Physics:Contracted Bianchi identities

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

[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. 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

[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

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