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