Is it valid to apply Einstein's Relativity to scenarios involving expansion of space? IIs it valid to apply Einstein's Relativity to scenarios involving expansion of space? For a practical example of this: Is it legitimate to speak of distant red-shift galaxies as experiencing time more slowly in relation to our experience of time? I appreciate that isn't sensible in other ways, but by explaining if it is legitimate if albeit not-sensibly, you are kindly helping me understand a bit more about Relativity and expansion of space :O)
 A: One can apply the underlying principle of relativity -- that all reference frames are valid and agree on the speed of light -- to expanding space, but one has to be careful.
In particular, special relativity assumes reference frames are these global things that cover all of space and time. Picture a uniform grid clocks and rulers stretching as far as the eye can see, existing for all time.
However, once space itself is expanding, curving, or doing anything other than sitting still and behaving nicely, special relativity is no longer adequate. This is where general relativity comes in. This is Einstein's extension of his theory, where the principle of relativity is taken to only apply locally. That is, two nearby observers can compare their results in the special-relativistic way, but distant ones can't do so quite so easily.
The problem is that it is ambiguous how to transport vector quantities from one location to another. Think of an arrow on the surface of the Earth, somewhere at the equator. You might ask, "How does its direction compare with this other arrow located at the North Pole?" To do the comparison, you slide the equatorial arrow on over to the pole, keeping its orientation the same. But its direction at the pole depends on the path you took to get there!
The same problem happens when comparing distant things in the universe. Many statements like "time is flowing slower over there" are actually devoid of meaning without more information, since it's unclear how to compare things like the flow of time between distant points.
A: 
Is it legitimate to speak of distant red-shift galaxies as experiencing time more slowly in relation to our experience of time?

No, it is not, since they are not moving relative to the hubble flow, which means that they are sitting on their comoving coordiantes and are therefore at rest relative to the CMB, just like we are (peculiar velocities neglected). Time dilatation only happens when objects are moving in space, not if they are flowing with the expanding space.

Is it valid to apply Einstein's Relativity to scenarios involving expansion of space?

Of course, but you have to keep in mind that you have to substract the recessional velocity due the hubble expansion from the total velocity relative to our galaxy to calculate the dime dilatation. 
For example, if your observed galaxy has a distance of $d$ and a total velocity of $v$ relative to our galaxy (which, for simplicity, we assume to be at rest to the CMB), the peculiar velocity $v_{pec}$ of this galaxy would be $v_{pec}=v-H\cdot d$ where $H$ is the hubble parameter with units of ${\text{sec}^{-1}}$. Now you can plug in the peculiar velocity into the formula for special relativistic time dilatation. Since peculiar velocities are rather small compared to the speed of light this effect is more or less neglectable.
For example: if you observe an object with redshift $z_{observed}+1=3$, but from the distance where you measure it you would expect redshift $z_{expected}+1=2$, then you know that $\frac{3}{2}=\sqrt{\frac{c+v}{c-v}}$ and the peculiar velocity would be $v_{pec}=0.3846 \, \text{c}$. Here you would get a time dilatation factor of $12:13$.
PS: Fraser Cain did a short video for laymen on this topic on Youtube
A: 
Is it legitimate to speak of distant red-shift galaxies as experiencing time more slowly in relation to our experience of time?

Yes, it is. Take a periodic source of light, acting as a clock. Consider these three scenarios:


*

*the source of light is close enough that the Minkowski approximation holds and moves away at a high velocity

*the source of light is far away and at rest relative to the Hubble flow

*the source of light sits lower in a gravitational potential at constant distance from the source of gravity


In any of these cases, the light will be redshifted and time will appear to tick slower. Observationally, the situations are equivalent, even though we attribute the effect to Doppler shift, cosmological and gravitational redshift and time dilation, respectively.
In any of these cases, we can parallel transport the source's velocity vector along the light path and treat the result as the source's velocity relative to the observer. It might seem paradoxical that objects can have a non-zero relative velocity even though they are both 'at rest' according to various interpretations of the term (no motion relative to Hubble flow in one case, a constant distance from the source of gravity in the other). This is no longer an issue once we reject distance parallelism.
