Euclidean we can use the Cartesian coordinate map, which is a global coordinate map (since Euclidean space is globally isomorphic to ##\mathbb{R}^{n}##), and as such, considering ##\mathbb{R}^{2}## for simplicity, for two points ##p=(x_{1},y_{1})## and ##q=(x_{2},y_{2})##, the distance between them is given by their so-called

*"coordinate distance"*$$\sqrt{\left(\Delta x\right)^{2}+\left(\Delta y\right)^{2}}$$, where ##\Delta x = x_{2}-x_{1}## and ##\Delta y = y_{2}-y_{1}##.

My question is, why is this not the case in general? I mean, even in Euclidean space this is not the case in other coordinate systems (for example, in polar coordinates, ##p=(r_{1},\theta_{1})## and ##q=(r_{2},\theta_{2})##, but the distance between them is not given by ##\sqrt{\left(\Delta r\right)^{2}+\left(\Delta\theta\right)^{2}}##). Is it simply to do with the coordinate transformation between Cartesian and polar and the coordinate scaling that it introduces, or the fact that the basis vectors are not constant in the polar coordinate system?

I get that on a more general manifold, where the geometry is non-Euclidean (and as such the metric ##g_{\mu\nu}(x)## is non-trivial), such that the parallel postulate no longer holds and the Pythagorean theorem holds only

*locally*, one must integrate the differential line element, ##ds^{2}## (of the form ##ds^{2}=g_{\mu\nu}(x)dx^{\mu}dx^{\nu}## along a geodesic connecting the two points ##p## and ##q## to determine the distance between them, but is there an argument for why the line element is not always of the form $$ds^{2}=(dx^{1})^{2}+(dx^{2})^{2}+\cdots +(dx^{n})^{2}$$

Finally, why can one simply not take the difference between the coordinates of two points ##p## and ##q## to gain the coordinates of another point ##p'##? Is this only true in Euclidean space because it is an affine space and so coordinate differences correspond to vectors (and hence satisfy the vector space axioms), and furthermore the Cartesian coordinate system is global, whereas, in a more general manifold, such a calculation is very much coordinate dependent and thus not well-defined, and indeed the difference between the coordinates of two points may not correspond to the coordinates of any other point on the manifold?!