*This is my original answer, which, sadly, missed the main point of the question. However, as I did invest some time to write it and it actually does answer at least part of the question, I'll leave it as-is.*

This is a fallacy of the hands-on approach to differential geometry using coordinate expressions only and one of the reasons why I prefer the abstract-geometric approach.

Let's assume for simplicity that out abstract manifold $M$ permits global coordinate systems
$$
\begin{align}
\varphi:M&\rightarrow\mathbb R^n \\
p&\mapsto x^\mu
\end{align}
$$
and
$$
\begin{align}
\varphi':M&\rightarrow\mathbb R^n \\
p&\mapsto x'^\mu
\end{align}
$$
The coordinate transformation from unprimed to primed coordinates is given by
$$
\begin{align}
\varphi'\circ\varphi^{-1}:\mathbb R^n&\rightarrow\mathbb R^n \\
x^\mu&\mapsto x'^\mu
\end{align}
$$
Now, a real transformation would be a diffeomorphism
$$
\begin{align}
f:M&\rightarrow M \\
p&\mapsto q
\end{align}
$$
which comes with a coordinate expression $f^\varphi=\varphi\circ f\circ\varphi^{-1}$
$$
\begin{align}
f^\varphi:\mathbb R^n&\rightarrow\mathbb R^n \\
x^\mu&\mapsto y^\mu
\end{align}
$$
where $x^\mu=\varphi(p)$ and $y^\mu=\varphi(q)$. Even though $f^\varphi$ *looks* like any other coordinate transformation, we remain in the same unprimed coordinate system.

A coordinate transformations won't change the value of scalar expressions - eg contraction of the metric tensor with two vectors to compute their inner product - by definition of the transformation laws for tensors.

This is not the case for real transformations: As we do not change coordinate systems, the components of the metric tensor won't transform and thus can't balance the change in coordinates of the vectors.

After a coordinate transformation, we're still computing the same quantity similar to using a different set of units, whereas after a real transformation, we'll actually compute a different quantity as we evaluate at different points on the manifold, ie move around in spacetime.

This post imported from StackExchange Physics at 2015-03-30 13:51 (UTC), posted by SE-user Christoph