I am in the process of trying to teach myself GR Maths, at the A101 level, and have been working through the idea of tensors as scalars, vectors and matrices, i.e. rank-0, 1 and 2 tensors. Think I have also acquired some idea of the concept of contravariance and covariance, which then seems to define the superscripting and subscripting on the tensor. Have also worked through the basic coordinate transforms associated with these concepts and have now arrived at the idea of a metric tensor, which seems pretty important in terms of GR, so wanted to try to clarify some issues before plodding on.

[1] [tex]ds^2 = g_{\alpha \beta} da^\alpha db^\beta [/tex]

I am using (da) and (db), not the usual (dx), to identify what I am assuming to be differential displacement vectors because I want to used (x), in the description below, to identify the unit vectors, which I believe in this context are called the basis vectors. As such, I am also assuming that (da) and (db) can be related to (a) and (b) in [2] below, which might be a mistake. Anyway, [1] above appears, to me, to be a generic description of any metric, e.g. 2D flat space through to 4D curved spacetime. For fairly obvious reasons I would like to start by trying to interpret this equation in terms of the simplest form, i.e. normal 2D flat space, as described in terms of Pythagoras’ theorem and Cartesian coordinates:

[2] [tex]s^2 = a^2 + b^2[/tex]

As indicated, the superscripted terms in [1] have been assumed to be contravariant displacement vectors [da, db] or rank-1 tensors, while (g) is an n-dimensional array or rank-2 tensor. In the 2D example associated with [2], (g) is a 2D array:

[3] [tex]g= \left(\begin{array}{cc}1&0\\0&1\end{array}\right) [/tex]

I will defer my questions about the nature of (g) to another time, as this post will probably end up being too detailed and too long to solicit any response. Anyway, if I expand [1] in terms of the multiplication of 2 contravariant 2D vectors, where the basis vectors are [x

^{1}, x

^{2}], I appear to end up with:

[4] [tex] a^\alpha b^\beta = a^1 b^1 x^1 x^1 + a^1 b^2 x^1 x^2 + a^2 b^1 x^2 x^1 + a^2 b^2 x^2 x^2 [/tex]

Now it would seem that [4] can be simplified by multiplying through by [3], which seems correct for the 2D flat space metric in [2]:

[5] [tex]s^2 = g_{\alpha \beta} a^\alpha b^\beta = a^1 b^1 + a^2 b^2 [/tex]

At this point, I wanted to try to anchor the notation in [5] to an actual geometric example, which would be a specific solution of [2], i.e. [a=3] and [b=4] giving [s=5]. My assumption was that the magnitude of (a

^{1}) is defined with respect to the basis axis (x

^{1}). Therefore, given that (a,b) in [2] actually align to the basis vector (x

^{1}, x

^{2}), I assumed the following values: (a

^{1}=3, a

^{2}=0, b

^{1}=0, b

^{2}=4). However, these values do not appear to prove any obvious equivalence between [2] and [5], therefore I must assume that I have misunderstood some fundamental concept and would appreciate any help on offer. Thanks