In general relativity and tensor calculus, the contracted Bianchi identities are:[1] where
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.
Start with the Bianchi identity[3] Contract 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 (changing dummy index n to m) 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
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