In the Newman-Penrose formalism one encodes the ten degrees of freedom of the Weyl tensor $C_{\alpha\beta\mu\nu}$ in the five complex scalar potentials $\Psi_0$, $\Psi_1$, $\Psi_2$, $\Psi_3$ and $\Psi_4$. The 10 degrees of freedom of the Ricci tensor $R_{\rho\sigma}$ are encoded in four real and three complex scalar potentials encoded in $\Phi_{00}$,$\Phi_{01}$, $\Phi_{02}$, $\Phi_{10}$, $\Phi_{11}$, $\Phi_{12}$, $\Phi_{20}$, $\Phi_{21}$, $\Phi_{22}$ and $\Lambda$.
If I understand correctly, all gravitational degrees of freedom are encoded in the Riemann tensor, which has 20 degrees of freedom. The Riemann tensor can be decomposed into the Weyl tensor (with 10 DoF) and the Ricci tensor (with the other 10 DoF). This fact makes me think that once we have the potentials for the Weyl and the Ricci tensors -As I described in the first paragraph- we should be done with degrees of freedom. However, in the NP formalism they then go on to define twelve spin coefficients encoding the degrees of freedom of the connection. My question is: Aren't these last degrees of freedom dependent on the degrees of freedom of the Weyl+Ricci tensors? I thought that Weyl and Ricci fully determined the Riemann tensor, and that the Riemann tensor fully determined gravity. What am I missing?
