How can we prove the covariant derivation of the Riemann-Christoffel tensor?

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In summary, the conversation discusses the definition of the covariant derivation of one order tensor and the proof of the covariant derivation of the Riemann-Christoffel tensor. The proof is necessary for defining the Einstein tensor and the poster is seeking help in the Special and General Relativity Room. They also mention a separate question about a geometric interpretation of the Laplacian on a manifold.
  • #1
Jinroh
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Given the definition of the covariant derivation of one order tensor :

[tex]\nabla _j v_k = \partial _j v_k - v_i \Gamma _{kj}^i[/tex]

How can we proove the covariant derivation of the Riemann-Christoffel tensor given by :


[tex]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[/tex]

is :

[tex]\nabla _t R_{i,rs}^l = \partial _{rt} \Gamma _{si}^l - \partial _{st} \Gamma _{ri}^l[/tex]

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.
 
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  • #2
hum...

perhaps by moving this post in Special and General Relativity Room I will have an answer... ?
 
  • #3
Jinroh said:
perhaps by moving this post in Special and General Relativity Room I will have an answer... ?

Maybe. I was looking at your problem last night, and I think I should be able to have something for you by this evening or tomorrow morning (party tonight, so no guarantees for today...)

However, while we're on the topic of soliciting responses, do you have any ideas about my question of a geometric interpretation of the Laplacian on a manifold? How's that for a shameless plug!
 

FAQ: How can we prove the covariant derivation of the Riemann-Christoffel tensor?

What is the Third Bianchi Identity?

The Third Bianchi Identity is a mathematical expression that relates the covariant derivatives of a symmetric tensor. It is also known as the Bianchi Identity of the third kind or the cyclic identity.

How is the Third Bianchi Identity used in physics?

The Third Bianchi Identity is used in physics to simplify equations and calculations involving symmetric tensors, such as the stress-energy tensor in general relativity. It is also used in the study of fluid dynamics and elasticity.

What is the significance of the Third Bianchi Identity?

The Third Bianchi Identity is significant because it is a fundamental mathematical expression that governs the behavior of symmetric tensors. It is also important in the development of mathematical models in physics and engineering.

How is the Third Bianchi Identity derived?

The Third Bianchi Identity is derived from the properties of the Riemann curvature tensor and its symmetries. It can also be derived using the properties of the covariant derivative and its commutation with other derivatives.

Are there any other Bianchi Identities?

Yes, there are three Bianchi Identities in total - the First, Second, and Third. They are all mathematical expressions that relate the covariant derivatives of tensors and play an important role in various fields of physics and mathematics.

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