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.

Source: http://www.mth.uct.ac.za/omei/gr/chap6/node14.html

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