Conformally flat s-t, includes implicit dependence in EoM?

[binbagsss]

1. Homework Statement

Question attached

2. The attempt at a solution

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 a conserved quantity associated with $z$ as there is no explicit z dependence.

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 thanks

 

[nrqed]

1. Homework Statement

Question attached

View attachment 224365

2. The attempt at a solution

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 a conserved quantity associated with $z$ as there is no explicit z dependence.

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 thanks

Can you explain what your ##\mu$$ stands for?

 

[binbagsss]

Can you explain what your ##\mu$$ stands for?

Given in the question posted in the op? Coordinate transformation on z ?

 

[nrqed]

Given in the question posted in the op? Coordinate transformation on z ?

Ah! You used $\mu$ for $u$ then? Ok, then what is your $i$ in your definition of the energy? And the dot over the time refers to a derivative with respect to $s$? What what do you mean by $s$ in your $d/ds$? Do you mean the proper time?