Jinroh
Jun11-04, 03:08 PM
Given the definition of the covariant derivation of one order tensor :
\nabla _j v_k = \partial _j v_k - v_i \Gamma _{kj}^i
How can we proove the covariant derivation of the Riemann-Christoffel tensor given by :
R_{i{\text{ }},{\text{ }}jk}^l = \partial _j \Gamma _{ik}^l - \partial _k \Gamma _{ij}^l + \Gamma _{ik}^r \Gamma _{jr}^l - \Gamma _{ij}^r \Gamma _{kr}^l
is :
\nabla _t R_{i,rs}^l = \partial _{rt} \Gamma _{si}^l - \partial _{st} \Gamma _{ri}^l
This could not be difficult but i'm afraid to make an error with the indices.
This proof will help me to define the Einstein tensor.
Thanks for your help.
\nabla _j v_k = \partial _j v_k - v_i \Gamma _{kj}^i
How can we proove the covariant derivation of the Riemann-Christoffel tensor given by :
R_{i{\text{ }},{\text{ }}jk}^l = \partial _j \Gamma _{ik}^l - \partial _k \Gamma _{ij}^l + \Gamma _{ik}^r \Gamma _{jr}^l - \Gamma _{ij}^r \Gamma _{kr}^l
is :
\nabla _t R_{i,rs}^l = \partial _{rt} \Gamma _{si}^l - \partial _{st} \Gamma _{ri}^l
This could not be difficult but i'm afraid to make an error with the indices.
This proof will help me to define the Einstein tensor.
Thanks for your help.