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Science articles often say that Andromeda galaxy is approaching us with speeds such and such. Sometimes we say that the universe is expanding away faster than light. Sometimes we talk about things that are light years away. We talk about stars forming billions of years ago.

How are these numbers still meaningful given that general relativity says there's no global inertial frame? I read that it's only meaningful to talk about local velocities. How is it still meaningful to talk about the speed of expansion of the universe?

And how is it meaningful to talk about how long ago stars formed, etc etc?

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There are two issues mixed up in your question.

First, there is a preferred cosmic rest frame: the frame of the cosmic microwave background. This does not imply any contradiction with the principle of relativity, because the rest frame is not an intrinsic property of general relativity, but just a property of the configuration of matter in the particular solution of general relativity we call home. There is a more detailed explanation of this point here: Is the CMB rest frame special? Where does it come from?

The cosmic rest frame is used to set a reference for many questions for cosmology. For example, to define the age of the Universe, we mean what is the time between now and the Big Bang as measured in the cosmic rest frame. When we talk about the expansion rate, we mean how fast distances between galaxies -- as measured in the cosmic rest frame -- are receding from one another per unit time -- as measured in the cosmic rest frame.

Second, you are correct that, strictly speaking, we should only talk about relative velocities in a locally intertial frame. In the context of cosmology, this means in practice that we should only talk about velocities of objects closer than about 100 Mpc away from us. For objects closer than 100 Mpc, the redshift is much smaller than one, and the effects of the expansion of the Universe can be ignored to a good approximation (or in other words, we can ignore the curvature of spacetime on these scales).

For galaxies closer than 100 Mpc, the component of the velocity due to the expansion of the Universe from us is determined by Hubble's law, $v = H_0 D = c z$; since $z \ll 1$, then $v \ll c$. Above about 100 Mpc, the redshift starts to become comparable to one, and the velocity you would infer from Hubble's law becomes relativistic. These very distant galaxies are the ones that are not in our same locally inertial frame, and you are correct that we shouldn't really talk about a velocity relative to them at all. To summarize: for galaxies close enough that it makes sense to talk about a relative velocity, the component of the relative velocity due to the expansion of the Universe is always much less than $c$. (This doesn't rule out that some object might be nearby and whizzing through space at a relativistic speed).

When people talk about the velocity of galaxies receding away from us at or above the speed of light due to the expansion of the Universe, more often than not what they are doing is extrapolating Hubble's law $v=cz$ outside its regime of validity $z\ll 1$. So long as you know what you mean, this is ok-ish, but it can be extremely confusing when you are first learning the subject.

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  • $\begingroup$ So we can measure times and distances across the entire universe in the cosmic rest frame? Wouldn't this imply the existence of a global co-ordinate system, whose co-ordinate are not abstract, but are actual spacetime measurements? $\endgroup$
    – Rain Deer
    May 5 at 1:31
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    $\begingroup$ @RainDeer Yes, these are called FLRW coordinates. They are the coordinates where the metric is homogenous (does not depend on spatial coordinates) and isotropic (the metric is diagonal and the spatial part of the metric is proportional to the identity matrix, assuming zero spatial curvature). Really these are a set of coordinates, since they are only defined up to a rotation. The main point is that the CMB defines a reference frame. $\endgroup$
    – Andrew
    May 5 at 1:48
  • $\begingroup$ So this frame can be used to define a "paused moment" for the entire universe? I mean the entire universe at a particular time. $\endgroup$
    – Rain Deer
    May 5 at 2:16
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    $\begingroup$ @RainDeer Essentially yes. Two caveats: (a) we can only observe a finite volume of the Universe, so there's no guarantee about what happens outside of what we observe. (b) The cosmic rest frame is defined in an average sense over large distances. There are fluctuations in the density and gravitational field, however. So, there can be local deviations which "detach" from the cosmic rest frame (eg near the event horizon of a black hole). And, if you want to define the rest frame extremely precisely, you will have to deal with ambiguities in how to average small perturbations ("gauge fixing"). $\endgroup$
    – Andrew
    May 5 at 2:42
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    $\begingroup$ This "paused moment" is often called a "spacelike hypersurface" or "spacelike slice" or "time slice" in the lingo. $\endgroup$
    – Andrew
    May 5 at 2:42

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