General relativity. Bianchi identities

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SUMMARY

The discussion focuses on proving that the divergence of the Einstein tensor is zero using the Bianchi identities. The participant seeks clarification on the transition from equation 2 to equation 3, specifically regarding the contraction of indices and the application of rules governing index manipulation. Additionally, they inquire about the distribution of the metric tensor in raising indices as shown in equations 5 and 6. The primary resource referenced is "Gravitation" by Misner, Thorne, and Wheeler.

PREREQUISITES
  • Understanding of Bianchi identities in general relativity
  • Familiarity with the Einstein tensor and its properties
  • Knowledge of tensor algebra and index notation
  • Basic concepts of covariant derivatives and metric tensors
NEXT STEPS
  • Study the derivation of the Bianchi identities in the context of general relativity
  • Learn about the properties of the Einstein tensor and its divergence
  • Explore tensor contraction rules and their applications in general relativity
  • Review the role of the metric tensor in raising and lowering indices
USEFUL FOR

Students and researchers in theoretical physics, particularly those studying general relativity and tensor calculus, will benefit from this discussion.

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Homework Statement



I have a problem. I need to prove that the divergence of Einstein tensor is 0 using the bianchi identities. I have looked to several sources and I have derived an answer, but I don't fully understand some steps.

Homework Equations


I have uploaded a document which shows a part of the solution, because I have no idea hot to write them in the post. The solution starts with eq0.
I don't understand how did they get from eq 2 to eq 3. It seems like some sort of contraction in the index a and u, but what kind or rule/law/equation is that? If it's a contraction between first and third slots, why didnt they contract also the third term in eq 2?
My second question is, when raising an index, how the metric in equation 5 is distributed to get eq6? It seems that the metric is distributed like simple algebra. Why is it like that? How the covariant derivative affect the index's raising.
If you can help me with those two questions, then I will be able to understand the rest.
Please be kind since I am very new at this stuff. The book I am using is gravitation by misner thorne wheeler.

The Attempt at a Solution

 

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I have attempted to answer the questions by looking at various sources. However, I am still not sure how to answer them.
 

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