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I would like help showing that the tensor, $$K_{\mu\nu}=a^2(g_{\mu\nu}+u_\mu u_\nu),$$ where $u^\mu =(1,0,0,0)$, is a Killing tensor of the spatially flat FRW metric,

$$ds^2=-dt^2+a(t)^2\left(dr^2+d\Omega^2\right).$$

Specifically it must satisfy $$\nabla_{(a}K_{\mu\nu)}=0.$$

I can see that the tensor is basically $a^2\times(\text{spatial projection matrix})$, but not sure if there is a trick or symmetry argument to show it is Killing?

Only source I can find is Carroll pg 344, claiming it is easy to check.

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