(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Show that all Robertson - Walker models are conformally flat.

2. Relevant equations

Robertson Walker Metric: [itex]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}[/itex]

Ricci Tensor: [itex]R_{\alpha\beta}=2Kg_{\alpha\beta}[/itex]

Ricci Scalar: [itex]R=8K[/itex]

Weyl Tensor: [itex]C_{\alpha\beta\gamma\delta}=R_{\alpha\beta\gammaÎ´}-\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})[/itex]

3. 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:

[itex]C_{\alpha\beta\gammaÎ´}=R_{\alpha\beta\gamma\delta}-\frac{2}{3}K(g_{\alpha\gamma}g_{\beta\delta}-g_{\alpha\delta}g_{\beta\gamma})[/itex]

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!

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# Homework Help: Proving the weyl tensor is zero problem

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