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Contracted Bianchi identities

{{Short description|Identities in general relativity}}

In general relativity and tensor calculus, the contracted Bianchi identities are:{{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 =

}}

: \nabla_\rho {R^\rho}_\mu = {1 \over 2} \nabla_{\mu} R

where {R^\rho}_\mu is the Ricci tensor, R the scalar curvature, and \nabla_\rho indicates covariant differentiation.

These identities are named after Luigi Bianchi, although they had been already derived by Aurel Voss in 1880.{{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}} 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{{cite book |author=Synge J.L., Schild A.|title=Tensor Calculus|year= 1949|pages=87–89–90}}

: 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^{bn} g^{am} (R_{abmn;\ell} + R_{ab\ell m;n} + R_{abn\ell;m}) = 0,

: g^{bn} (R^m {}_{bmn;\ell} - R^m {}_{bm\ell;n} + R^m {}_{bn\ell;m}) = 0,

: g^{bn} (R_{bn;\ell} - R_{b\ell;n} - R_b {}^m {}_{n\ell;m}) = 0,

: 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_{;\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_{;\ell} = 2 R^m {}_{\ell;m},

which is the same as

: \nabla_m R^m {}_\ell = {1 \over 2} \nabla_\ell R.

Swapping the index labels l and m on the left side yields

: \nabla_\ell R^\ell {}_m = {1 \over 2} \nabla_m R.

See also

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Notes

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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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