It is well-known that the equation for stationary solutions of the Einstein-Hilbert functional (without matter and cosmological constant, which is irrelevant here):  

$$S = \int\_M R \mu\_g,$$
is given by the Einstein's field equation:
$$Ric -\frac{1}{2}R g = 0, $$
where $\mu\_g$ is the canonical volume form given by the metric $g$, $Ric$ is the Ricci curvature and $R$ is the Ricci scalar.


The standard derivation of the above statement seems to be a not so hard but not so pleasant direct calculation, either in coordinates or abstract indices, expanding everything in terms of the Christoffel symbol and eventually in terms of $g$ and then calculus.


My questions is: is there a more geometric and coordinate-free way to derive this? 



Coordinate-free derivation of the Einstein's field equation from the Hilbert action.
This can be found in Besse "Einstein Manifolds", in chapter 4.


The idea is to use Koszul formula for the Levi-Civitta connection to compute the derivative of the curvature with respect to the metric. Bianchi identities also help.


