Identities
In general relativity and tensor calculus, the contracted Bianchi identities are:
formula_1
where formula_2 is the Ricci tensor, formula_3 the scalar curvature, and formula_4 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.
Proof.
Start with the Bianchi identity
formula_5
Contract both sides of the above equation with a pair of metric tensors:
formula_6
formula_7
formula_8
formula_9
The first term on the left contracts to yield a Ricci scalar, while the third term contracts to yield a mixed Ricci tensor,
formula_10
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,
formula_11
which is the same as
formula_12
Swapping the index labels "l" and "m" on the left side yields
formula_13
See also.
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Notes.
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