Contracted Bianchi identities

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

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

where ${\displaystyle {R^{\rho }}_{\mu }}$ is the Ricci tensor, ${\displaystyle R}$ the scalar curvature, and ${\displaystyle \nabla _{\rho }}$ indicates covariant differentiation.

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.