B How can the Universe be infinite and yet have a finite age?

[timmdeeg]

Topology has nothing to say about that. A spacelike slice could have topology $R^3$, or $T^3$, the 3-torus, for that matter, and still not have the same curvature everywhere. There are some particular restrictions that topology can place on possible curvatures: for example, no sphere topology ($S^n$ for any $n$) can be flat (have zero curvature everywhere), and the 2-torus, $T^2$, also cannot be flat (but the 3-torus can). But those restrictions don't restrict very much--for example, there's nothing preventing a manifold with topology $S^3$ from having curvature that varies from point to point, or even from being flat in some finite region (just not everywhere).

Thank you very much for this detailed and enlightening answer.

 

[kimbyd]

There are some particular restrictions that topology can place on possible curvatures: for example, no sphere topology ($S^n$ for any $n$) can be flat (have zero curvature everywhere), and the 2-torus, $T^2$, also cannot be flat (but the 3-torus can).

I don't think you can extend this to the observable universe. These topologies cannot be globally flat. But any topology can be locally flat. As we can only see a finite distance away, I'm pretty sure it's possible that any topology combined with a universe significantly larger than the observable universe can be flat within the observable universe.

 

[timmdeeg]

A sphere might seem to be flat locally. But doesn’t that depend on the precision of our measurement? Isn’t flat in this context true only for an infinitesimally small area?

 

[PeterDonis]

I don't think you can extend this to the observable universe.

Agreed; the observable universe could be perfectly flat without excluding a global topology like $S^3$. That's why I specified that $S^3$ cannot be flat everywhere; it can be flat in a finite region, just not everywhere.

 

[PeterDonis]

A sphere

You are confusing two meanings of the word "sphere". The relevant one for this thread is "topology $S^n$". But the one you are implicitly using is "topology $S^n$ plus constant curvature". The two are not the same, and only the former meaning is intended in this thread.

 

[kimbyd]

A sphere might seem to be flat locally. But doesn’t that depend on the precision of our measurement? Isn’t flat in this context true only for an infinitesimally small area?

To expand a bit upon what PeterDonis said, a blobby ball has a spherical topology. So does a sea cucumber. Or a sea urchin. Or an octopus. It is very possible for a local region with spherical topology to be perfectly flat, at any measurement accuracy.

The main argument against such a thing is that it requires fine tuning: if there's a spherical topology, there will be many regions on the surface that will not be flat. So you would need fine-tuning to land in a place on the surface that is.

 

[rootone]

,,, if there's a spherical topology, there will be many regions on the surface that will not be flat. So you would need fine-tuning to land in a place on the surface that is.

Apollo 11.

 

[timmdeeg]

You are confusing two meanings of the word "sphere". The relevant one for this thread is "topology $S^n$". But the one you are implicitly using is "topology $S^n$ plus constant curvature". The two are not the same, and only the former meaning is intended in this thread.

Thanks. Yes indeed, I was thinking of the latter. Is that, "topology $S^n$ plus constant curvature", restricted to a FRW universe or generally to a universe which obeys the cosmological principle?

 

[timmdeeg]

To expand a bit upon what PeterDonis said, a blobby ball has a spherical topology. So does a sea cucumber. Or a sea urchin. Or an octopus. It is very possible for a local region with spherical topology to be perfectly flat, at any measurement accuracy.

The main argument against such a thing is that it requires fine tuning: if there's a spherical topology, there will be many regions on the surface that will not be flat. So you would need fine-tuning to land in a place on the surface that is.

That’s helpful, thanks!

 

[Ibix]

a blobby ball has a spherical topology. So does a sea cucumber. Or a sea urchin. Or an octopus.

Point of pedantry - an octopus at least has a digestive tract, so is topologically a torus (or possibly something more complex if it has something like gills as well - I'm no marine biologist) since it effectively has a hole right through it. A stuffed toy octopus, though, lacking such biological messiness, would be topologically a sphere.

 

[PeterDonis]

Is that, "topology $S^n$ plus constant curvature", restricted to a FRW universe or generally to a universe which obeys the cosmological principle?

It's restricted to a spacetime that has spatial topology $S^n$ and constant spatial curvature. Such spacetimes are a subset of FRW spacetimes or spacetimes that obey the cosmological principle (since there are other topologies besides $S^n$ that allow constant spatial curvature).

 

[kimbyd]

Point of pedantry - an octopus at least has a digestive tract, so is topologically a torus (or possibly something more complex if it has something like gills as well - I'm no marine biologist) since it effectively has a hole right through it. A stuffed toy octopus, though, lacking such biological messiness, would be topologically a sphere.

Ah, yes, you're right. I didn't realize it had a full digestive tract. I misremembered.

 

[Fervent Freyja]

If the universe does indeed bounce in between cycles of beginning and ending, then it can be considered infinite, with a finite "time" in between each cycle. Conceptually, this makes sense, more so than one universe having a single beginning and end, and nothing else existing after that one cycle. With so many possible combinations with particles at each beginning, each new cycle can be different than prior.