In this topic, we shall give a natural way to construct 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
.
Theorem. The tensor
is divergence free in the sense that
.
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. Some authors denote the Einstein tensor by
. We will see later that Einstein’s field equations for General Relativity in the vacuum case are
where is the stress- energy tensor. The Bianchi identities then imply
which is the conservation of energy and momentum.