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Contracted Bianchi Identities

Contracting tensors lambda with nu in the Bianchi identities

R_(lambdamunukappa;eta)+R_(lambdamuetanu;kappa)+R_(lambdamukappaeta;nu)=0
(1)

gives

R_(mukappa;eta)-R_(mueta;kappa)+R^nu_(mukappaeta;nu)=0.
(2)

Contracting again,

R_(;eta)-R^mu_(eta;mu)-R^nu_(eta;nu)=0,
(3)

or

(R^mu_eta-1/2delta^mu_etaR)_(;mu)=0,
(4)

or

(R^(munu)-1/2g^(munu)R)_(;mu)=0.
(5)

See also

Bianchi Identities

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Cite this as:

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

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