Einstein's field equations are:
$R_{ab} - {1 \over 2}g_{ab}\,R + g_{ab} \Lambda = {8 \pi G \over c^4} T_{ab}$
And since the Ricci curvature tensor is "less information" than the Riemann curvature tensor because:
$R_{ab} = R^c{}_{acb}$
and the Riccie curvature scalar is even "less information" than the Ricci curvature tensor because:
$R = R^a{}_{a}$
This appears to indicate the GR field equations don't contain enough information to specify how the full Riemann curvature tensor evolves. So it looks like something missing. Is it even possible to say how the full Riemann curvature tensor evolves in GR?
Since the Riemann curvature tensor can be obtained from the metric, if we can solve for how the metric evolves using the Ricci curvature tensor, we can then get the Riemann curvature tensor evolution. So the equation is equivalent to the title question, what all do we need to solve for the metric in GR?
In books where they obtain the solution outside a static spherical object, they seem to always refer back to the Newtonian limit and compare to Newtonian gravity to fully fix the answer. This seems a bit scary, as it seems to suggest indeed that GR needs some other equations specified to get the answer.
If I choose a coordinate system and specify the metric everywhere at some initial time, is that initial metric + the GR field equations enough to solve for the metric everywhere in spacetime? Or is there some way to use GR to get the metric without any prior geometry put in?