The Einstein Tensor

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

so

These equations are called the contracted Bianchi identities .

Let us now contract a second time on the indices and :

This gives

so

or

Since , we get

Raising the index with we get

Defining

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.