Английская Википедия:Contracted Bianchi identities
In general relativity and tensor calculus, the contracted Bianchi identities are:[1]
- <math> \nabla_\rho {R^\rho}_\mu = {1 \over 2} \nabla_{\mu} R</math>
where <math>{R^\rho}_\mu</math> is the Ricci tensor, <math>R</math> the scalar curvature, and <math>\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> 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> g^{bn} g^{am} (R_{abmn;\ell} + R_{ab\ell m;n} + R_{abn\ell;m}) = 0,</math>
- <math> g^{bn} (R^m {}_{bmn;\ell} - R^m {}_{bm\ell;n} + R^m {}_{bn\ell;m}) = 0,</math>
- <math> g^{bn} (R_{bn;\ell} - R_{b\ell;n} - R_b {}^m {}_{n\ell;m}) = 0,</math>
- <math> 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> 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> R_{;\ell} = 2 R^m {}_{\ell;m},</math>
which is the same as
- <math> \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> \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
Шаблон:Mathematics-stub
Шаблон:Relativity-stub