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From the Earth If we observe two galaxies that are diametrically opposed and each 1000ly far from the earth. the separation distance between the galaxies will be 20000ly? really question is: if the galaxies were separated from Earth 10Gly. then the separation distance between galaxies would 20Gyr?

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For nearby distances, yes, you can just add them up in the usual way. (Of course, if two galaxies are just a few thousand light-years apart, then they're right on top of each other -- galaxies are much bigger than 1000 light-years!)

For large distances (i.e., distances comparable to the Hubble length, as in your last example), you have to be careful. The best answer is that there is no unique, well-defined notion of distance over such large distances. On cosmological distances, spacetime is curved, and what that means is that there are no inertial reference frames covering these large distances. Different people may choose different (non-inertial) reference frames, and as a result they'll disagree about the distance, but no one is necessarily "right."

When people talk about the distance to a faraway galaxy, they most often mean distance as measured in a particular coordinate system, namely comoving coordinates, with distances evaluated at the present (cosmic) time. With that specific definition, the answer to your question is yes: two galaxies that are 10 Gly from earth in opposite directions are 10 Gly from each other.

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true. I put "1000ly" just do the question. but in reality the question is: if the above situation is true, someone who is in the galaxy "A", could see the galaxy, "B"? –  jormansandoval Aug 26 '11 at 16:25
    
The answer to that is that it depends! It depends on the distances you're talking about, and on when the observer is trying to look. In general, the maximum distance an observer can see at any given time depends not just on the age of the Universe at that time, but also on what the expansion rate was like as a function of time in the past. You can see further than you might naively expect (i.e., if the Universe is 14 Gyr old today, you can see further than 14 Gly), simply because, at the time the light left on its way to you, the distances were smaller. –  Ted Bunn Aug 26 '11 at 17:15
    
(continuing ...) But it's certainly true that there are some objects so far away that we can't see them now. And it's possible to find a pair of galaxies, in opposite directions on the sky, that are visible from Earth but which are too far from each other to be visible to each other at the present time. To explore the relation between ages and distances, you might want to play around with Ned Wright's cosmology calculator, astro.ucla.edu/%7Ewright/CosmoCalc.html . To get the maximum distance you can see, plug in very large values of z (the maximum distance is really $z=\infty$). –  Ted Bunn Aug 26 '11 at 17:19
    
Mr Bunn. Thank you very much!!! –  jormansandoval Aug 27 '11 at 14:31
    
In your last sentence, don't you mean that two galaxies that each are 10 Glyr away from us in opposite directions are 20 Glye apart, not 10 Glyr apart...? –  Thriveth Jun 9 '13 at 18:59
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