Tensors and Relativity: Chapter 6
The Ricci tensor
Next: The Einstein Tensor Up: The Bianchi identities; Ricci Previous: The Bianchi identities; Ricci
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