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As you look at circle cross-sections around a sphere it's obvious that as you move the circles along the z-axis, the circles get bigger and smaller. When you're at the farthest points on the $z$-axis, the cross-sectional circles shrink to a point, and reach their largest around the middle of the circle.

I'm curious why the third-dimension shouldn't work this way along a fourth-dimensional axis, like time?

Wouldn't it stand that the spatial dimension simply "grow" along the axis of the fourth dimension (time), such that at the beginning of time the three dimensions would be of infinitesimal dimension, a point, but as you move from three dimensional cross-section to three dimensional cross-section along the fourth-axis time away from the first point on that fourth axis that the three dimensions rapidly expand?

Would the big bang and big crunch then, rather than being a surprising quality of our universe, be simple and geometrically provable qualities of a four-dimensional space with a time axis -- the exact thing you'd expect as you move three dimensions along a fourth dimension -- at the beginning of time, point, rapidly expanding to three dimensions, and at the far end of time, collapsing back down to a point?

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  • $\begingroup$ You described a version of a closed universe where the "sphere" was your assumption. Another assumption may be a "straight cone" that would describe a closed universe that expands forever (not FLRW). As Javier mentions in his answer, another yet scenario is that the universe has been infinite from the beginning. This model is singular (infinite) on mass and distance (in addition to density) and thus has no physical meaning, but still is accepted by many for some reason. Also note that the spacetime geometry is hyperbolic. Check out the de Sitter and anti deSitter geometries. $\endgroup$ – safesphere Sep 21 '17 at 4:15
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Well, the particular model of the universe that was mostly accepted until some 20 years ago was more or less like this, with a Big Bang and a Big Crunch. The problem with your argument is that you have identified a possible way the universe could be, but how do you know it must be this way? A two dimensional space can be a sphere and have the property you describe, but it could also be a plane, and be infinite and eternal. Similarly, the universe could be a four-dimensional sphere (sort of) and hence be finite and have a beginning and an ending, but the math shows that there are other possible shapes the universe could take.

In fact, current observations seem to show that the universe is infinite in the spatial direction, and had a beginning but no ending, which means that your "provable qualities" seem to be false in reality.

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  • $\begingroup$ I believe, more precisely, the observation is that space is flat. It is then a conclusion of the Friedmann model that flat space must be infinite. $\endgroup$ – safesphere Sep 21 '17 at 4:12

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