Bianchi Identities

The covariant derivative of the Riemann tensor is given by


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


which can be written concisely as


(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

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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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.

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