In his book "Road to Reality" section 19.7 Roger Penrose asks the question:
What is the appropiate analogue of the Maxwell field tensor $F_{ab}$ describing the gravitational degrees of freedom? According to Penrose "... it is more appropiate to choose what is called the Weyl tensor".
I don't really understand this assertion. It might be that Einstein's field equations (EFE) are here only considered as constraint on the Ricci tensor
$$R_{ab} = \kappa \left( T_{ab} - \frac{1}{2}g_{ab} T\right).$$
In Maxwell theory $\nabla \cdot \mathbf{E} = 4\pi \rho$ is sometimes considered as a constraint, but we have in free Maxwell theory at least $\partial_a F^{ab} =0$ whereas in gravitational theory the Weyl-tensor is completely free (there is no field equation for it, at least I don't know any). It is actually this complete freedom which confuses me.
The gravitational physics seems to be independent of the value of the Weyl tensor, but can this be ? The Weyl tensor is known be responsible for tidal forces on matter, without rule for the Weyl tensor they could be arbitrary (large).
