Twin paradox and the size of the universe

[Grinkle]

I am probably over-simplifying this, but it seems to me that they would agree on the size of the observable universe, and they would disagree on the rate of expansion of the universe. I am assuming that they infer the age of the universe by mathematically running the trajectory of observable matter backwards until we see it converges to a point, and calculating how long that takes. That might be way off base.

Alice would say that the cosmological constant (I am dropping a phrase hear that I am not qualified to drop, please help me if its being used in a nonsensical context) had a different value during her trip than before and after her trip.

Am I missing something in taking that perspective?

 

[PeterDonis]

it seems to me that they would agree on the size of the observable universe

Since Alice and Bob are at rest relative to each other when they make the comparison, yes, they would. But while Alice is moving relative to Bob, she would not assign the same "size" to the observable universe, at least not if she defined "size" as "size in the frame in which Alice is at rest". (There are plenty of technicalities involved here, but I don't think we need to go into them.)

and they would disagree on the rate of expansion of the universe.

If they calculate "rate of expansion" as "amount of proper time it took by my clock to get to its current size", then yes, they would. However, this is not necessarily the correct way to calculate "rate of expansion". See below.

I am assuming that they infer the age of the universe by mathematically running the trajectory of observable matter backwards until we see it converges to a point, and calculating how long that takes.

That's basically what is done in cosmology, yes (although there are plenty of complications which we don't need to go into here). However, as was noted in earlier posts in this thread, "how long" is defined as "how long according to a comoving observer's clock". The reason for that is that comoving observers have a special property: they are the only ones who see the universe as homogeneous and isotropic, all the time. Alice is not a comoving observer (because during the time when she moves relative to Bob, she does not see the universe as homogeneous and isotropic), so if she runs the same calculation but defines "how long" as "how long according to Alice's clock", she will get a different answer.

However, there is another way to define the "rate of expansion" of the universe which does not have this problem. There is a quantity called the "expansion scalar" which is frame invariant; it's the same for all observers. This quantity has the right units to be a "rate of expansion" (it basically is the fractional rate of increase of the scale factor), so it can be used to give an invariant meaning to that term. If Alice measures the expansion scalar while she is moving relative to Bob, she will get the same answer Bob does; so in this sense, they will agree on the rate of expansion of the universe.

Alice would say that the cosmological constant (I am dropping a phrase hear that I am not qualified to drop, please help me if its being used in a nonsensical context) had a different value during her trip than before and after her trip.

No, she wouldn't. The cosmological constant is frame invariant, like the expansion scalar above; it has the same value in all frames and for all observers.

 

[Grinkle]

Thanks for the very digestible reply. You have given me some interesting hooks for further reading - much appreciated.