Contracted Bianchi Identities

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

[math]\displaystyle{ \nabla_\rho {R^\rho}_\mu = {1 \over 2} \nabla_{\mu} R }[/math]

where [math]\displaystyle{ {R^\rho}_\mu }[/math] is the Ricci tensor, [math]\displaystyle{ R }[/math] the scalar curvature, and [math]\displaystyle{ \nabla_\rho }[/math] 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.

Proof

Start with the Bianchi identity^[3]

[math]\displaystyle{ R_{abmn;\ell} + R_{ab\ell m;n} + R_{abn\ell;m} = 0. }[/math]

Contract both sides of the above equation with a pair of metric tensors:

[math]\displaystyle{ g^{bn} g^{am} (R_{abmn;\ell} + R_{ab\ell m;n} + R_{abn\ell;m}) = 0, }[/math]

[math]\displaystyle{ g^{bn} (R^m {}_{bmn;\ell} - R^m {}_{bm\ell;n} + R^m {}_{bn\ell;m}) = 0, }[/math]

[math]\displaystyle{ g^{bn} (R_{bn;\ell} - R_{b\ell;n} - R_b {}^m {}_{n\ell;m}) = 0, }[/math]

[math]\displaystyle{ R^n {}_{n;\ell} - R^n {}_{\ell;n} - R^{nm} {}_{n\ell;m} = 0. }[/math]

The first term on the left contracts to yield a Ricci scalar, while the third term contracts to yield a mixed Ricci tensor,

[math]\displaystyle{ R_{;\ell} - R^n {}_{\ell;n} - R^m {}_{\ell;m} = 0. }[/math]

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,

[math]\displaystyle{ R_{;\ell} = 2 R^m {}_{\ell;m}, }[/math]

which is the same as

[math]\displaystyle{ \nabla_m R^m {}_\ell = {1 \over 2} \nabla_\ell R. }[/math]

Swapping the index labels l and m on the left side yields

[math]\displaystyle{ \nabla_\ell R^\ell {}_m = {1 \over 2} \nabla_m R. }[/math]

See also

• Bianchi identities
• Einstein tensor
• Einstein field equations
• General theory of relativity
• Ricci calculus
• Tensor calculus
• Riemann curvature tensor

Notes

References

• Lovelock, David; Hanno Rund (1989) [1975]. Tensors, Differential Forms, and Variational Principles. Dover. ISBN 978-0-486-65840-7. 
• Synge J.L., Schild A. (1949). Tensor Calculus. first Dover Publications 1978 edition. ISBN 978-0-486-63612-2. https://archive.org/details/tensorcalculus00syng. 
• J.R. Tyldesley (1975), An introduction to Tensor Analysis: For Engineers and Applied Scientists, Longman, ISBN 0-582-44355-5 
• D.C. Kay (1988), Tensor Calculus, Schaum’s Outlines, McGraw Hill (USA), ISBN 0-07-033484-6 
• T. Frankel (2012), The Geometry of Physics (3rd ed.), Cambridge University Press, ISBN 978-1107-602601 

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