# What is the fate of a 3-Torus universe?

Since it is flat, will it expand forever like a flat and open universe or collapse like a closed and curved universe?

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Huh? What is the equation you are assuming? Are you asking in the context of the FLRW solutions? –  Willie Wong Apr 11 '12 at 11:29
Yes. Can the FLRW solutions be applied to a 3-Torus? If not, is there any equivalent way to model a 3-Torus topology universe's history? –  Ocsis2 Apr 11 '12 at 12:04

Starting with $\mathbb{T}^3$ with the standard metric, it is just $\mathbb{R}^3/\mathbb{Z}^3$. In particular, taking the FLRW ansatz $\mathrm{d}s^2 = -\mathrm{d}t^2 + a(t)^2 \mathrm{d}\Sigma^2$ where $\mathrm{d}\Sigma^2$ is the flat Euclidean metric, you see that modding out the spatial slice by translations you get immediately a solution with spatial slice being the 3-torus. So the geometry of the universe (locally in space but globally in time) will be identical to that of the flat FLRW solution.

In other words, if you take the flat FLRW solution with Euclidean coordinate system $(x,y,z,t)$ such that the metric is

$$-\mathrm{d}s^2 = - \mathrm{d}t^2 + a(t)^2 \left(\mathrm{d}x^2 + \mathrm{d}y^2 + \mathrm{d}z^2\right)$$

and restrict the coordinates $x,y,z \in [0,2\pi)$, this will give you a coordinate representation of the universe with flat $\mathbb{T}^3$ slices.

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Indeed, the istropic and homogeneity assumptions for the FLRW model implies that the spatial slices are symmetric spaces. And so for any quotient of the 3-sphere, Euclidean space, or the hyperbolic space by discrete symmetry you get a corresponding quotient of the FLRW solution. So in particular you can also get compact hyperbolic 3 manifolds as slices, or stuff like lens- or prism- spaces. It is implicit that by the symmetry assumption, understanding their universal covers (which are precisely the three model geometries) is enough to understand all of them. –  Willie Wong Apr 11 '12 at 13:42
From your previous questions I'd guess Willie's answer is a bit technical. The answer is basically that there is no difference between a universe that is globally a torus and one that is globally an infinite flat sheet. You'd expect this because the field equations are local i.e. they say nothing about the large scale topology. –  John Rennie Apr 11 '12 at 13:45
in any case the isotropy property is only a local one; a 3-torus has always 3 preferred directions, just like a regular 2-torus has 2 preferred directions (maximal/minimal geodesics). They might not be manifest but on the largest scale structures –  lurscher Apr 11 '12 at 14:59