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Thinking about the universe as a whole. One could imagine that the three spatial dimensions each have the same, say positive, curvature, making space spherical, while time is negatively curved, making spacetime hyperbolic (saddle shaped). Is this possible? If it were possible, would $k$, the curvature term in the FWLR model, be referring to the curvature of space or of spacetime?

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  • $\begingroup$ De Sitter spacetime is of constant positive curvature. In such a spacetime one can have flat spatial sections, or sections with positive or negative 3d curvature. I don't know, if this is the same for anti de Sitter, but I would think so. $\endgroup$
    – p6majo
    Commented Apr 10, 2021 at 12:39

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The idea of constant spatial curvature comes from the idea, i.e. observation, that the spatial part of the universe at a certain instance in time (!) should be homogenous and isotropic. This leaves the three possibilities of $k=1,0,-1$, if all spatial coordinates are normalised correspondingly, because these three spatial metrics fulfill the conditions.

The idea of assigning ONE curvature to all four space-time coordinates, including time, does not make a lot of sense in a universe that we expect to be different at different points in time, since including the time dimension, the universe is NOT homogenous and isotropic (it DOES matter at which time you look, and it DOES matter if you look in the direction of the past or towards the future).

Mathematically you cannot assign a curvature to ONLY the time component, because a one-dimensional space cannot have curvature (there are no angles between different points that you could measure).

You could think of a universe where time loops, being a circle, I think this would correspond to your idea of time having $k=1$ (which, again, it has not, since mathematically a 1D circle still has no curvature). I think this would make it harder to solve the Friedmann equations compared to Standard cosmology, since you would have boundary conditions for $a(t)$ at the beginning and everytime the time component loops, which for example forbids exponential growth as in a dark energy dominated universe.

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  • $\begingroup$ Thank you. So k in FLWR refers to space, not spacetime. $\endgroup$ Commented Apr 10, 2021 at 15:18
  • $\begingroup$ Yes, in FLRW metric, k only affects the spatial part of the metric. $\endgroup$
    – Koschi
    Commented Apr 10, 2021 at 15:40
  • $\begingroup$ I am not sure that "a one-dimensional space cannot have curvature (there are no angles between different points that you could measure)" is correct. Intrinsic topological curvature does not require measurement of angles, while Gaussian curvature specifically relates the curvatures of orthogonal lines. Consider for example the pseudosphere, which has a circular cross-section and constant negative Gaussian curvature. One needs care in identifying which kind of curvature one means. $\endgroup$ Commented Apr 11, 2021 at 18:19

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