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It would seem that measuring an age of the universe from the big bang requires separating spacetime into a 3D coordinate system and a time track. I fail to understand why it is appropriate to take the reference of a comoving observer as being stationary to the cosmic microwave background when this seems to be a choice based upon our subsequent detection of that entity.

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    $\begingroup$ Welcome to Physics Stack Exchange. I think this question could use some more detail. Can you explain why you think measuring time from the big bang separates space-time into a singularity and a time track? $\endgroup$ – DanielSank Jul 1 '15 at 3:20
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    $\begingroup$ What singularity would that be? There is no singularity in the current cosmological model. $\endgroup$ – CuriousOne Jul 1 '15 at 5:31
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    $\begingroup$ Hang on, back up. I don't understand the nature of your grievance stated in that last sentence. We detected the CMB, we noticed that we were not at rest wrt it, we discovered a frame that has no peculiar velocity wrt the CMB, then we called it "comoving". Why do you find it confusing that we would call the comoving frame at rest wrt the CMB when it was a frame designed to be that way. And we could only design the frame to be that way after having discovered the CMB, right? $\endgroup$ – Jim Jul 2 '15 at 14:02
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You're quite correct that measuring time from the Big Bang does separate spacetime into a time bit and a space bit, but this isn't arbitrary.

When we want to describe the universe around us we need to choose some coordinate system that we can use to record physical quantities. Whatever coordinate system we choose will have one coordinate that behaves like time and three that behave like space. General relativity tells us that any coordinate system can be used, so there is no unique way to split our coordinates between time and space. However some coordinate systems are more convenient than others and it's often the case that one coordinate system is obviously easier to work with than others.

Another basic principle of relativity is that locally spacetime always looks like Minkowski space i.e. regular flat spacetime. And an obvious choice of coordinates for flat spacetime are the Cartesian coordinates $(t, x, y, z)$ with the origin centred on ourself. By choosing these coordinates we are choosing to split spacetime up into a $t$ coordinate with a family of spatial hypersurfaces for each value of $t$, but this is a natural splitting that's intuitively easy to understand. Even this simple splitting isn't unique. Effects like time dilation happen because observers moving relative to us split their coordinates in a different way and their time coordinate doesn't match ours.

Anyhow, it turns out that for describing the expanding universe we can make a choice of coordinates that is very close to the simple coordinates used in flat spacetime. We call these comoving coordinates. The metric for flat spacetime in Cartesian coordinates is:

$$ ds^2 = -c^2dt^2 + dx^2 + dy^2 + dz^2 $$

and the metric for an expanding universe in comoving coordinates is:

$$ ds^2 = -c^2dt^2 + a^2(t)(dx^2 + dy^2 + dz^2) $$

So the only difference is that factor of $a(t)$, which we call the scale factor. This split between time and space is very similar to the split we use in flat spacetime, and it chooses a time coordinate we call comoving time. The nice feature of comoving time is that all comoving observers share this time coordinate i.e. they all agree on its value. As a rough guide, a comoving observer is one who is stationary with respect to the cosmic microwave background, and most objects in the universe are roughly stationary wrt the CMB. That means most observers in the universe are comoving, and therefore share the same time coordinate. That's why choosing to split spacetime into space and comoving time is so convenient.

So to return to your question, yes we are splitting spacetime into separate time and spatial bits. However this is a natural split that turns out to be very useful for describing the universe around us, so it is far from arbitrary.

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  • $\begingroup$ Thanks for your useful answer. I have edited the question based upon your input $\endgroup$ – A Monroe Jul 1 '15 at 17:42
  • $\begingroup$ @AlanMonroe: by editing the question you have rendered my answer pointless. Ideally you should revert the question to its original form then post a new question as a follow-up. $\endgroup$ – John Rennie Jul 1 '15 at 19:32
  • $\begingroup$ I take issue with the term "comoving time". Do you have a reference for that? I have never heard it used, myself. I have heard conformal time used, but that isn't what you defined. You seem to have defined proper time in the comoving frame. Calling it comoving time, I think, is somewhat misleading $\endgroup$ – Jim Jul 2 '15 at 14:06
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    $\begingroup$ Well, it's the time component of the comoving coordinates. Is abbreviating this to comoving time such a sin? You are of course correct that it is also the proper time of a comoving observer. $\endgroup$ – John Rennie Jul 2 '15 at 14:22
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    $\begingroup$ @JimsBond FWIW, in my experience comoving time is part of the usual jargon. $\endgroup$ – Kyle Oman Jul 2 '15 at 15:53

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