Why is an interval invariant on coordinate transformations?

Question

I learned that in (semi-)Riemannian geometry an interval $ds^{2}=g_{ab}dx^{a}dx^{b}$ is invariant on coordinate transformations. But there is something I do not get in the concept. Consider the simple coordinste transformation $x'^{a}={x^{a}}/{2}$, where both x and x' are Cartesian coordinate systems. Even intuitively, since the interval, afaik, is the distance between two infinitesimally close point, how can it then be invariant, and why does it not shrink to it's half in this case? And the calculations also show this:
$ds^2=\delta_{a}^{b} dx^a dx^b$
$ds'^2=\delta_{a}^{b} dx'^a dx'^b=\delta_{a}^{b} \frac{\partial x'^a}{\partial x^a}\frac{\partial x'^b}{\partial x^b}dx^a dx^b=\delta_{a}^{b} *\frac{1}{2}*\frac{1}{2}*dx^a dx^b$
$ds'^2=\frac{1}{4}*ds^2$

(Using Einstein's summation convencion above. I am learning from M. P. Hobson, G. P. Efstathiou, A. N. Lasenby: General relativity.)

Answer 1

The numbers $g_{ab}$ are not unchanging under coordinate switches. General relativity is formulated using the language and concepts of differential geometry, in this context the metric $g$ is a smoothly varying bilinear form on $T_pM$ of a certain signature (usually $(+,+,+,-)$ ).

If one has a coordinate chart around $p$ this induces a basis of $T_pM$ via $\partial_{x^a}$, and given a vector $v\in T_pM$ one can expand it vectors in coordinates: $v=v^a\partial_{x^a}$. If one defines: $$g_{ab}:=g(\partial_{x^a},\partial_{x^b})$$ One can see from the bilinearity of $g$ that: $$g(v,w)=g_{ab}v^aw^b$$ for any $v,w\in T_pM$. This is the same statement as $g=g_{ab}\,dx^a\otimes dx^b$ where $dx^a$ is the dual basis of $\partial_{x^a}$. In other words the $g_{ab}$ are the components of the bilinear form $g$ in the coordinate system. If one takes a different coordinate system one will get different numbers for $g_{ab}$.

With this context the statement that $g_{ab}\, dx^a\otimes dx^b$ doesn't change under coordinate switch follows quickly since the numbers $g'_{ab}$ in the new coordinate system are defined precisely so that the expression remains the same bilinear form.