Contracted Bianchi identities

In general relativity and tensor calculus, the contracted Bianchi identities are:
where formula_2 is the Ricci tensor, formula_3 the scalar curvature, and formula_4 indicates covariant differentiation.
A proof can be found in the entry Proofs involving covariant derivatives.
These identities are named after Luigi Bianchi, although they had been already derived by Aurel Voss in 1880. In the Einstein field equations, the contracted Bianchi identity ensures consistency with the vanishing divergence of the matter stress–energy tensor.