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  tex2html_wrap_inline3874 :

equation3211

It is the contraction of tex2html_wrap_inline3876 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 tex2html_wrap_inline3781 is antisymmetric on tex2html_wrap_inline3432 and tex2html_wrap_inline3434 and on tex2html_wrap_inline3456 and tex2html_wrap_inline3886 , all these contractions either vanish or reduce to tex2html_wrap_inline3888 . Therefore the Ricci tensor is essentially the only contraction of the Riemann tensor.

Similarly, the Ricci scalar  is defined as

equation3213