redwards said:
...If, say, the most distant body in the universe from our current position is 13.7 billion light years away...
This is something to get straight. The
presentday distance to the furthest material we can see is 45 billion lightyears. That is, if you could freeze expansion, light would take 45 billion years to get from that material to us.
I am talking about the matter which is the source of microwave background radiation, the "oldest light" we can see. It has a redshift of 1090 (its wavelength has been extended by about that factor.)
The travel time has been nearly the entire expansion age---13 some billion years.
When the light from that material was emitted and started out on its way here, that material was about 41 million lightyears from us (or the material that eventually condensed to form us). If expansion could have been frozen at that point. The light would only have needed 41 million years to get here.
Distance then: 41 million LY
Distance now: 45 billion LY
Travel time: 13-some billion years.
You see expansion gets in there and confuses the relation between travel time and actual
now distance. At first, when the light sets out on its way, expansion seems to make travel harder because it is extending the distance the light has to go. Later it seems to have facilitated travel because it increases the distance the light has already traveled.
Because the spatial distance has expanded at different rates at different times in history, there is no regular relation between light travel time and actual distance. Maybe now the answers to your questions are obvious.
1. If, when we examine the distant edges of the universe, what we're really doing is looking back in time roughly 13.7 billion years then why has it taken 13.7 billion years for that light to reach us considering that at the moment that photon began its trip the universe was quite young and small?
I already explained, using the CMB (cosmic microwave background) light as an example.
...does that imply the both the Earth and that most distant body are traveling away from the center of the universe at 1/2 the speed of light?
According to the standard cosmo model that nearly all professionals use, there is no "center of the universe". The main standard of motion and rest that people use is the CMB itself, according to which none of the galaxies are moving significantly, they have negligible individual motion compared with the rates that distances are expanding.
To get the picture, google "wright balloon model" and watch it for a while. In that toy model of the universe all existence is concentrated on the 2D surface of an expanding sphere---there is no inside or outside surrounding--the whole world is 2D and on that surface. Galaxies are the white dots. They stay always at the same latitude and longitude, which means fixed location relative to CMB. They don't move although they do get farther apart. The other things, the little wigglers,
do move. They are the photons of light. They change latitude and longitude. You can also see them gradually get their wavelength stretched out. You can watch the redshift happen to them.
Ned Wright is a prof at UCLA who teaches cosmology. That one little computer animation movie of his is worth a lot of words.
When all the distances are expanding, nobody is
traveling because there is no destination you or anyone else is getting closer to. So expansion is not ordinary motion as we know it. It is geometry changing (according to the Einstein rules for geometry change that were set out in 1915, called gen. rel.) In a world governed by those rules you don't expect either geometry or distance to always remain fixed. Small percentage increases of largescale (extragalactic) distances occur in accordance with gen. rel.
I'll google "wright balloon model" for you to make it easy:
http://www.astro.ucla.edu/~wright/Balloon2.html
Remember that is a 2D toy model. The corresponding model of space would be a 3D hypersphere. The 3D analog. There would be no center of expansion in the 3D hypersphere, just as there is no center of expansion on the balloon surface. Or if you like to think of it that way, every galaxy is the center of expansion because all the others are getting farther from it. The center in the case of the toy model is not in the 2D universe. The 2D creatures imagined living on the balloon surface would have no notion of it. Likewise for us there is no center of expansion anywhere in 3D space. No direction we can point and say "it is over there". The existence of a center in higher dimension is entirely conjectural and supported by no empirical evidence, so it is a non-issue.
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The point should be made that both the actual distance and the light travel time are derived quantities which depend on the parameters of the model.
The redshift is what one measures and then one appeals to a model (which has been fitted to other data) to calculate an estimate of the time the light has been traveling and the presentday distance to source.
One is as "speculative" as the other, which is to say that neither presentday distance or light travel time are speculative, they simply are sensitive to the parameters of the model (like the value of the Hubble rate, something that much effort goes into getting right.)
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