- See also
- Notes
- References
In general relativity and tensor calculus, the contracted Bianchi identities are[1]:
where 
is the Ricci tensor, 
the scalar curvature, and 
indicates covariant differentiation.
A proof can be found in the entry Proofs involving covariant derivatives.
These identities are named after Luigi Bianchi, although they had been already derived by Aurel Voss in 1880.[2]
See also
- Einstein tensor
- Ricci calculus
- Tensor calculus
- Riemann curvature tensor
Notes
1. ^{{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 = }}
2. ^{{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|url=}}
2. ^{{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|url=}}
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
| origyear = 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}}
- {{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}}
3 : Concepts in physics|Tensors|General relativity
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