Contraction of the Bianchi identity

[whatisreality]

Homework Statement

I've been given the Bianchi identity in the form

$\nabla _{\kappa} R^{\mu}_{\nu\rho\sigma} + \nabla _{\rho} R^{\mu}_{\nu\sigma \kappa} + \nabla _{\sigma} R^{\mu}_{\nu\kappa\rho} =0$

Homework Equations

The Attempt at a Solution

In order to get from this to the Einstein tensor, I've seen online that you perform a contraction so you get the Ricci tensor and go from there. No two of my indices are the same though, can I just relabel an index? How does the contraction work? Apologies, the bottom three indices should be offset to the right but I can't make latex do it.

 

[Orodruin]

You can put a contravariant index to the samd as a covariant one and sum over them. This is what it means to make a contraction and it results in a tensor of two ranks lower. The canonical example of this being the product of a tangent and a dual vector.

To be honest, if you are expecting to learn more advanced subjects and applications, you should make sure that you master the basics first. Based on your question, I would suggest going back to repeat the basics of tensor analysis rather than trying to understand curvature and the Einstein tensor. It will serve you better in the long run.

 

[whatisreality]

When I say how does the contraction work, that makes it sound worse than it is, I think. I do know what a contraction is, what I wanted to double check is that I can relabel a contravariant index to be the same as the covariant. I'll definitely go back and re-read my notes though, it's always worthwhile. Thank you, I appreciate your help!