View source for Contracted Bianchi identities

{{Short description|Identities}}
In [[general relativity]] and [[tensor calculus]], the '''contracted Bianchi identities''' are:<ref>{{Citation
|author-first=Luigi
|author-last=Bianchi
|author-link =Luigi Bianchi
|title = Sui simboli a quattro indici e sulla curvatura di Riemann
|trans-title=
|journal = Rend. Acc. Naz. Lincei
|volume =11
|issue=5
|pages =3–7
|year =1902
|language =Italian
|url =https://archive.org/stream/rendiconti51111902acca#page/n9/mode/2up
|doi =
|jfm =
}}</ref>
:<math> \nabla_\rho {R^\rho}_\mu = {1 \over 2} \nabla_{\mu} R</math>
where <math>{R^\rho}_\mu</math> is the [[Ricci tensor]], <math>R</math> the [[scalar curvature]], and <math>\nabla_\rho</math> indicates [[covariant differentiation]].
These identities are named after [[Luigi Bianchi]], although they had been already derived by [[Aurel Voss]] in 1880.<ref name="voss" >{{citation|title=Zur Theorie der Transformation quadratischer Differentialausdrücke und der Krümmung höherer Mannigfaltigketien |last=Voss|first=A.|author-link=Aurel Voss|journal=Mathematische Annalen|volume=16|pages=129–178|year=1880|issue=2 |url=https://zenodo.org/record/2440927|doi=10.1007/bf01446384|s2cid=122828265}}</ref> 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 [[Curvature form#Bianchi identities|Bianchi identity]]<ref name="syngeandschild">{{cite book |author=Synge J.L., Schild A.|title=Tensor Calculus|year=  1949|pages=87–89–90}}</ref>
:<math> R_{abmn;\ell} + R_{ab\ell m;n} + R_{abn\ell;m} = 0.</math><!-- The symbols \,\! exist to force PNG rendering; do not remove them. -->
[[Tensor contraction|Contract]] both sides of the above equation with a pair of [[metric tensor]]s:
:<math>  g^{bn} g^{am} (R_{abmn;\ell} + R_{ab\ell m;n} + R_{abn\ell;m}) = 0,</math>
:<math> g^{bn} (R^m {}_{bmn;\ell} - R^m {}_{bm\ell;n} + R^m {}_{bn\ell;m}) = 0,</math>
:<math> g^{bn} (R_{bn;\ell} - R_{b\ell;n} - R_b {}^m {}_{n\ell;m}) = 0,</math>
:<math> R^n {}_{n;\ell} - R^n {}_{\ell;n} - R^{nm} {}_{n\ell;m} = 0.</math>
The first term on the left contracts to yield a Ricci scalar, while the third term contracts to yield a [[mixed tensor|mixed]] Ricci tensor,
:<math> R_{;\ell} - R^n {}_{\ell;n} - R^m {}_{\ell;m} = 0.</math>
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,
:<math> R_{;\ell} = 2 R^m {}_{\ell;m},</math>
which is the same as
:<math> \nabla_m R^m {}_\ell = {1 \over 2} \nabla_\ell R.</math>
Swapping the index labels ''l'' and ''m'' on the left side yields
:<math> \nabla_\ell R^\ell {}_m = {1 \over 2} \nabla_m R.</math>
==See also==
{{div col}}
* [[Bianchi identities]]
* [[Einstein tensor]]
* [[Einstein field equations]]
* [[General theory of relativity]]
* [[Ricci calculus]]
* [[Tensor calculus]]
* [[Riemann curvature tensor]]
{{div col end}}
==Notes==
{{Reflist}}
==References==
*{{cite book
| last = Lovelock
| first = David
|author2=Hanno Rund
| title = Tensors, Differential Forms, and Variational Principles
| year=  1989
| publisher = Dover
| isbn = 978-0-486-65840-7
| orig-date = 1975
}}
* {{cite book |author=Synge J.L., Schild A. |title=Tensor Calculus |publisher=first Dover Publications 1978 edition |year=1949 |isbn=978-0-486-63612-2 |url-access=registration |url=https://archive.org/details/tensorcalculus00syng }}
* {{citation | author=J.R. Tyldesley| title = An introduction to Tensor Analysis: For Engineers and Applied Scientists| publisher=Longman| year=1975|isbn=0-582-44355-5}}
* {{citation | author=D.C. Kay| title = Tensor Calculus| publisher=Schaum’s Outlines, McGraw Hill (USA)|year=1988|isbn=0-07-033484-6}}
* {{citation | author=T. Frankel| title = The Geometry of Physics| publisher=Cambridge University Press|edition=3rd|year=2012|isbn=978-1107-602601}}
[[Category:Concepts in physics]]
[[Category:Tensors]]
[[Category:General relativity]]
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