Contracted Bianchi identities

Short description: Identities in general relativity

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

∇ρRρμ=12∇μR

where Rρμ is the Ricci tensor, R the scalar curvature, and ∇ρ indicates covariant differentiation.

These identities are named after Luigi Bianchi, although they had been already derived by Aurel Voss in 1880,[2] and independently by Gregorio Ricci-Curbastro in 1889.[3]: 275–6 In the Einstein field equations, the contracted Bianchi identity ensures consistency with the vanishing divergence of the matter stress–energy tensor.

## Contents

* 1 Proof
* 2 See also
* 3 Notes
* 4 References

## Proof

Start with the Bianchi identity[4]

Rabmn;ℓ+Rabℓm;n+Rabnℓ;m=0.

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

gbngam(Rabmn;ℓ+Rabℓm;n+Rabnℓ;m)=0,

gbn(Rmbmn;ℓ−Rmbmℓ;n+Rmbnℓ;m)=0,

gbn(Rbn;ℓ−Rbℓ;n−Rbmnℓ;m)=0,

Rnn;ℓ−Rnℓ;n−Rnmnℓ;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;ℓ−Rnℓ;n−Rmℓ;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;ℓ=2Rmℓ;m,

which is the same as

∇mRmℓ=12∇ℓR.

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

∇ℓRℓm=12∇mR.

## See also

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

## Notes

1. ↑ "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
2. ↑ 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
3. ↑ Pais, Abraham (1982). 'Subtle is the Lord..': The Science and the Life of Albert Einstein. Oxford: Oxford University Press. ISBN 0-19-853907-X.
4. ↑ Synge J.L., Schild A. (1949). Tensor Calculus. pp. 87–89–90.

## 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

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