Question

Why is the Ricci tensor diagonal for isotropic spacetime?

Answer 1

Zee's Einstein Gravity in a Nutshell and while calculating the Ricci tensor for FRW spacetime he claims that because the spacelike slices of constant tt are rotationally invariant, the off-diagonal elements of the Ricci tensor vanish (working in tt, rr, θθ, ϕϕ coordinates). Without calculating the Ricci tensor from the Christoffel symbols element-by-element, can someone provide an explanation of why this is true? More generally is there a good way to see how symmetries of the metric, like rotation or time translation, et