Physics:Contracted Bianchi identities

Short description: Identities in general relativity

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

${\mathrm{\nabla }}_{\rho }{R^{\rho }}_{\mu }=\frac{1}{2}{\mathrm{\nabla }}_{\mu }R$

where ${R^{\rho }}_{\mu }$ is the Ricci tensor, $R$ the scalar curvature, and ${\mathrm{\nabla }}_{\rho }$ indicates covariant differentiation.

These identities are named after Luigi Bianchi, although they had been already derived by Aurel Voss in 1880, and independently by Gregorio Ricci-Curbastro in 1889. 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

$R_{abmn;\ell }+R_{ab\ell m;n}+R_{abn\ell ;m}=0.$

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

$g^{bn}{g}^{am}(R_{abmn;\ell }+R_{ab\ell m;n}+R_{abn\ell ;m})=0,$

$g^{bn}(R^m{}_{bmn;\ell }-R^m{}_{bm\ell ;n}+R^m{}_{bn\ell ;m})=0,$

$g^{bn}(R_{bn;\ell }-R_{b\ell ;n}-R_b{}^m{}_{n\ell ;m})=0,$

$R^n{}_{n;\ell }-R^n{}_{\ell ;n}-R^{nm}{}_{n\ell ;m}=0.$

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

$R_{;\ell }-R^n{}_{\ell ;n}-R^m{}_{\ell ;m}=0.$

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,

$R_{;\ell }=2R^m{}_{\ell ;m},$

which is the same as

${\mathrm{\nabla }}_m{R}^m{}_{\ell }=\frac{1}{2}{\mathrm{\nabla }}_{\ell }R.$

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

${\mathrm{\nabla }}_{\ell }{R}^{\ell }{}_m=\frac{1}{2}{\mathrm{\nabla }}_{m}R.$

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). 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