# How is the full Riemann curvature tensor determined if the Einstein field equations only present the Ricci curvature tensor? [duplicate]

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?

• They are calculated from the christoffel symbols which are calculated from the metric tensor. I encourage you to look at the riemann curvature section in this article. medium.com/@trigress09/… Commented Jan 8, 2023 at 18:35
• The linked posts don’t answer OP’s question, so it’s not a dupe and I don’t see why it is closed. To me it seems more like OP is having an issue understanding what the Einstein equations actually tell us/require us to solve for. OP is right that it is possible to be Ricci-flat ($R_{ab}=0$) without being flat ($R^a_{bcd}=0$) (e.g minkowski metric vs Schwarzschild, a fact mentioned in the links as well). But the point to be emphasized is that Einstein’s equations are a system of PDEs for the metric $g_{ab}$. Once we solve the equations, we get $g_{ab}$. Using this, we get $R^a_{bcd}$. Commented Jan 10, 2023 at 6:51