(this question was already done on the physics stackexchange but it will probably get closed for some reason I still can't understand)
I need some help with the divergence of the time component of the stress energy tensor for dust $\nabla_{\mu}T^{0\mu}$ given the stress energy tensor for dust is $T^{\mu\nu}=(\rho+p)u^{\mu}u^{\nu} + g_{\mu\nu}p$. According to the internet, this is, using the FLRW metric (defined as $ds^2=-dt^2+a(t)^2(dr^2/(1-kr^2)+r^2d\Omega^2$ with $d\Omega^2= d\theta^2=sin^2(\theta) d\varphi^2$) :
$$\nabla_{\mu}T^{0\mu}=\dot{\rho} +3(\rho +p)\frac{\dot{a}}{a}$$
I have no idea how should I proceed but I'll show my progress:
Using the definition of the Stress-energy tensor for dust and knowing that the only non zero component in this case is $T^{00}$ we have:
$$\nabla_{\nu}T^{0\nu}=(\nabla_{\nu}\rho+\nabla_{\nu}p)u^{0}u^{\nu} + g_{0\nu}\nabla_{\nu}p=\nabla_0T^{00}$$
Using the definition for covariant derivatives and given that the partial derivative of $g_{00}$ wrt time is 0, we have the Christoffel symbols
$\Gamma^r_{rt}=\frac{\dot{a}}{a}=\Gamma^{\theta}_{\theta t}=\Gamma^{\varphi}_{\varphi t}$
That we can place in the definition of the covariant derivative:
$$\nabla_0T^{00}=\dot \rho +3\rho\frac{\dot{a}}{a}$$
Where or how can I find the p component?