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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.)

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    The numbers $g_{ab}$ are not unchanging under coordinate switches. The correct language of GR is differential geometry in which the metric is a smoothly varying bilinear form of signature $+++-$ on $T_pM$. It is an object that doesn't depend on any coordinates, but if you pick a coordinate frame around $p$ you get a basis of $T_pM$ via $\partial_{x^a}$ and with this identification you can write $g(v,w)=g_{ab}v^aw^b$ for any $v,w\in T_pM$ where $v^a,w^b$ are the components. Changing the coordinates will give different numbers $g_{ab}$. – 2017-01-14
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    By taking the dual basis $dx^a$ of $\partial_{x^a}$ you can write $g(v,w)=g_{ab}dx^a(v)dx^b(w)=(g_{ab}dx^a\otimes dx^b)(v,w)$. So $g_{ab}$ are the components of $g$ when it is written in the dual basis coming from some coordinate choice, meaning $g=g_{ab}dx^a\otimes dx^b$ is an equality between bilinear forms on $T_pM$. $ds^2$ is a strange notation for $g$ that physicists use. – 2017-01-14
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    @s.harp You are right. I realized since I should've applied the coordinate transform to g_{ab} as well. What you wrote is a good answer to the question, you might want to turn it into one. :) – 2017-01-14

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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.