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Given 2 objects moving at some velocity $v$ relative to one another, is it possible to determine whether they are moving or whether the space between them is expanding?

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Firstly, remember what it means for space to be expanding - it doesn't mean that space is some rubbery fabric that gets pulled, it means that the metric of space is expanding. That is, the scale factor (which essentially represents the relative distance of objects) is growing in time. So, if all galaxies were separated by some distance, this distance would then be larger at a later time.

So, how could we tell that the expansion is really metric expansion, and not relative velocity ? Well, the first reason is Hubble's Law. Since the universe is undergoing metric expansion, it appears that galaxies have an apparent velocity, given by $V=H_{0}D$. So, we see that galaxies that are further away are moving away with a higher velocity. This makes sense - photons travelling from more distant galaxies must traverse more expanding space, and therefore have their wavelengths expanded by a larger amount, so that they are more redshifted. This wouldn't be the case if galaxies were just moving away from us, since we would see a wide variety of redshifts, not a pattern.

The second reason is general relativity. GR predicts that certain metrics that contain homogeneous distributions of matter (i.e. the galaxies that make up the universe) will cause metric expansion to occur. Since GR is supported by the evidence, so therefore is metric expansion.

Third is the cosmic microwave background. We observe a uniform microwave radiation that fills the universe, that has a temperature of 2.73 degrees Kelvin. Satellites that have studied it (COBE, WMAP) have determined that it has a blackbody curve, and that it has the redshift that dates it back to a time very soon after the big bang. We now know that this represents the first radiation ever emitted, 380,000 years after the big bang at the recombination, when hydrogen atoms formed.

Another reason is that when a quasar's spectrum shows absorption lines due to neutral hydrogen clouds, the redshifts of the hydrogen lines are always found to be less than the redshift of the quasar. Furthermore, examples have been found in which the absorption spectrum shows a feature called the Gunn-Peterson trough, which had been predicted earlier as a consequence of the reionization of hydrogen in the early universe.

However, your example considers only two galaxies. In that case, there is nothing that they can do to determine if they are moving apart just due to velocity, or expansion. Fortunately, we don't live in such a universe.

For the evidence for metric expansion (and the big bang) see here:

http://www.talkorigins.org/faqs/astronomy/bigbang.html

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+1, this is the right answer. But i couldn't help to mention my personal views about this topic. –  lurscher Aug 15 '12 at 3:18
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thinking about this question, one is eventually led to think of the initial attempts to formulate Mach principle; in that hypothesis, which predates the creation of General Relativity, but after Special Relativity was established, Ernst Mach especulated that, if there was no special frame of reference from where to measure absolute velocities, there should not be absolute accelerations either; objects will have an inertial response only when they are tried to be accelerated relative to the "distant background of stars". In a sense, General relativity captures that sort of behavior in the frame dragging effect confirmed by Gravity Probe B (although the data from that probe is still unclear regarding if we live in an universe with non-zero torsion or not)

But i think it is worth to mention that the dilemma of your question has a somewhat similar flavor to the Mach's dilemma, and Mark' answer is a proof of that; indeed, because all the objects are moving consistently, we classify it as a space expansion. Even if the movements are not perfectly uniform, we would say that the space expansion describes the long scale behavior of spacetime, while individual galactic motions describe the granular scale.

Who knows? maybe there is some hidden symmetry that is still eluding us, that is relevant to inertia and cosmology.

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If the relative motion is due to the metric expansion of space, the relative motion is a function of the distance between the objects, i.e., the relative motion will change as the distance between the objects changes.

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Since we can say that $v$=$v$(t) and the distance can be a function of $v$(t) and t, then I don't see how we can rule out one from the other even if the relative motion due to metric expansion of space is dependent upon distance between objects. Actually the more I think about it, the more this reminds me of the equivalence principle. –  mcFreid Aug 14 '12 at 17:19
    
If $v$ is an arbitrary function of $t$ then all bets are off. I understand your question to be in the context of two objects in inertial (free falling) relative motion in a very simplified context, i.e., ignore everything else. If we have the metric of SR, the relative motion is constant with time. If we have an expanding metric, the relative motion is a function of distance. –  Alfred Centauri Aug 14 '12 at 18:56
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If you are allowed to make repeated measurements long stretches of time apart, then you could differentiate between a galaxy receding due to the Hubble flow and an object traveling away at a constant velocity.

For the galaxy, you'd notice that its velocity is changing between your measurements, whereas for an object traveling away at a constant velocity (in an otherwise static Universe), you would measure the same velocity at different times.

It's interesting to note that for the galaxy, Hubble's law holds to an excellent approximation, which says that

velocity = Hubble's constant x distance

However, if you perform two measurements some time apart, and the measured distance is 2x the second time around, the measured velocity would not necessarily be 2x, as Hubble's constant is in fact time dependent (not really constant). Its value depends on how much expansion has taken place and the rate at which the Universe expands at the given cosmological time.

http://en.wikipedia.org/wiki/Hubble%27s_law

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Given 2 objects moving at some velocity v relative to one another, is it possible to determine whether they are moving or whether the space between them is expanding?

No, the difference between the two interpretations is fundamentally not testable. It's a verbal distinction that doesn't appear anywhere in the actual mathematical formalism of GR. GR doesn't have global frames of reference, and therefore doesn't have any meaningful way to describe the velocity of object A relative to cosmologically distant object B. That means that the choice of one description or the other is purely one of convenience or pedagogy. Bunn 2009 and Francis 2007 give two opposing views.

That doesn't mean that cosmological observations can be explain simply by having galaxies moving inertially in flat spacetime. They can't. But what's measurable is the curvature of spacetime, not the relative velocities.

E.F. Bunn and D.W. Hogg, "The kinematic origin of the cosmological redshift," American Journal of Physics, Vol. 77, No. 8, pp. 694, August 2009, http://arxiv.org/abs/0808.1081v2

Francis et al., "Expanding Space: the Root of all Evil?," 2007, http://arxiv.org/abs/0707.0380v1

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