Let us apply the Ricci contraction to the Bianchi identities
Since and , we can take in and out of covariant derivatives at will. We get:
Using the antisymmetry on the indices and we get
These equations are called the contracted Bianchi identities .
Let us now contract a second time on the indices and :
Since , we get
Raising the index with we get
The tensor is constructed only from the Riemann tensor and the metric, and it is automatically divergence free as an identity. It is called the Einstein tensor , since its importance for gravity was first understood by Einstein. We will see in the next chapter that Einstein's field equations for General Relativity are
where is the stress- energy tensor. The Bianchi Identities then imply
which is the conservation of energy and momentum.