# 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.