# Proving the weyl tensor is zero problem

## Homework Statement

Show that all Robertson - Walker models are conformally flat.

## Homework Equations

Robertson Walker Metric: $ds^{2}=a^{2}(t)\left(\frac{dr^{2}}{1-Kr^{2}}+r^{2}(d\theta^{2}+(sin\theta)^{2}d\phi^{2} )\right)-dt^{2}$

Ricci Tensor: $R_{\alpha\beta}=2Kg_{\alpha\beta}$

Ricci Scalar: $R=8K$

Weyl Tensor: $C_{\alpha\beta\gamma\delta}=R_{\alpha\beta\gammaδ}-\frac{1}{2}(g_{\alpha\gamma}R_{\beta\delta}-g_{\alpha\delta}R_{\beta\gamma}-g_{\beta\gamma}R_{\alpha\delta}+g_{\beta\delta}R_{\alpha\gamma}) + \frac{R}{6}(g_{\alpha\gamma}g_{\beta\delta}-g_{\alpha\delta}g_{\beta\gamma})$

## The Attempt at a Solution

In order for the models to be conformally flat the weyl tensor must vanish, therefore that is what I have tried to show. By subbing in the values for the Ricci tensor and the Ricci scalar (both of which were given in a lecture by my professor) I arrived at the following expression:

$C_{\alpha\beta\gammaδ}=R_{\alpha\beta\gamma\delta}-\frac{2}{3}K(g_{\alpha\gamma}g_{\beta\delta}-g_{\alpha\delta}g_{\beta\gamma})$

However, as you can see I am left with the Riemann tensor undefined and I cannot show the weyl tensor to be zero. Any help is greatly appreciated, thanks!

$R_{ij} = R^k_{\, ikj}$,