Schwarzchild metric solution: Einstein's field equations seem redundant Einstein went to a lot of trouble linking the stress-energy tensor to the Ricci tensor (and curvature scalar).
Fast forward to using General Relativity for the Schwarzschild solution. 
- Those beautiful field equations are reduced to zero on both sides i.e. Ricci goes to zero.
- Symmetry arguments are used to create some limits for the metric.
... And we simply plug in the answer! (weak field approximation)
Is that deeply unsatisfying to anybody else? I had thought of General Relativity as: Energy (T tensor) generates the curvature (R tensor) that makes the geodesics change and, heyho, we see gravity. But the Schwarzschild solution has gravity with Ricci at zero. How can that be? And what was the point of Einstein perfecting his field equations?
 A: The Einstein field equations are,
$$R_{\mu\nu} -\frac12 g_{\mu\nu}R = 8\pi \, T_{\mu\nu}$$
for some matter described by a stress-energy tensor, $T_{\mu\nu}$. The Schwarzschild solution describing a non-rotating, neutral black hole corresponds to a Ricci-flat ($R_{\mu\nu} = 0$) solution of the Einstein field equations and can be derived with a spherically symmetric ansatz.
None of this means the Einstein field equations are redundant; remember that $R_{\mu\nu} = 0$ which is the Einstein field equation for a vacuum solution, imposes conditions on the ansatz for the Schwarzschild metric and is required for the derivation.
So what would make you think the general Einstein field equations are redundant? Say I give you some stress-energy $T_{\mu\nu}$; what are you going to use to find $g_{\mu\nu}$? In general, it will be the Einstein field equations alongside perturbation theory. 
There are other advanced techniques to describe solutions, but these are based on for example Lie symmetries of the Einstein field equations, or generation techniques which rely on knowledge of the behaviour and characteristics of solutions to the Einstein field equations.
Regardless of what method you may use, they can all be somehow linked to something requiring the fact that $G_{\mu\nu} = 8\pi \, T_{\mu\nu}$
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