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If I understand correctly, Einstein's theory of General Relativity predicted the expansion of space itself, and Hubble confirmed this prediction by observing the red shift of receding galaxies.

I have always wondered how physicists know that this red shift indicates that space itself is expanding, as opposed to simply indicating that these galaxies are moving through space and away from Earth. I can accept that Einstein's theory predicts the expansion of space itself, but as far as confirming this prediction by observation, if space was static (neither expanding nor contracting), but the observed galaxies were moving through space at a high velocity away from Earth, wouldn't that also produce a red shift when observed from Earth?

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duplicate of physics.stackexchange.com/q/34164/4552 –  Ben Crowell Jun 19 '13 at 4:27
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marked as duplicate by Ben Crowell, Waffle's Crazy Peanut, John Rennie, Qmechanic Jun 19 '13 at 12:11

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The cosmic microwave background provides a convenient reference frame for measuring motion (called the co-moving frame). If you are moving relative to the CMB then the doppler shift means the CMB looks slightly hotter in the direction you are moving (the dipole anisotropy), and slightly cooler in the other direction. This motion is called the peculiar motion. If we measure the CMB from Earth we find that in fact we are moving relative to the CMB, but at only 368 km/s (you might think 368 km/s is pretty fast, but it's peanuts compared to galactic red shifts) so we conclude the Earth is roughly stationary wrt the CMB.

General Relativity, or more precisely the solution called the FLRW metric, predicts that every star/galaxy/whatever finds itself to be roughly stationary wrt the CMB, so the fact that the Earth appears to be stationary is no surprise.

But if the red shift is caused by galactic motions then those galaxies would not be stationary wrt the CMB. You'd have to conclude the Earth is stationary and all the galaxies are moving away from it i.e. have non-zero peculiar motions, and you'd have to conclude that this peculiar motion increases with distance from the Earth. This would make the Earth a very special place, which seems improbably because the Sun seems to be a fairly standard star, in a fairly standard galaxy, in a fairly standard cluster, and so on. It's much more likely that the red shift is due to the expansion of the universe and not peculiar motions of galaxies.

So given that it's extremely unlikely that the Earth just happens, by chance, to be at the centre of the universe, and that we have a well tested theory (General Relativity) that predicts the expansion of space, few of us doubt that the expansion of space is the cause of the red shift.

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I think I don't quite get your question.

If those galaxies had been moving through space, the redshift from every single object (for instance, the Proxima Centauri and some other star say, Sigma Orionis) would've been the same. It wouldn't have been such a big headache. But, it's not. Hubble's observation itself showed that the whole universe is accelerating. First of all, there's no direction or center (well, every single position can be a center) for the expansion (which you may already know). It's not expanding from Earth in some particular direction. As per the observation, nearer objects move at slightly lower velocities than those farthest objects. This can be seen from the classical Hubble's law $v=Hd$. A slight thought can show that if velocity increases with every increment in distance, then it's accelerating...


More interesting explanation can be found on Wiki articles on Metric expansion of space, Hubble's law and our currently accepted $\Lambda-\text{CDM}$ model.

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