Here is a simple approach that might work. Start by defining

$$F_{\mu\nu} \equiv G_{\mu\nu} - T_{\mu\nu}$$

where $G_{\mu\nu} = R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R$ is the Einstein tensor. Now from what you know we have
$$F_{00} = 0$$

$$\nabla^{\mu}F_{\mu0} = 0$$

You must show that $F_{\mu\nu} = 0$. Writing out the last equation gives

$$0 = \partial_{i} F^{i 0} + \Gamma^{\mu}_{\mu \alpha}F^{\alpha 0} + \Gamma^{0}_{\mu \alpha}F^{\mu \alpha}$$

Homogenous and isotropic implies that gradients vanish and that $F^{11}=F^{22}=F^{33}$ so

$$0 = a^2 H \delta_{ij} F^{i j}$$

This shows that $F_{00} = F_{11} = F_{22} = F_{33} = 0$. Now to show $F_{ij} = 0$ for $i\not =j$ you might need some additional assumptions on the energy momentum tensor $T^{\mu\nu}$, for example $T^{ij} = 0$.

This post imported from StackExchange Physics at 2014-06-21 08:52 (UCT), posted by SE-user Winther