8.2 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Contracted Bianchi identities | 1/1 | https://en.wikipedia.org/wiki/Contracted_Bianchi_identities | reference | science, encyclopedia | 2026-05-05T12:04:39.641021+00:00 | kb-cron |
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