In general relativity and tensor calculus, the contracted Bianchi identities are:
∇ ρ R ρ μ = 1 2 ∇ μ R {\displaystyle \nabla _{\rho }{R^{\rho }}_{\mu }={1 \over 2}\nabla _{\mu }R}
where R ρ μ {\displaystyle {R^{\rho }}_{\mu }} is the Ricci tensor, R {\displaystyle R} the scalar curvature, and ∇ ρ {\displaystyle \nabla _{\rho }} indicates covariant differentiation. These identities are named after Luigi Bianchi, although they had been already derived by Aurel Voss in 1880, and independently by Gregorio Ricci-Curbastro in 1889. In the Einstein field equations, the contracted Bianchi identity ensures consistency with the vanishing divergence of the matter stress–energy tensor.
Proof Start with the Bianchi identity
R a b m n ; ℓ + R a b ℓ m ; n + R a b n ℓ ; m = 0. {\displaystyle R_{abmn;\ell }+R_{ab\ell m;n}+R_{abn\ell ;m}=0.}
Contract both sides of the above equation with a pair of metric tensors:
g b n g a m ( R a b m n ; ℓ + R a b ℓ m ; n + R a b n ℓ ; m ) = 0 , {\displaystyle g^{bn}g^{am}(R_{abmn;\ell }+R_{ab\ell m;n}+R_{abn\ell ;m})=0,}
g b n ( R m
b m n ; ℓ − R m
b m ℓ ; n + R m
b n ℓ ; m ) = 0 , {\displaystyle g^{bn}(R^{m}{}_{bmn;\ell }-R^{m}{}_{bm\ell ;n}+R^{m}{}_{bn\ell ;m})=0,}
g b n ( R b n ; ℓ − R b ℓ ; n − R b
m
n ℓ ; m ) = 0 , {\displaystyle g^{bn}(R_{bn;\ell }-R_{b\ell ;n}-R_{b}{}^{m}{}_{n\ell ;m})=0,}
R n
n ; ℓ − R n
ℓ ; n − R n m
n ℓ ; m = 0. {\displaystyle R^{n}{}_{n;\ell }-R^{n}{}_{\ell ;n}-R^{nm}{}_{n\ell ;m}=0.}
The first term on the left contracts to yield a Ricci scalar, while the third term contracts to yield a mixed Ricci tensor,
R ; ℓ − R n
ℓ ; n − R m
ℓ ; m = 0. {\displaystyle R_{;\ell }-R^{n}{}_{\ell ;n}-R^{m}{}_{\ell ;m}=0.}
The last two terms are the same (changing dummy index n to m) and can be combined into a single term which shall be moved to the right,
R ; ℓ = 2 R m
ℓ ; m , {\displaystyle R_{;\ell }=2R^{m}{}_{\ell ;m},}
which is the same as
∇ m R m
ℓ = 1 2 ∇ ℓ R . {\displaystyle \nabla _{m}R^{m}{}_{\ell }={1 \over 2}\nabla _{\ell }R.}
Swapping the index labels l and m on the left side yields
∇ ℓ R ℓ
m = 1 2 ∇ m R . {\displaystyle \nabla _{\ell }R^{\ell }{}_{m}={1 \over 2}\nabla _{m}R.}
See also
Notes
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. {{cite book}}: ISBN / Date incompatibility (help) 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