GR: conformally flat S-T cosmology

[binbagsss]

Homework Statement

Homework Equations

look up, look down, your pants fell down

The Attempt at a Solution

[/B]

Time-like killing vector is associated with energy.

$\frac{d}{ds} (\frac{\mu^2\dot{t}}{R^2})=0$

Let me denote this conserved quantity by the constant $E=\frac{\mu^\dot{t}}{R^2}$
where $\mu=\mu(z)$ . similarly we for sure have a conserved quantity associated with $x$ and $y$ , I am unsure about $z$ however...

a)$ds^2=\frac{\mu^2}{R^2}(-dt^2+dx^2+dy^2+dz^2)$

b) it is conformally flat since it is the flat space-time (in Cartesian coordinates) metric multiplied by a factor which is a function of the coordinates
c ) since it says to 'write down' the geodesic equations it is clear that the metric components have no dependence on $t,y,x$ and so there is a KVF associated with each of these coordinates. I am unsure whether you treat $\mu(z)$ as $z$ dependence or not.
So if when writing the Euler Lagrange equation associated with $z$ i was going to do:
$\frac{dL}{dz}=\frac{dL}{d/mu}\frac{d\mu}{dz}$ .If I don't do this then there is not a conserved quantity associated with $z$ .
However since the question asks you to 'write down' I suspect we ignore the implicit depdence of $z$ via $\mu$ so that there is a conserved quantity associated with $z$ and are only interested in any explicit dependence when differentiating $L$. Is this correct? I don't really understand why you'd ignore the implicit dependence of $z$

many thansk

 

[mighty2000]

Thank you for your post. It appears that you are attempting to solve a problem related to time-like killing vectors and conformally flat metrics. However, the equations you have provided do not seem to be related to the problem at hand. Additionally, the homework statement and equations are not clearly stated, making it difficult to provide a helpful response.

I suggest reviewing the problem and providing clear and concise statements of the homework statement and equations. Additionally, please provide your attempt at a solution with relevant equations and explanations. This will help me and other forum members to better understand the problem and provide a more accurate response.

In general, it is important to clearly state the problem and provide all relevant information in order to receive a helpful response. I look forward to seeing your revised post and assisting you with your problem.