Contracted Bianchi identities

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

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

where $\nabla _{\rho }$ indicates covariant differentiation.

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

These identities are named after Luigi Bianchi.

See also

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

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