Recipes for $g_{ij}$

A non-euclidean surface with coordinate system

Metrics are important in both the calculus of surfaces and in General Relativity.  These objects are essentially NxN matrices whose values are defined by a) the coordinate system used to parametize the space, and b) the particular point $\boldsymbol{x} = (x^i)$.  In general, theorems can be categorized as either intrinsic or extrinsic, where extrinsic means that information about how the space is embedded in a higher dimensional space is used, and intrinsic means that nothing other than the metric and it's partial derivatives were used.  The word intrinsic comes from the idea that a being which "lives" in the space can calculate the metric.

So how does such a being find the metric? 

 2 & 3 dimensional spaces

In three dimensions it's pretty straightforward:

1. Impose a coordinate system $x^i = x^1, x^2, x^3$ on the space
2. Define the covariant basis $\boldsymbol{e_i} = \frac{\partial{\boldsymbol{R}}}{\partial{x^i}}$
3. The metric is $g_{ij} = \boldsymbol{e_i}.\boldsymbol{e_j}$

I.e. the metric is $\lVert \boldsymbol{e_i} \rVert \lVert \boldsymbol{e_j} \rVert cos\theta$ where $\theta$ is the angle between the two vectors. So all the being that lives in the space needs to be able to do is find the angle between two vectors and the magnitude of vectors.  It's then  able to calculate the metric at each point in space.

<aside> That's all from the point of view of a being living in the space.  Mathematically it's a bit more subtle than that.  Just because you have parametised your set using $\mathbb{R}^3$ it doesn't follow there's a natural way to take partial derivatives $\frac{\partial{\boldsymbol{R}}}{\partial{x^i}}$ let alone define magnitudes of such objects or angles between them.

Some courses, such as the International Winter School on Gravity and Light  take the mathematically rigorous route, starting off with set theory, moving on to topological and then metric spaces, and imposing more and more structure.  However, in my opinion this abstraction completely obscures the underlying concepts.  I much prefer the route taken by Pavel Grinfeld in his tensor course.  This approach starts off with the euclidean space $\mathbb{R}^3$ in which differences, limits, angles and absolute values are already well understood.  Given this framework it is possible to first study arbitrary coordintate systems, and then to study non-euclidean embedded surfaces.  The beauty of embedding surfaces within a 3D euclidean space is that there is no need to invoke an enormous amount of abstract pure mathematics in order to get the building blocks.  And all the maths applies to higher dimensional surfaces too (embedded in even higher dimensional euclidean ambient spaces).  Furthermore, the resulting theory is no less general because it is always possible to embed a manifold in a higher dimensional euclidean space.  </aside>

4 dimensional spacetime

Okay, so we know a being living in a 3 dimensional space should be able to calculate $g_{ij}$.  But how do we extend this to General Relativity in which the space is 4 dimensional and comprised of 3 dimensions of space and 1 of time?  The recipe provided above does not work.  To see this, imagine trying to take the dot product of two vectors $\overrightarrow{AB}$ and $\overrightarrow{AC}$  (where A,B, and C are spacetime events, e.g. bulbs flashing.)  Neither $\lVert \overrightarrow{AB}\rVert$ or $\lVert \overrightarrow{AC}\rVert$ have obvious interpretations, and $\theta$ - the angle between $\overrightarrow{AB}$ and $\overrightarrow{AC}$ is particularly meaningless.

Actually, $\lVert \overrightarrow{AB}\rVert^2$ does have an interpretation.  It is the spacetime interval.  If this is positive then it's the square of the time experienced by an observer travelling from A to B, and if negative it is minus the square of the distance between A and B, as seen by an observer for whom A and B are contemporaneous. Either way these are values which can be obtained by a being which lives in the spacetime.

So, we can now find a recipe for $g_{\mu\nu}$ (using greek letters for indices because we are now referring to any of the four dimensions of spacetime) .  All we need to do rewrite the recipe we have in 3D, but using only squares of magnitudes of vectors.  Here goes:

$$
\begin{align}
g_{\mu\nu} &= \boldsymbol{e_\mu}.\boldsymbol{e_\nu} \\
&= \frac{1}{2}(2\boldsymbol{e_\mu}.\boldsymbol{e_\nu}) \\
&= \frac{1}{2}((\boldsymbol{e_\mu}+\boldsymbol{e_\nu}).(\boldsymbol{e_\mu}+\boldsymbol{e_\nu}) - \boldsymbol{e_\mu}.\boldsymbol{e_\mu} - \boldsymbol{e_\nu}.\boldsymbol{e_\nu})\\
&= \frac{1}{2}(\lVert \boldsymbol{e_\mu}+\boldsymbol{e_\nu} \rVert^2 - \lVert \boldsymbol{e_\mu} \rVert^2 - \lVert \boldsymbol{e_\nu} \rVert^2)
\end{align}
$$

Hey presto, we've written the metric in terms of squares of norms, and these are things which can be measured by a being that lives in the space whether that space is a 2D or 3D surface, or the 4D spacetime of General Relativity.  In the former case they are just squares of distances, in the latter they are spacetime intervals, e.g. squares of proper times.

Note the relation to Pythagoras.  Suppose $\boldsymbol{e_\mu}=\overrightarrow{AB}$ and $\boldsymbol{e_\nu}=\overrightarrow{BC}$. Then $g_{\mu\nu} = 0$ if and only if ABC satisfies the equation for a right angled triangle with the right angle at B.