I'm currently learning general relativity following Leonard Susskind's lectures, and I was very surprised by the fact that the components of the full Riemann curvature tensor are relevant even though the EFE only presents the Ricci curvature tensor and scalar.
For instance, it is shown that in a vaccum:
$$G^{\mu \nu} = 0 \Rightarrow R^{\mu \nu} = 0$$
but gravitational waves can exist in the components of $R^{\rho}_{\sigma \mu \nu}$.
So how is it possible to determine the Riemann tensor if the field equations seem to not give any additional information about it other than the fact that the Ricci tensor is zero?
