On the assumption of an infinite universe
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elcaro
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Since the measured value of the curvature is very near a flat universe, it indeed includes the possibility our universe is infinite in extent.love_42 said:
elcaro
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Wouldn't a flat finite universe have an edge?kimbyd said:
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no, for example a toruselcaro said:
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How can you have a spatially infinite universe with a small positive curvature (since you said "large" positive curvature instead of just positive curvature period).kimbyd said:
elcaro
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That is not exactly flat, unless you deform it into a pancake with a hole in it...ergospherical said:
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Ah, so you are considering models that are not homogeneous.kimbyd said:
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A 2-D torus cannot be flat, but a 3-D torus can be. The 3-D flat torus is the spatial geometry being referred to.elcaro said:
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Not true. You are thinking of the geometry on the torus induced by its typical embedding in three-dimensional Euclidean space. He is not.elcaro said:
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Yes, it can.PeterDonis said:
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Ah, yes, I was forgetting the "Asteroids" arcade game.Orodruin said:
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When I was an undergrad we used to play a lot of Go during breaks. Eventually we invented the game of toroidal Go by identifying the sides. It got very confusing, but fun. Very different game when you cannot cling to the borders.PeterDonis said:Ah, yes, I was forgetting the "Asteroids" arcade game.
elcaro
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I can't curve my head around that...Orodruin said:
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A simple example of the difference between intrinsic and extrinsic curvature is the cylinder. Take a flat sheet of paper and bend it into an open cylinder. The 2D differential geometry in terms of intrinsic flatness has not changed, although it is now extrinsically curved in 3D.elcaro said:
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Central to the notion of a metric space is the idea of a metric. The metric is a function that tells you how far it is from one point in the space to another (along the shortest path, of course). The 3-dimensional Euclidean metric is, of course, ##d=\sqrt{(\Delta x)^2+(\Delta y)^2 + (\Delta z)^2}## for the distance between two points given with cartesian coordinates.elcaro said:
If we have a the two-dimensional surface of a torus embedded in this three-dimensional space, we can measure the three dimensional path length of any path that stays on the surface. This allows us to induce a metric on the two dimensional space -- the length of the shortest path that stays on the surface.
But we are not required to use this metric. We can discard the connection to three dimensional Euclidean space and use a different metric. We can subtly shift the metric so that points on the outside of the torus are "closer" to one another and so that points on the inside of the torus are "farther apart". So that it becomes like a tube made from rolled up paper (still flat) and yet the two ends of the tube still meet so that the space is closed.
Still head-curving, though.