Kruskal coordinates from GR field equations The "usual" way (as I've seen in most text books) of dealing with Schwarzschild spacetime is to start with Schwarzschild coordinates, then explore the nature of coordinate singularity at $r=2M$ and eventually introduce Kruskal coordinates. This approach seems a bit weird -- sure there must be a way to get complete solution right from the field equations?
I've found a wonderful article which follows this approach. Interesting part is that to get Kruskal coordinates you have to assume that relation between "radial" function $r$ and coordinates $u,v$ is not 1-1 and can be implicit $f(r) = h(u,v)$. This is great, but leaves me wondering if there are more assumptions that are left unmentioned when we do the "usual" derivation? Why do we assume that other functions are explicit?
 A: A space-time $(M,g)$ is spherically symmetric implies that the metric $g$ is invariant under spatial rotation. These generators of symmetry which preserves $g$ are called the Killing Vector Fields (KVFs). An unit sphere ($S^2$, $d\Omega^2$) enjoys symmetry under proper rotation and let $R$, $S$, $T$ denote three linearly independent KVFs for $S^2$. Now, given that ($M$, $g$) is a spherically symmetric space-time, we expect that KVFs for unit sphere $S^2$ should also be KVFs for our space-time metric $g$. If we choose $R$, $S$, $T$ such that they form a complete basis for a Lie-algebra, then it follows from Frobenius theorem that these KVFs will be tangent to the submanifold $S^2\subset M$. In fact, we can always decompose a spherically symmetric space-time $M$ as a Cartesian product $M=U^2\times S^2$ where $U^2$ is a 2-dim manifold with an indefinite metric. So we can write our product metric $g=g_U\oplus g_S$. In other words, we can always write a generic spherically symmetric space-time metric as:
$$ds^2=\underbrace{d\tau^2(u,v)}_{U^2\;part}+\underbrace{e^{2\mu_3(u,v)}d\Omega^2}_{S^2\;part}$$where $u$ and $v$ are coordinates on $U^2$. This decomposition of metric has been motivated in chapter 7 of "Relativity: The General Theory" by J. L. Synge. If the coordinates u ,v are defined such that null rays $u=$const. and $v=$const. exists then our metric $ds^2$ in null coordinates reads (for derivation, refer to chapter 3 of "The mathematical theories of black holes" by S. Chandrasekhar) :
$$ds^2=4f(u,v)dudv-e^{2\mu_3(u,v)}d\Omega^2$$For simplicity, define $Z=e^{\mu_3}$. Now, the vacuum Einstein's equations $R_{ab}=0$ implies
\begin{align}
   f(u,v)&=-A(u)B(v)\left(1-\frac{2M}{Z}\right)\\
  dZ&=-\left(1-\frac{2M}{Z}\right)[A(u)du+B(v)dv]
\end{align}where $A(u)$ and $B(v)$ are some arbitrary functions and $2M$ is a constant. We can now choose our coordinates such that metric is free from coordinate singularities. One such example is the Kruskal frame, where we choose $Z=r$, $A(u)=-\frac{2M}{u}$ and $B(v)=-\frac{2M}{v}$. The metric simplifies to
$$ds^2=-\frac{32M^3}{r}e^{-r/2M}dudv-r^2d\Omega^2$$ and the apparent coordinate singularity at $Z=2M$ vanishes. The double valuedness in $(r, uv)$ relation follows from the solution: $$uv=(1-r/2M)e^{r/2M}$$If $M$ is allowed to have negative values, then we have two branches : $r>0$ and $r<0$ corresponding to a given value for $uv=$const. For Schwarzschild solution, we introduce a time coordinate $t$ defined as : $t=2M\log(u/v)$
