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Bianchi Identities

The covariant derivative of the Riemann tensor is given by

R_(lambdamunukappa;eta)=1/2partial/(partialx^eta)((partial^2g_(lambdanu))/(partialx^kappapartialx^mu)-(partial^2g_(munu))/(partialx^kappapartialx^lambda)-(partial^2g_(lambdakappa))/(partialx^mupartialx^nu)+(partial^2g_(mukappa))/(partialx^nupartialx^lambda)).
(1)

Permuting nu, kappa, and eta (Weinberg 1972, pp. 146-147) gives the Bianchi identities

R_(lambdamunukappa;eta)+R_(lambdamuetanu;kappa)+R_(lambdamukappaeta;nu)=0,
(2)

which can be written concisely as

R^alpha_(beta[lambdamu;nu])=0
(3)

(Misner et al. 1973, p. 221), where T_([a_1...a_n]) denoted the antisymmetric tensor part. Wald (1984, p. 39) calls

del _([a)R_(bc]d)^e=0
(4)

the Bianchi identity, where del is the covariant derivative, and R_(abc)^d is the Riemann tensor.

See also

Contracted Bianchi Identities, Einstein Field Equations, Ricci Curvature Tensor, Riemann Tensor

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References

Misner, C. W.; Thorne, K. S.; and Wheeler, J. A. Gravitation. San Francisco: W. H. Freeman, 1973.Wald, R. M. General Relativity. Chicago, IL: University of Chicago Press, 1984.Weinberg, S. Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity. New York: Wiley, 1972.

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Bianchi Identities

Cite this as:

Weisstein, Eric W. "Bianchi Identities." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BianchiIdentities.html

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