Proper time elapsed between a photon being sent and received

[Bailey2013]

Homework Statement

The problem I am trying to solve is the proper time elapsed along A's worldline between a photon being emitted and sent to B (which is a distance L away from it along the x axis) and being reflected and detected by A again. The question is the second part of the question which I have attached. The variables for the question are best understood by reading the question itself.

Homework Equations

The metric for this question is shown in the tagged files.

The Attempt at a Solution

I have used the fact that the photon's worldline is a null geodesic and therefore that the geodesic equation is just includes dt and dx (dy and dz are assumed to be constant as is seen on the question) and then tried to derive by integration the time of reflection, which I hope to then use to do the same to find the time when the photon comes back to A. However, I seem to not be getting towards the answer and so wondered if someone could tell me if my approach is right.

 

[TSny]

Welcome to PF!

I have used the fact that the photon's worldline is a null geodesic and therefore that the geodesic equation is just includes dt and dx (dy and dz are assumed to be constant as is seen on the question)

I'm not sure what you mean by "dy and dz are assumed to remain constant". A photon cannot travel purely along the x-axis from A to B and still satisfy the condition of traveling on a null geodesic. The presence of the small quantity $h(t, z)$ in the line element requires the null geodesic from A to B to deviate from a straight line in the given coordinate system. So, you cannot assume that both $dy$ and $dz$ will be zero along the null geodesic. Without getting bogged down in details, you should be able to construct an argument for why $dy^2$ and $dz^2$ can nevertheless be neglected in the line element for this problem.

So, I believe your equation for the outward trip from A to B, which involves the integral from $t = t_e$ to $t = t_r$, is OK. I would recommend writing a similar equation for the return trip. The two equations can then be combined into one equation involving a single integral over the time of the entire round trip. Then make approximations to first order in $h$. I think it will work out and give you the desired expression.