I know that in Euclidean geometry, where the manifold is "flat" (such that it is isomorphic to an open subset of $\mathbb{R}^{n}$), $M\cong\mathbb{R}^{n}$, one can use Cartesian coordinates, $\phi (p)\equiv x^{\mu}:M\rightarrow\mathbb{R}^{n}$, to cover the entire manifold, and further more, if one has two points $p$ and $q$, then their coordinate difference, $\sqrt{\left[x^{\mu}(p)-x^{\mu}(q)\right]^{2}}$ (summation implied), corresponds to the actual distance between the two points on the manifold.
My question is, what is the reasoning for why this is not true in general? i.e. Why doesn't so-called "coordinate distance" between two points correspond to the actual distance between them on the manifold?
For example, the coordinate difference between two points on a sphere does not correspond to the actual distance between two points on the sphere (considering the Earth, one cannot simply naively take the difference between two points on a map an equate the reply to the actual physical distance between them). Is it simply that the coordinate maps will in general be highly non trivial (and not the simple identity map as in Cartesian coordinates), and so simply subtracting the coordinate values of one point from another will not map back to differences between points on the manifold? Is it also to do with the fact that one needs to define a metric on the manifold in order to measure distances between points on the manifold and such a metric will in general be non-Euclidean, and so the simple subtraction of coordinate values does not equate to the actual metric distance?
Apologies if this post is a bit confused, but I'm a bit stuck on how to understand this concept correctly. Any help would be much appreciated.
Best Answer
The "formal Pythagorean distance" is not invariant under change of coordinates, i.e., it depends on the choice of coordinate system (e.g., multiply Cartesian coordinates by a positive constant, or consider polar coordinates).
The Riemannian-geometric way to interpret the Pythagorean distance is via arc length with respect to a metric (smooth field of inner products) $g$: If $\gamma$ is a piecewise $C^{1}$ path satisfying $\gamma(a) = p$ and $\gamma(b) = q$, the length of $\gamma$ over $[a, b]$ is $$ \ell(\gamma) = \int_{a}^{b} \sqrt{g(\dot\gamma, \dot\gamma)}; $$ the distance from $p$ to $q$ is the infimum of $\ell(\gamma)$ over all such paths joining $p$ to $q$.
The Pythagorean formula comes from the constant field $g$ whose value at each point is the Euclidean inner product, with components $g_{\mu\nu} = \delta_{\mu\nu}$; the shortest paths are easily shown to be line segments, and the length of a line segment is given by the Pythagorean distance.
For a general metric $g$, however, the distance is not given by the formal Pythagorean distance.