Contracting Tensors: Multiply by g^αρgασ?

[pleasehelpmeno]

When contracting $R^{\sigma}_{ \mbox{ }\mu\nu\rho}$ to $R_{\mu\nu}$

Should one multiply by $g^{\alpha\rho}g_{\alpha\sigma}$? I often get confused with ordering of indices and such like

 

[Fredrik]

Why are you starting a third thread about this?

https://www.physicsforums.com/showthread.php?t=678141
https://www.physicsforums.com/showthread.php?t=678235

 

[pleasehelpmeno]

there was no real answer, ignore the ricci tensor i just mean in general then, I don't really understand the order in which the indices should go, if it even matters at all.

 

[Fredrik]

If you're not satisfied with the answers, you should say that in the same thread, not start a new one.

 

[blue_raver22]

I understand your confusion with the ordering of indices when contracting tensors. To answer your question, yes, you should multiply by g^{\alpha\rho}g_{\alpha\sigma} when contracting R^{\sigma}_{ \mbox{ }\mu\nu\rho} to R_{\mu\nu}. This is known as the metric tensor and it is used to raise and lower indices in tensor equations. In this case, the g^{\alpha\rho} term will raise the lower index \rho in R^{\sigma}_{ \mbox{ }\mu\nu\rho}, while the g_{\alpha\sigma} term will lower the upper index \sigma in R_{\mu\nu}. This will result in the indices being in the correct order for the contraction to be performed. It is important to pay attention to the ordering of indices in tensor equations, as it can greatly affect the final result. I suggest practicing with simpler tensor equations to become more familiar with the ordering of indices and the use of the metric tensor.