Contracted Bianchi identities

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

${\displaystyle \nabla _{\rho }{R^{\rho }}_{\mu }={1 \over 2}\nabla _{\mu }R,}$

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

A proof can be found in the entry Proofs involving covariant derivatives.

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.

See also

• Bianchi identities
• Einstein tensor
• Ricci calculus
• Tensor calculus
• Riemann curvature tensor

Notes

1. Bianchi, Luigi (1902), "Sui simboli a quattro indici e sulla curvatura di Riemann", Rend. Acc. Naz. Lincei (in Italian), 11 (5): 3–7
2. Voss, A. (1880), "Zur Theorie der Transformation quadratischer Differentialausdrücke und der Krümmung höherer Mannigfaltigketien", Mathematische Annalen, 16: 129–178, doi:10.1007/bf01446384, S2CID 122828265

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.
• 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