Why Are Off-Diagonal Ricci Tensors Zero in Symmetric FRW Metrics?

[pleasehelpmeno]

Hi does anyone know a formal definition for why off diagonal ricci tensors are equal to zero in a symmetric standard FRW metric?

 

[bcrowell]

Is this in a particular set of coordinates?

 

[pleasehelpmeno]

yeah sorry (t,r,\theta,\phi)

 

[bcrowell]

I think it follows from symmetry. For instance, $R_{\theta t}$ flips signs under a change of coordinates $\theta\rightarrow \pi-\theta$, so if it was nonzero it would violate isotropy. Under the same transformation $R_{\theta\theta}$ stays the same.