How are different metrics for spacetime constructed? is a metric something derived? for example where does the extremal Reissner Nordstrom metric come from?
 A: A metric in general relativity is a solution to a wave equation called the Einstein field equation (EFE). If you're familiar with Maxwell's equations for electromagnetism, the EFE is very similar. It relates the field to its sources. In a vacuum, its solutions are waves.
The Reissner Nordstrom metric is a general solution to the Einstein field equations that satisfies certain other conditions. It's asymptotically flat, i.e., it looks like empty space at large distances. It's static and stationary. It contains an electric field. It's axially symmetric.
I don't know if that's what you were asking, or if you understood all that and wanted to know how people would find closed-form solutions. There is a variety of techniques for finding closed-form solutions, and none of them is a general technique -- all of them are just tricks that work in certain cases. For example, one way of guessing certain solutions in spherical coordinates is to write down a metric of the form $ds^2=Adt^2-Bdr^2-r^2d\Omega^2$, then apply the EFE to get a differential equation for $A$ and $B$ as functions of $r$. This is probably how Reissner and Nordstrom did it, since they were working in the very earliest years of GR. Circa 1970, I think people developed new, coordinate-independent techniques that didn't depend on writing down a metric in a particular coordinate chart. I believe these methods were more algebraic, but I don't know much about them.
