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#55
Nov2212, 10:51 PM

P: 15




#56
Nov2312, 07:28 PM

P: 398

Ω > 1, Closed => Elliptical space: parallel lines intersect, the value of Ω changes, but is always > 1. Space contracts, leading to "Big Crunch" Ω = 1, Flat => Euclidean space: Parellel lines do not intersect, and parallel lines only come in unique pairs. Geometry does not change. Space expands, but decelerates, so that space is constant after an infinite amount of time. Ω < 1, Open => Hyperbolic space: Parallel lines do not intersect, but to contrast with Euclidean geometry, there are infinitely many unique, parallel, non intersecting lines. I believe all parallel lines diverge. Geometry changes, space expands at an increasing rate forever. We live in Ω = 1 with Dark Energy. The difference between this and an Ω<1 universe is that there are not infinitely many unique parallel lines ones can draw in a moment in time, even with expansion due to D.E. 


#57
Nov2312, 10:23 PM

P: 398

By "unique" parallel lines, I mean this: In Euclidean geometry, we can draw many lines that are parallel to one another, but they are simply translations of one another, which doesn't mean much. If we took a eucliden geodesic and rotated it by ANY angle theta, it will intersect once, at some point. In Hyperbolic geometry, if we have two geodesics of some finite separation, we can rotate one of those geodesics by some finite angle theta such that the lines STILL do not even intersect. In fact, there are infinitely many such lines, as you can easily see, which are all UNIQUE lines. This is the difference between flat space with dark energy and open spaceboth have expanding geodesics through time, but geodesics also diverge in slices of constant time in a hyperbolic universe.
Inflation complicates the picture. In the typical description of inflation, the universe started as a point, of Ω > 1, and as it expanded, instead of quickly reaching a maximum and recollapsing, it reached a point where Dark Energy was 27 orders of magnitude stronger than it is today, and rapidly pushed the universe to be flat, that is, Ω is so close to one, we don't notice any difference, and after that very early inflationary period, the geometry of space has not changed any measurable amount. I don't know what effect this would have on the horizon of the observable universe. 


#58
Dec2912, 03:45 AM

P: 110

I respond frequently to questions that are covered in the FAQ. Any way I apologize I derailed the thread here. And there is a subforum for your new ideas that might change the world.



#59
Dec2912, 10:30 AM

Mentor
P: 22,244

Starting off in a new forum by criticizing the way they do things is not a good way to start, but there is a bigger problem here:



#60
Dec2912, 09:17 PM

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P: 22,244




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