Spacetime curvature and curvature index

[Ranku]

TL;DR

What is the difference between spacetime curvature and curvature index of space?

The presence of the cosmological constant produces a flat spacetime universe with Ω = 1. There is also the curvature index of space k, which can be +1, 0, -1. But it is possible to have any of these values of k with Λ > 0 or Λ < 0. How is the curvature of spacetime determined by Λ different from that of the curvature index of space k?

 

[Orodruin]

Spacetime curvature is a tensor. The curvature index is a normalised description of the curvature of a homogeneous and isotropic spatial hypersurface.

 

[Ranku]

Spacetime curvature is a tensor. The curvature index is a normalised description of the curvature of a homogeneous and isotropic spatial hypersurface.

Could you elaborate a bit more, in terms of how is it that spacetime curvature and curvature index of space don't have to agree, and yet each is supposedly uniquely describing the curvature and fate of the universe?

 

[Orodruin]

They do not have to agree because they are completely different things.

 

[Ranku]

They do not have to agree because they are completely different things.

So, with Ω = 1, which describes a flat universe, and suppose k= +1, which describes a closed universe, what would be the fate of such a universe?

 

[Orodruin]

If $\Omega = 1$, then $k = 0$. You cannot say what is the fate unless you know how the energy content splits into components.

 

[Ranku]

If $\Omega = 1$, then $k = 0$. You cannot say what is the fate unless you know how the energy content splits into components.

If Ω = 1, then k = 0, then that implies a consistent correlation, even if they are "completely different things". I am trying to understand that.

 

[Orodruin]

If Ω = 1, then k = 0, then that implies a consistent correlation, even if they are "completely different things". I am trying to understand that.

$\Omega$ is not the spacetime curvature. It is the energy density divided by the critical energy density. The critical energy density is defined as the energy density at which $k = 0$.