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Our perception of the Universe is from observation at a local point.
Stars are both moving towards and away from us. Nearby galaxies are both moving towards and away from us.
On a large scale, distant galaxy clusters appear to be moving away from each other.
From special relativity, objects cannot travel faster than the speed of light but space itself can expand faster than the speed of light.
The Big Bang occurred 13.7 billion years ago but the edge of the observable Universe is now 46.5 billion light years.
In fact, the observable Universe can be smaller than the entire Universe as objects move beyond the cosmic event horizon.
Consider travel on the Earth. Coordinates determine our exact position on the globe and our local time. We can fly in a "straight" line and end up back where we started. We can even fly above the Earth and get a picture of our three-dimensional habitat. However, our every day experiences are in terms of a flat (two-dimensional) Earth e.g. local street directory. "Down" means to the centre of the Earth, a straight line, but north means to the North Pole, which is a actually a curved line. It's not too hard to extrapolate our local two-dimensional existence to a three-dimensional Earth.
However, a non-Euclidean model of the Universe is something way beyond our experience. The problem is that space-time is at least a four-dimensional concept and we can't get outside our Universe and get a picture of it. The analogy of ants living on the surface of an expanding sphere is usually used to grasp some concept of an expanding Universe. In reality, we have three dimensional space expanding with time. Up and down have some meaning locally but not on a Universe scale. In a closed Universe, space wraps back around on itself. A non-diverging laser beam could theoretically cross the whole Universe and return to its starting point. In practice, the Universe is not perfectly uniform and the laser would miss itself. In any case, the laser, traveling at light speed, will most likely lose its chase to get back as the Universe continues to expand.
The topology of the Universe is a theoretical description of the way matter influences space. The curvature of space is not something we can see other than in secondary phenomena such as the bending of starlight around the Sun or gravitational lensing. The higher the mass, the more the contraction of the space around it. In normal circumstances, we cannot tell whether the light from a star got to us in a straight line or a curved line. In fact, other than in a flat geometry, the very idea of a straight line is meaningless. Although the type of topology may be supported by scientific observations, there is nothing in our simple observations as space travelers that would lead us to distinguish between the models. This is because local experiences do not reveal the overall picture. To all intents and purposes we may seem to be traveling in a straight line while actually following a geodesic resulting from both local and Cosmic gravitational influences.
The topology of the Universe is important in terms of understanding its evolution and direction. The amount of matter (average matter density) determines the geometry of space and whether the Universe continues to expand or contract in the long term, whether expansion accelerates or decelerates. Dark energy is of course another added complication.
My viewpoint and interpretation.
Regards, Rob
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