Proper distance problem/interpretation

[deneve]

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!

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(t₀)=1 so that d_p(t) = a(t₀)r = r at t₀. Ryden then considers the light ray moving from the distant galaxy by setting ds²=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 t₀ 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 t₀ 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.

 

[PeterDonis]

If you look at my diagram though, the RHS integral should start at a value smaller than r

No. $r$ isn't a distance, it's a coordinate. For a "comoving" object, $r$ never changes; all that changes is the scale factor $a(t)$, so a "comoving" object at $r$ is a distance $a(t) r$ away from a "comoving" object at the spatial origin at time $t$. So the object emitting the light is always at coordinate $r$, and the integral on the RHS is over coordinates, not distances.

 

[deneve]

No. $r$ isn't a distance, it's a coordinate. For a "comoving" object, $r$ never changes; all that changes is the scale factor $a(t)$, so a "comoving" object at $r$ is a distance $a(t) r$ away from a "comoving" object at the spatial origin at time $t$. So the object emitting the light is always at coordinate $r$, and the integral on the RHS is over coordinates, not distances.

Hi PeterDonis Thank you for that but I'm still puzzled as to how I should change my diagram to make what you say more clear.

Kind regards. Thank you.

 

[PeterDonis]

I'm still puzzled as to how I should change my diagram to make what you say more clear.

Remove the dotted lines marked "r"; they're wrong. The worldline of the galaxy emitting the light is the "grid line" marking coordinate location $r$. The "grid" expands as the universe expands. The distances marked on the diagram, corresponding to $a(t) r$ at different times $t$, are correct; but they don't correspond to a change in where the galaxy is relative to the "grid line" marking coordinate location $r$--the galaxy is always at coordinate location $r$, so it is always on the "grid line" $r$.

 

[deneve]

Thank you PeterDonis. I think I get this now thanks to your kind help. I'll try and have another think through it this evening and then see if it still makes sense. I'm really grateful for your help. Many thanks.