What is the role of gravity in the cosmological redshift?

[Jorrie]

Jorrie, I appreciate your focused response here.

Thanks. Now, to focus the discussion even more, I have PM'd you a spreadsheet (link) with my solution to the Peacock, B&H, 1/a and your models, aggregated over some 8000 steps, z=0 to 1000. When you have the time, please investigate and let me know if you agree. Especially, my spreadsheet fails to show the "second element of error" that you refer to here:

However, B&H fail to articulate that their use of the SR velocity addition formula introduces a second element of error even if the spacetime curvature is zero. When the light travel distances measured locally for each segment are summed, they do not exactly equal the FRW 'global' light travel distance, instead they only asymptotically approach it. This is because the SR formula includes an element of time dilation, yet any non-zero amount of time dilation will cause the sum of the local light travel times to diverge from the FRW 'global' light travel distance.

Despite your many explanations, I still fail to understand what "FRW 'global' light travel distance" means. AFAIK, there is only one definition for light travel distance: $D_{lt} = c(t_{observe}-t_{emit})$, where t_emit is expressed in the reference frame of the observer. This is essentially the SR distance. Do you have a reference for FRW global light travel distance?

 

[Ich]

Ich, what don't you understand about this part, which is referring to summing up locally observed light travel distances in Minkowski coordinates?

"This should be obvious: no comoving Milne observer sees the distances to galaxies near his own location to be contracted; only distances to galaxies far away from his location are contracted. When each such Milne observer contributes to the group calculation his own measurement of only the local part of the worldline as he observers it, each such local measurement is non-contracted, and therefore the global sum of all such local measurements cannot be length contracted."

I understand this part. Do you understand that each observer will actually improve her contribution if he includes time dilation in its (maximum pc ) frame?
I don't know how often I repeated that point: it's actually relativistic doppler shift, but can be approximated as a "classical" shift for small distances.

 

[nutgeb]

Despite your many explanations, I still fail to understand what "FRW 'global' light travel distance" means. AFAIK, there is only one definition for light travel distance: $D_{lt} = c(t_{observe}-t_{emit})$, where t_emit is expressed in the reference frame of the observer. This is essentially the SR distance. Do you have a reference for FRW global light travel distance?

Jorrie, the SR 'global' light travel distance is Lorentz-contracted, which is obviously different from the definition you give. It is illustrated for the empty Milne model in Ned Wright's lower diagram, as the vertical distance along the t axis.

The formula you cite is in fact the formula for FRW 'global' light travel distance. It is illustrated for the empty Milne model in Ned Wright's upper diagram, again as the vertical distance along the t axis. It is not Lorentz-contracted; therefore it is longer than its SR counterpart. There is no problem expressing t in the observer's local frame, because as I've said many times, in FRW coordinates t = t', the observer's t is exactly the same as the emitter's t', there is no time dilation as between fundamental comovers.

If you sum up locally observed (SR or FRW) light travel segments, you will automatically calculate (asymptotically) the FRW 'global' light travel distance, not the SR "global light travel distance. As I said in an earlier post, due to the nature of the Lorentz contraction, locally observed segments have asymptotically approaching zero Lorentz contraction. That's why you can't sum up local non-Lorentz-contracted segments to calculate a Lorentz-contracted total.

 

[Jorrie]

If you sum up locally observed (SR or FRW) light travel segments, you will automatically calculate (asymptotically) the FRW 'global' light travel distance, not the SR "global light travel distance. As I said in an earlier post, due to the nature of the Lorentz contraction, locally observed segments have asymptotically approaching zero Lorentz contraction. That's why you can't sum up local non-Lorentz-contracted segments to calculate a Lorentz-contracted total.

Sorry, but I'm still totally lost as to what you mean by "FRW 'global' light travel distance" (and I've read everything that you wrote in this thread). Ned Wright's tutorial does not help either, probably because we interpret the diagrams differently. I will discuss this with you in a PM first, to try and understand what you say. Then we can come back here if we wish.