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GR: Quick Q, conservation energy, stuck on algebra

  • Thread starter binbagsss
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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.
 
Last edited:

Answers and Replies

  • #2
Dick
Science Advisor
Homework Helper
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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})##.
 
Last edited:

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