Contracted Bianchi identities

In general relativity and tensor calculus, the contracted Bianchi identities are[1]:

${\displaystyle \nabla _{\rho }{R^{\rho }}_{\mu }={1 \over 2}\nabla _{\mu }R,}$

where ${\displaystyle {R^{\rho }}_{\mu }}$ is the Ricci tensor, ${\displaystyle R}$ the scalar curvature, and ${\displaystyle \nabla _{\rho }}$ 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.[2] In the Einstein field equations, the contracted Bianchi identity ensures consistency with the vanishing divergence of the matter stress-energy tensor.