Tensors and Relativity: Chapter 6
The Ricci tensor
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Before looking at the consequences of the Bianchi identities, we need to define the Ricci tensor 
:
It is the contraction of 
on the first and third indices. Other contractions would in principle also be possible: on the first and second, the first and fourth, etc. But because 
is antisymmetric on 
and 
and on 
and 
, all these contractions either vanish or reduce to 
. Therefore the Ricci tensor is essentially the only contraction of the Riemann tensor.
Similarly, the Ricci scalar is defined as
