Contracted Bianchi identities
In general relativity and tensor calculus, the contracted Bianchi identities are:
where is the Ricci tensor, the scalar curvature, and indicates covariant differentiation.
These identities are named after Luigi Bianchi, although they had been already derived by Aurel Voss in 1880, and independently by Gregorio Ricci-Curbastro in 1889. In the Einstein field equations, the contracted Bianchi identity ensures consistency with the vanishing divergence of the matter stress–energy tensor.
Proof
Start with the Bianchi identityContract both sides of the above equation with a pair of metric tensors:
The first term on the left contracts to yield a Ricci scalar, while the third term contracts to yield a mixed Ricci tensor,
The last two terms are the same and can be combined into a single term which shall be moved to the right,
which is the same as
Swapping the index labels l and m on the left side yields