general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics. ...

and

tensor calculus In mathematics, tensor calculus, tensor analysis, or Ricci calculus is an extension of vector calculus to tensor fields (tensors that may vary over a manifold, e.g. in spacetime). Developed by Gregorio Ricci-Curbastro and his student Tullio Lev ...

, the contracted Bianchi identities are: :

\nabla_\rho _\mu =  \nabla_ R

where

_\mu

is the

Ricci tensor In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measur ...

R

the

scalar curvature In the mathematical field of Riemannian geometry, the scalar curvature (or the Ricci scalar) is a measure of the curvature of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the geometry ...

, and

\nabla_\rho

indicates covariant differentiation. These identities are named after

Luigi Bianchi Luigi Bianchi (18 January 1856 – 6 June 1928) was an Italian mathematician. He was born in Parma, Emilia-Romagna, and died in Pisa. He was a leading member of the vigorous geometric school which flourished in Italy during the later years of the ...

, although they had been already derived by Aurel Voss in 1880. In the

Einstein field equations In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it. The equations were published by Einstein in 1915 in the form ...

, the contracted Bianchi identity ensures consistency with the vanishing divergence of the matter

stress–energy tensor The stress–energy tensor, sometimes called the stress–energy–momentum tensor or the energy–momentum tensor, is a tensor physical quantity that describes the density and flux of energy and momentum in spacetime, generalizing the stress t ...

Proof

Start with the Bianchi identity :

R_ + R_ + R_ = 0.

Contract A contract is a legally enforceable agreement between two or more parties that creates, defines, and governs mutual rights and obligations between them. A contract typically involves the transfer of goods, services, money, or a promise to ...

both sides of the above equation with a pair of

metric tensor In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allo ...

s: :

g^ g^ (R_ + R_ + R_) = 0,

g^ (R^m _ - R^m _ + R^m _) = 0,

g^ (R_ - R_ - R_b ^m _) = 0,

R^n _ - R^n _ - R^ _ = 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_ - R^n _ - R^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_ = 2 R^m _,

which is the same as :

\nabla_m R^m _\ell =  \nabla_\ell R.

Swapping the index labels ''l'' and ''m'' yields :

\nabla_\ell R^\ell _m =  \nabla_m R.

Notes

References

* * * * * Concepts in physics Tensors General relativity {{relativity-stub

Proof

See also

Notes

References