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When the red shifts of galaxies are large, why do we think that they are far away?

I know about Hubble's law, Tully-Fisher relation of spiral galaxies, Faber-Jackson relation of elliptical galaxies, and so on , but they seem to the empirical rules.

I think that the value of red shift has only the information on speed, so they might be the objects which move with the high speed near by our Galaxy.

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Comment to the question(v2): OP's underlying implicit question seems to be How do we experimentally measure radial distances from Earth to various galaxies? Answer: By using so-called standard candles. – Qmechanic♦ Aug 10 '12 at 13:00

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The reason why think a large red shift corresponds to large distance is that this is what is predicted by General Relativity.

If you make a few simplifying assumptions about the universe you can solve Einstein's equation for the universe to give a result called the FLRW metric. This predicts the universe is expanding and predicts the red shift increases with distance.

However we have experimental evidence for the red shift-distance relation. For example there is a type of supernova called SN1a for which we can calculate the brightness. Because we know what the brightness should be we can measure the brightness as seen from Earth and use this to calculate how far away the supernova is. Then we can measure the red shift and test the red shift-distance relationship.

The measurement of SN1a red shifts is the way dark energy was detected, because if you do the experiment you find the supernovae are actually slightly farther away than the red-shift predicts they should be.

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Kind of unrelated, but can you explain how SN1a red shift has anything to do with dark matter? I was under the impression that dark energy was "detected" when they had trouble explaining the orbital velocities of stars in galaxies. – Captain Claptrap Aug 10 '12 at 11:55
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@CaptainClaptrap I think you're mixing up terms. One of the early signs of dark matter did come from rotation curves of galaxies. Dark energy and the accelerating expansion of the universe are signaled by SNe Ia, which have had no bearing on our understanding of dark matter. Despite the common use of the word "dark," these things have nothing to do with one another. If there's still any confusion, perhaps you can open another question. – Chris White Aug 10 '12 at 12:07
I think that clears it up. Thanks. – Captain Claptrap Aug 10 '12 at 12:43
@John Rennie: The most important reason to think they are far away is that they are small and faint, but otherwise similar to close objects. So they are obviously far away. If I see a tiny airplane outside my window, I don't assume it's actually a tiny airplane very close. – Ron Maimon Aug 10 '12 at 19:17

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