# Homework Help: General Relativity - FRW Metric - FRW Equations show that ..

1. May 16, 2017

### binbagsss

1. The problem statement, all variables and given/known data

2. Relevant equations
see above
3. The attempt at a solution
Using the conservation equation for $p=0$

I find: $\rho =\frac{ \rho_0}{a^3}$; (I am told this is $\geq0$ , is $a\geq0$ so here I can conclude that $\rho_0 \geq =0$ or not?)

Plugging this and $p=0$ into the first Einstein equation I get:

$\dot{a^2}+k-\Lambda a^2=\frac{8\pi G \rho_0}{a}$

So a stationary solution is to solve for $a$ and get no time independence, so don't we need something of the form:

$\frac{da}{dt} a^k =0$ or can I find a more general expression to this?

This is ofc not possible to get since $\Lambda$ and $k$ are constants and can not depend on $a$?

Many thanks

2. May 16, 2017

### phyzguy

In a time independent solution, a is constant, so Λ and k can depend on a and still be constant.

3. May 16, 2017

### binbagsss

mmm

so $k=8 \pi G \rho_0 / a$ and $\Lambda =0$ ?

4. May 16, 2017

### phyzguy

How did you conclude Λ=0? Remember k = 0, +/-1.

5. May 16, 2017

### binbagsss

$k=0$ and $\Lambda = \frac{-8\pi G \rho_0}{a}$ ?

Basically need to solve for $8\pi G \rho_0/a +\Lambda a^2 -k = 0$ subject to the above values of $k$ only allowed ?

6. May 16, 2017

### phyzguy

Aren't there two equations?

7. May 16, 2017

### binbagsss

equating $a$ coefficients, but baring in mind that $\Lambda$ can depend on $a$ and $k$ can not, so $k$ for the 0th order equation?

8. Apr 10, 2018

### binbagsss

apologies to re-bump but
re-looking at this , $\rho =\frac{ \rho_0}{a^3} \geq 0$
$\rho=\frac{\rho_0}{a} \frac{1}{a^2} \geq 0$ and so since $\frac{1}{a^2}>0$ it must be that $\rho=\frac{\rho_0}{a} \geq 0$

so from the other equation $0+k-\Lambda a^2 \geq 0$ is the required constraint ?

9. Apr 11, 2018

### binbagsss

I'm getting a bit confused since it says $k=-1,0,1$ , but since the question doesn't specify whether $\Lamda$ is $>0$ or $<0$ there is no need to/no way to narrow down this further