 #1
 109
 14
Homework Statement:

Given the FRW metric
$$d s^{2}=d t^{2}+a(t)^{2}\left[\frac{d r^{2}}{1k r^{2}}+r^{2}\left(d \theta^{2}+\sin ^{2} \theta d \varphi^{2}\right)\right]$$
and the Stressenergy tensor of an ideal fluid,
$$T_{a b}=(P+\rho) u_{a} u_{b}+P g_{a b}, \quad \text{where}\quad P=\omega \rho\quad \text{and}\quad u^a:=\frac{dx^a}{d\tau},$$
My Prof. claimed during yesterdays lecture that one can show that the zero component of the convervation equation of ##T_{ab}## results in
$$\frac{d}{d t} \log \rho\propto\frac{d}{d t} \log a(t)\tag{1}.$$
I've been trying to show this now for a while and I just can't figure it out...
Relevant Equations:
 All given above.
My attempt:
I suspect that my problem lies in the fact that I'm not sure if
 Realize we can work in whatever coordinate system we want, therefore we might as well work in the rest frame of the fluid. In this case ##u^a=(c,\vec{0})##.
 The conservation law reads ##\nabla^a T_{ab}=0##. Let us pick the LeviCivita connection so that we don't have to worry about ##\nabla g##. We then get $$\begin{align*}\nabla^a T_{ab} &= \nabla^a\left((1+\omega)\rho u_{a} u_{b}+\omega\rho g_{a b}\right)\\&= (1+\omega)\nabla^a(\rho u_au_b)+ \omega g_{ab}\nabla^a\rho\\&\overset{(*)}{=} (1+\omega) u_au_b\nabla^a\rho+ \omega g_{ab}\nabla^a\rho\\&= \left((1+\omega) u_au_b+ \omega g_{ab}\right)\nabla^a\rho,\end{align*}$$ where in ##(*)## we use the fact that ##u^a## are constant in the comoving frame.
 For the zero component I now get $$\begin{align*}\nabla_a T^{a0} &= \left((1+\omega) u^au^0+ \omega g^{a0}\right)\nabla_a\rho = \left((1+\omega)c u^a+ \omega g^{a0}\right)\nabla_a\rho\\&= (1+\omega)c^2 \nabla_0\rho \omega \nabla_0\rho\\&= (1+\omega)c^2 \partial_t\rho \omega \partial_t\rho\\&\overset{!}{=}0 \end{align*}$$ I don't really think there is a way to get to the claimed result from here, since I'm completely missing the scale factor ##a##.
I suspect that my problem lies in the fact that I'm not sure if
 I'm actually allowed to pick a specific frame of reference and work in it and
 I understand correctly how the covariant derivative acts on scalars (##\rho##) and constant values (##u^a##, since they are only zero or ##c## in the comoving frame).