Measurement in spacetime begins with distance. We have to declare some rules of distances, so called axioms. Let us call $d()$ distance function, that accepts two points that accepts two points as parameters, and the rules are:
Distance of a point $a$ from itself equals zero:
$d(a,a)=0$
$d(a,b)$= $d(b,a)$
Let us now consider a curvilinear co-ordinate system $(u_{1},u_{2},...u_{n})$. Let $\vec{r}$ be any position vector of the curvilinear system. We then have that $\vec{r}=\vec{r}(u_{1},u_{2},...u_{n})$. Now, from the vector differential calculus we have the differential element $d(\vec{r})$ as equal to the following;
(Nb: we are considering for the case of $n=3$);
Here, $e_{1},e_{2}, e_{3}...$ are the direction vector of the unit tangent vectors of the respective coordinate, "$h_{n}$" is simply the scale factor(magnitude) of the tangent vector.
More elegantly,
Here, we have made use of the Einstein's summation convention. Einstein summation convention is just a notational convention that implies summation over a set of indexed terms in a formula, thus achieving notational brevity [3]. This then, is the equation of the arc length of the curvilinear coordinate system. Here,$g_{\alpha }=\alpha_{p} \cdot \alpha_{q}$ is called the metric coefficient and $\alpha_{n}$ the tangent vector (not necessarily the unit vector) of the concerned axes in a curvilinear coordinates system. The value of $g_{pq}$ are different in different curvilinear coordinate system. In fact, we can say that the metric tensor of any particular coordinate system is the signature of the system concerned.
The tensor $g_{pq}$ satisfies the following property:
(that is, it is symmetric) because the multiplication in the Einstein summation is ordinary multiplication and hence commutative.
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