Physics: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
- ↑ "Sui simboli a quattro indici e sulla curvatura di Riemann" (in Italian), Rend. Acc. Naz. Lincei 11 (5): 3–7, 1902, https://archive.org/stream/rendiconti51111902acca#page/n9/mode/2up
- ↑ Voss, A. (1880), "Zur Theorie der Transformation quadratischer Differentialausdrücke und der Krümmung höherer Mannigfaltigketien", Mathematische Annalen 16 (2): 129–178, doi:10.1007/bf01446384, https://zenodo.org/record/2440927
- ↑ Synge J.L., Schild A. (1949). Tensor Calculus. pp. 87–89–90.
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
![]() | Original source: https://en.wikipedia.org/wiki/Contracted Bianchi identities.
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