Undergrad Curvature of space in large regions: zero or not?

[harrylentil]

I have read numerous times that the overall curvature of space in extremely large regions -1000s of megaparsecs say - is zero. I also keep reading that the expansion rate of the universe is increasing, and that the universe is resultantly positively curved. I would be interested in a reconciliation of these apparently contradictory facts.

 

[lomidrevo]

One should distinguish between curvature of space and curvature of spacetime.

 

[harrylentil]

One should distinguish between curvature of space and curvature of spacetime.

I would be happy if that was part of the explanation. I don't understand the concept of the curvature of spacetime so I didn't use the phrase.

 

[Orodruin]

I would be happy if that was part of the explanation.

It is not part of the explanation. It is the explanation. If you do not understand the concept of curvature of spacetime vs curvature of space, then that is where you have to start learning.

When we talk about curvature of space, we talk about the curvature of a three-dimensional submanifold of spacetime, a "surface of simultaneity". When we talk about the curvature of spacetime we talk about the curvature of a 4-dimensional manifold.

 

[PeterDonis]

I don't understand the concept of the curvature of spacetime so I didn't use the phrase.

Not explicitly, but you did implicitly. The "curved" in the bolded part of the quote from you below is curvature of spacetime, not space.

I have read numerous times that the overall curvature of space in extremely large regions -1000s of megaparsecs say - is zero. I also keep reading that the expansion rate of the universe is increasing, and that the universe is resultantly positively curved.

In other words, the 4-dimensional spacetime of the universe has positive curvature, but it is "sliced up" into 3-dimensional spatial surfaces (of constant time) that are flat. It's no different from cutting flat slices out of a roughly spherical apple. No contradiction at all.

 

[timmdeeg]

In other words, the 4-dimensional spacetime of the universe has positive curvature, ...

Hm, what determines the sign of spacetime curvature, the sign of the stress-energy tensor ? Would negative curvature of the the 4-dimensional spacetime of the universe account for a negative stress-energy tensor? And if yes, would this be true if repelling gravity dominates attractive gravity?

 

[PeterDonis]

what determines the sign of spacetime curvature

Spacetime curvature in general doesn't have a single sign, at least not in any dimension higher than two. The Riemann curvature tensor in 4 dimensions has 20 independent components.

the sign of the stress-energy tensor ?

The SET in general doesn't have a single sign either; in 4 dimensions it has 10 independent components.

The positive curvature the OP referred to is the positive curvature of de Sitter spacetime, which is an idealized model of a universe containing nothing but dark energy, which is expanding exponentially. (Our actual universe isn't quite like this now, but it will get more and more like it as time goes on.) This spacetime is maximally symmetric, which is why its curvature can be given a single sign.

 

[timmdeeg]

The positive curvature the OP referred to is the positive curvature of de Sitter spacetime, which is an idealized model of a universe containing nothing but dark energy, which is expanding exponentially

Ah, I see, only in this special case does it make sense to talk about a sign of thr SET. Thanks for clarifying. How about the special case the universe contains nothing but energy density represented by $T^{00}$? Does the SET have a single sign then?

 

[PeterDonis]

How about the special case the universe contains nothing but energy density represented by $T^{00}$? Does the SET have a single sign then?

$T^{00}$ is a component of the SET. If it's the only one that is not zero, as you have proposed, what do you think the answer to your question is?

 

[timmdeeg]

$T^{00}$ is a component of the SET. If it's the only one that is not zero, as you have proposed, what do you think the answer to your question is?

$T^{00}$ and thus the sign of the SET should be positiv then.

I should try to get some basic understanding of the mathematical background of tensors as I’m stumbling across such questions again and again.