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Contracted Bianchi identities
Identities in general relativity From Wikipedia, the free encyclopedia
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In general relativity and tensor calculus, the contracted Bianchi identities are:[1]
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.[2] In the Einstein field equations, the contracted Bianchi identity ensures consistency with the vanishing divergence of the matter stress–energy tensor.
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Proof
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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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