[SOLVED] Distance actually travelled by a photon

vikter

First I just want to say thanks to everyone on this site that take time to answer questions like mine. This is my first post but I've used the forum many times for help on other problems.

1. The problem statement, all variables and given/known data
Consider two observers separated by a fixed, comoving distance r (which we can set to be their distance today). The true physical distance between the galaxies of course changes with time due to cosmic expansion; for each small comoving distance increment dr, the corresponding increment in physical distance is just given by [tex]^{dl}phys[/tex] = a(t)dr.
Consider a light ray which moves from galaxy A to galaxy B. It's speed relative to the observers it passes is v = [tex]\frac{dl}{dt}[/tex] = a(t)dr/dt = c. Using this, show that a light ray emitted at time t = 0, and detected at time t, travels a physical distance
[tex]^{l}hor[/tex](t) = c * a(t) * [tex]\int d\tau/a(\tau)[/tex]. Sorry, that integral should be taken from 0 to t, but I'm not seeing how to add the code for that. And by this point I'm quite frustrated with not only the problem but trying to present it here in a coherent matter haha so I'm sorry if it's hard to follow. The actual version can be found here starting at the bottom of the second to last page.

a(t) being the scale factor.

The attempt at a solution
I have about two pages worth of algebra etc which has led me nowhere so I'm not going to even bother. I understand the problem but really am quite lost as to how to get to where I need to go, so any suggestions will be GREATLY appreciated.

Thanks.

vikter

I'm trying to work with this equation: [tex]\Delta[/tex][tex]t_{obs}[/tex] = [tex]\Delta[/tex][tex]t_{em}[/tex] [tex]\frac{1}{a_e_m}[/tex]