Hi I'm trying to put some notes together but have run into an anomaly which I seem to have overlooked in the past but puzzles me now. I've included a jpg file of the page I've written up so far with the problem indicated right at the end. I'm using Barbara Ryden's book as my source, but it doesn't really matter because all the other texts I've looked at concur with her and not me so I must be wrong!(adsbygoogle = window.adsbygoogle || []).push({});

In the attached file you'll see my picture of the observer (moving up the ct axis) and the curved line of a galaxy slowly moving away from the observer due to the scale factor. (galaxy is commoving).

d_{p}(t) = a(t)r is what I am using and derive this by integrating over the commoving distance r which I've fixed at a(t_{0})=1 so that d_{p}(t) = a(t_{0})r = r at t_{0}. Ryden then considers the light ray moving from the distant galaxy by setting ds^{2}=0 in the metric to get cdt/a = -dr. Now comes my problem. She integrates the left hand side of this from t_{e}to t_{0}and the right hand integral from r to zero - so far so good but If you look at my diagram though, the RHS integral should start at a value smaller than r - namely at the position marked with an A and corresponding to not r, but a(t_{e})r. Ryden ignores this difference on the limits cf her equation 3.39 on p40 if you have the book. she clearly states (after removing the - sign and switching the limits) that

integral from t_{e}to t_{0}of cdt/a = integral 0 to r of dr

I'm clearly missing something really obvious here. It may be that my interpretation of the spacetime diagram is wrong. I'm really struggling to make any more progress. Any help would be gratefully received. I think I'm making a conceptual error of som sort that needs straightening out. Kind thanks to anyone who responds.

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# Proper distance problem/interpretation

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