Does the age of the universe depend on the way it is defined?

Question

I was reading some earlier posts on this question in stack exchange. These posts give me the impression that the age of the universe is defined in a certain way following prescriptions such as that:

• you should be co moving with the galaxy

• avoid strong gravitational fields

• choose the longest proper time recorded in all the frames

Defined this way we get the feeling that the age of the universe is not quite objective. Because we will get a different age if we change the definition. To highlight this arbitrariness, let us consider two space-time travellers A and B equipped with rockets and enough fuel who start at the same space point after setting their respective clocks to zero. Let us say that they started long time ago and took very different space-time paths and meet now and compare their clock readings T(A) and T(B). Each one can in principle claim that his clock showed the age of the universe. One may go a bit further and send a large number of travellers and collect all their clock readings at the end and declare the longest reading as the age of the universe. In this case we can say that the traveller who did not fire his rocket is the winner.

The implication in accepting this definition is that the time shown in this clock is the objective or absolute time of the universe while the times recorded in the other clocks are subjective. This way this definition will bring in a preferred reference frame and make the age of the universe an absolute time and not relative. In relativity theories we have not so far invoked preferred reference frames or absolute time concepts. Should the age of the universe be an exception?

I see that my post has some overlap with some earlier posts.However the important difference is that, as agreed by all who responded to my post,the earlier posts did not indicate that the age of the universe could be as arbitrary as the choice of a reference frame and no reference frame has any privileged status compared to the others.If any one wants to use a particular frame for some reason or another we can not question him.But one can not and should not object to the use of a different frame which computes a much younger universe, say an universe only 2000 years old !!! Lo and Behold! 2000 years can not be true as we know that we existed from much earlier days onwards. Hence the interesting question is: what may be a more realistic lower bound for the age of the universe?

Answer 1

When we are doing a calculation in general relativity we usually have to choose some coordinates, and one of these will be the time coordinate. The time measured using our coordinates is then called the coordinate time.

It is important to understand that the coordinate time is just a label we use to identify points in spacetime, and it need not and frequently does not have any physical meaning. A quick search of the site reveals this has been discussed already in the question Does coordinate time have physical meaning? The only times that have an observer independent meaning are proper times. We can calculate the proper time for some trajectory using our coordinates, and no matter what coordinates we choose we will get the same answer. The proper time is a scalar invariant.

This applies when we apply GR to the universe. While it would be a bit eccentric to use the coordinates for an observer boosted relative to the CMB, it is common to use conformal time as well as comoving time as the time coordinate. Either is a perfectly good time coordinate.

So if you are asking about the age of the universe measured in coordinate time then yes this is an arbitrary measure and will have different values in different coordinate systems.

But while we are free to choose any coordinates we want, some coordinates have more physical relevance than others. For example if we are considering the age of the universe we expect this to be related to the age of the objects within it in a straightforward way. For example if we know the age of the oldest star in the universe then we expect the age of the universe to be equal to the age of the oldest star plus a bit. And this is where the comoving coordinates come in because most of the stuff in the universe is approximately comoving, and that means the age in proper time is equal to the age in the comoving coordinate time.

And this is why the age of the universe is generally taken to mean the age in comoving time. It's because for the vast majority of the stuff in the universe the comoving time is equal to the proper time and this makes the comoving time an obvious and convenient choice.

Answer 2

You are correct in stating that the age of the universe depends on the observer/frame of reference. This is unavoidable because of the theory of relativity.

However, when it comes to discussing cosmology there is a preferred frame of reference that makes sense to consider. One of the cosmological assumptions is that the universe is homogeneous and isotropic. This is assumed in order to avoid believing that we occupy a special place in the universe, a mistake similar to the one that Ptolemy made.

Now comes the interesting bit: If the universe is homogeneous for one observer, it does not mean it is homogeneous for all of them! Indeed, if an observer is moving with respect to an observer who sees a homogeneous universe, because of length contraction the universe will look very different to her.

So the assumption becomes that "There is an observer (reference frame) for whom the universe is homogenous and isotropic". When we talk about the age of the universe, we refer to the proper time of that observer.

For more information and depending on how much depth you would like to go, you could read more about the FRW metric.

Answer 3

The proper age of the universe is the one that is measured the longest out of all frames of reference. It can be shown that it is the comoving time. Thus the comoving time is the age of the universe not as an arbitrary or convenient choice, but because it matches the proper age.

In the $\Lambda\text{CDM}$ cosmology, the Friedmann equations have an analytic solution for a flat universe dominated by matter and dark energy. This closely describes the universe since a very young age of about when the Cosmic Microwave Background radiation was emitted. In this solution, the scale factor of the expansion depending on time is given by

$$ a(t)=\sqrt[3]{\dfrac{\Omega_{m,0}}{1-\Omega_{m,0}}}\sinh^{\frac{2}{3}}{\left(\dfrac{3}{2}\sqrt{1-\Omega_{m,0}}\,H_o t\right)} $$

Where $\Omega_{m,0}$ is the current total matter density, $H_o$ is the current Hubble parameter, and $t$ is the current age of the universe.

Because the current scale factor is unity, the ratio of the current age of the universe to the current Hubble time based on the formula above is

$$ \dfrac{t}{t_h}=H_ot\approx 0.99$$

Answer 4

I think this question has may have been answered before, but perhaps not in the way you are asking.

From the perspective of our experience we consider that the dimensions of space and time are fixed but relativity tells us that this is not so. Both space and time can bend (as you clearly understand).

So when our space and our time are not uniform how do we find an absolute measurement of them? Does it even make sense to ask for an absolute measurement of them?

Any measurement we take must be from the point of view of a frame of reference, but how can we say that one frame of reference is the one for 'absolute' answers?

It's easier to think of space rather than time first. If, in your example, your two travellers start and finish at the same point, and travel in (from their perspective) a straight line, but one of them experienced space bending, which of them has travelled further? They have started and finished at the same points, but the space itself that they have travelled through may not be the same.

So if on a journey one of them experiences time bending, which of them has travelled longer? They start and finish at the same time, but the amount of time they have travelled through may be different.

Our experience tells us that space and time are different - we experience motion through time at a uniform rate. Mathematically though we can see that time is not uniform in the way we expect.

Our experience wants us to have an absolute answer and for most of things we do we can ignore any relativistic spacetime bending effects so we don't have to grapple with this question.

But for both relatvistic and quantum effects the universe does not behave the way our experience tells us it does - but our natural instinct is that we still want to operate from what our experience tells us is true.

So, not a deinitive answer to your question, but I hope it is helpful in helping you to grapple with the concepts you are raising.