- #1
binbagsss
- 1,266
- 11
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