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Is all of the visible Universe exactly the same age everywhere? What about the Universe beyond what we can see?

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Ben Hocking is right that the Universe we observe is not the same age everywhere: the further away we look, the younger things are, simply because the light from faraway objects takes time to reach us.

But you may be wondering about more than that: what if we could remove those light-travel-time delays and consider the Universe as it is now, as opposed to the way we see it (with greater delays for more distant objects)? Would it all be the same age in that case?

That turns out to be a more subtle question than you might think, because of relativity. Relativity says, among other things, that different choices of reference frame lead to different notions of time -- things that are simultaneous in one reference frame are not simultaneous in another. When you consider what a distant object is like now, you have to be careful to specify what reference frame you're talking about, because different frames lead to different "nows."

At each point in spacetime, there is a reference frame that seems like the "most natural" one to use, namely the one in which the expansion of the Universe looks roughly the same in all directions. Cosmologists tend to use that reference frame to define and synchronize their clocks. That is, the time coordinate at any point in spacetime is, by definition, the amount of time that would have elapsed according to an observer who'd been at rest in that reference frame since the Big Bang. That time coordinate is often called "cosmic time."

If you use cosmic time as your time coordinate, then it is true that, at any fixed moment of time, all points in the Universe have the same age. But it's true pretty much because of the way we defined our time coordinate, so this is kind of a vacuous statement!

Update: Based on the comments, I realize I should have explained some things explicitly. I'm considering here "cosmological" effects, meaning I'm ignoring things that are due to small-scale inhomogeneities and thinking about large scales, on which the Universe is approximately homogeneous. If, for instance, you hang around near the horizon of a black hole, you'll age at a different rate from someone else. Even if you avoid such extreme cases, small-scale variations in the gravitational potential will lead to small-amplitude age variations. In fact, the definition of cosmic time only really makes sense in the approximation where we're willing to average over small-scale inhomogeneities. If you're not willing to do that, then different prescriptions for synchronizing your clocks, no one of which is obviously more natural than another, will lead to small differences in ages of things.

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@Ted Bunn: Nice addition. I question your last paragraph, though. Wouldn't general relativistic effects make that not true, since time flows slower in a gravitational field? (As an extreme case, consider the adventures of an observer near the event horizon of a black hole who can manage to watch the universe age billions of years while mere minutes elapse for himself.) –  Ben Hocking Jun 21 '11 at 15:15
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@Ben Hocking -- I probably should've been more explicit here. I was considering only "large-scale" or "cosmological" effects. You're quite right that small-scale, local differences will still lead to differences in age. –  Ted Bunn Jun 21 '11 at 15:38
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I'm not sure I understand your comment. In a sense, it is circular (I used the word "vacuous," but I meant the same idea) to say that all parts of the Universe are the same age. But here are some statements that aren't vacuous (and appear to be at least approximately true): 1. At every point in spacetime, there is a local reference frame in which things look roughly the same in all directions. 2. If we define the age at any point to be the elapsed time since the Big Bang, as measured by a clock at rest in this frame, then points with the same age also are also at the same density. –  Ted Bunn Jun 23 '11 at 20:22
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The assertion that different kinds of clock run at the "correct" rate is a content-ful, non-vacuous assertion about the world. It's essentially the principle of equivalence: a clock in a sealed box won't "notice" the expansion around it and will run at the correct rate. Of course, I can't check this assertion directly as applied to the early Universe, but people can and do check it in lots of other circumstances, and it seems to be a good assumption. [More in next comment.] –  Ted Bunn Jun 26 '11 at 15:10
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You could if you prefer define time in terms of the density of the Universe. Then the statement that any given type of clock correctly tracks the time as defined in this way becomes a content-ful, in principle testable assertion. That way of formulating the theory feels less natural to me. Our model of the Universe is based on general relativity, in which there is a notion of (proper) time independent of any particular solution to the equations, so defining time in a way that relies on properties of a particular spacetime feels like a bad idea. Anyway, it's not what people usually do. –  Ted Bunn Jun 26 '11 at 15:12
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No, the further away we look, the younger the Universe becomes. That's why we're able to see galaxies that were forming when the Universe was approximately a half billion years old. Although it's possible there is an unknown portion of the Universe beyond what we can see, we currently have no reason to believe such a region exists. Rather, the boundaries of what we can see are dictated by when light decoupled from matter.

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