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Flat Universe as a Torus

  1. Dec 20, 2015 #1
    A lot of cranks on the net make a lot of the torus - of course magnetic fields are toroidal - but not space time right? Or could it be - a torus is flat (zero gaussian curvature) which is how we observe the universe to be as far as we have measured it's curvature on large scales - and it would also be a way for the universe to be finite while still having zero curvature - my question is - does anyone know of any legit research that posits toroidal solutions to Einstein's equators? Do these solutions exist and are they stable?
  2. jcsd
  3. Dec 20, 2015 #2


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    You need to distinguish between curvature (i.e., spacetime curvature) and spatial curvature. Our universe has nearly zero spatial curvature, but its curvature is not zero.

    A torus with zero curvature is a solution to the Einstein field equations without matter. This is just Minkowski space with a nontrivial topology, and it's a universe without gravity.

    Yes, it's stable, but stability isn't something we expect in a cosmological solution.

    The zero-curvature torus is not a possible model of the universe we live in, because it's incompatible with observations such as the cosmic microwave background.

    Standard cosmological models have nearly zero spatial curvature, but not necessarily exactly zero. Because measurements always have a range of uncertainty, any measurement of the curvature that's consistent with zero spatial curvature is also consistent with positive or negative spatial curvature. If the spatial curvature is positive, then the universe does have the topology of a torus (direct product of a three-sphere with a line).
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