If we observe two galaxies that are diametrically opposed from the Earth, and each $1000\,{\rm ly}$ away from the Earth, is the separation distance between the galaxies $2000\,{\rm ly}$? Really the question is: if the galaxies were separated from Earth by $10\,{\rm Gly}$ each, then would their separation be $20\,{\rm Gly}$?
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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 20 Gly from each other.
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$\begingroup$ 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"? $\endgroup$ Aug 26, 2011 at 16:25
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$\begingroup$ 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. $\endgroup$– Ted BunnAug 26, 2011 at 17:15
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$\begingroup$ (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 $z=\infty$). $\endgroup$– Ted BunnAug 26, 2011 at 17:19
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$\begingroup$ 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...? $\endgroup$– ThrivethJun 9, 2013 at 18:59
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1$\begingroup$ Btw. @jormansandoval You might consider accepting Ted Bunns answer if you liked it, so he can get proper rep for it. $\endgroup$– ThrivethJun 9, 2013 at 19:22