Possible contractions of the Riemann curvature tensor

Question

The Riemann curvature tensor is R_αβγδ. I'm curious, what are the other (apparently 12) possible contractions?

Contractions meaning raising any index and contracting with any other (4 choices for raised each followed by 3 choices of lower index). Also if you could show me how you got the contractions, thank you

Answer 1

So, there are 4 possible ways to raise the index in the R tensor, using the contravariant metric tensor g^ij_, resulting in a mixed tensor of rank 4 with 1 superscript and 3 subscripts. Then, for each of these resulting tensors, you can contract the superscript with any of the 3 subscript. For example, for the first index, α:

 

R^σ_βγδ= g^ασ R_αβγδ

 

Now, you can contract sigma with each of β, γ, and δ. This makes 3 results.

 

Then, you can repeat this with each of the other indices in the R tensor (with β, γ, and then δ). There will be a total of 12 results.

 

Hope this helps.

 

- Ben