GR: Quick Q, conservation energy, stuck on algebra

[binbagsss]

I'm stuck on deriving the follow. Context: FRW Metric, dust dominated universe, use of conservation of energy-momentum tensor. I'll now state all the equations I can see are relevant. (But just in case the source is http://arxiv.org/pdf/gr-qc/9712019.pdf and it's eq 8.43).

$\frac{d}{dt}(\rho a^{3}) = a^{3}(\dot{\rho}+3\rho\frac{\dot{a}}{a}) = -3pa^{2}\dot{a}$, and it's the last equation I am stuck on. I am getting it to be $-3 \rho a^{2}\dot{a}$...

2. Homework Equations :

$0 = \bigtriangledown_{a}T^{a}_{0}=-\partial_{0}\rho=3\frac{\dot{a}}{a}(\rho +p)$ [1]

and using $p=w\rho$
this becomes:

$\frac{dot{\rho}}{\rho}=-3(1+w)\frac{\dot{a}}{a}$, [2]
where for dust w takes the value 0.

3. The Attempt at a Solution

So I believe the last equality comes from [1], and for dust we have P=0, which is why I get $-3 \rho a^{2}\dot{a}$. (the second equality I can see is just differetiating).

Thanks in advance.

 

[Dick]

I'm stuck on deriving the follow. Context: FRW Metric, dust dominated universe, use of conservation of energy-momentum tensor. I'll now state all the equations I can see are relevant. (But just in case the source is http://arxiv.org/pdf/gr-qc/9712019.pdf and it's eq 8.43).

$\frac{d}{dt}(\rho a^{3}) = a^{3}(\dot{\rho}+3\rho\frac{\dot{a}}{a}) = -3pa^{2}\dot{a}$, and it's the last equation I am stuck on. I am getting it to be $-3 \rho a^{2}\dot{a}$...

2. Homework Equations :

$0 = \bigtriangledown_{a}T^{a}_{0}=-\partial_{0}\rho=3\frac{\dot{a}}{a}(\rho +p)$ [1]

and using $p=w\rho$
this becomes:

$\frac{dot{\rho}}{\rho}=-3(1+w)\frac{\dot{a}}{a}$, [2]
where for dust w takes the value 0.

3. The Attempt at a Solution

So I believe the last equality comes from [1], and for dust we have P=0, which is why I get $-3 \rho a^{2}\dot{a}$. (the second equality I can see is just differetiating).

Thanks in advance.

I really don't see what you are stuck on. $\frac{d}{dt}(\rho a^{3})$ is not the same thing as $\frac{d}{dt}(\rho)$. Which one do you want in the case p=0? If it's the first just substitute the result from [1] into the expression for $\frac{d}{dt}(\rho a^{3})$.