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In summary, the conversation discusses the possibility of a universe that is infinite in extent within the standard framework of cosmology. While the standard model allows for an infinite universe, it is not taken seriously beyond our horizon. The measured value of the curvature suggests the potential for an infinite universe, but it is currently unknown. The spatial curvature and its relation to the universe's size is also discussed, with the possibility of a flat, finite universe without an edge. The concept of a 3-D torus as a flat space is mentioned, and the difference between intrinsic and extrinsic curvature is explained. Ultimately, the conversation highlights the complexity and potential for different interpretations within the standard cosmological framework.

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kimbyd

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The standard model is infinite in extent, but it's not a good idea to take the standard model seriously far beyond our horizon.love_42 said:

Whether or not the universe can be infinite in extent is currently unknown.

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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:

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kimbyd

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The spatial curvature of the observable universe isn't necessarily related. With the exception of a large positive curvature, any curvature value permits either finite or infinite solutions.elcaro said:Since the measured value of the curvature is very near a flat universe, it indeed includes the possibility our universe is infinite in extent.

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elcaro

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Wouldn't a flat finite universe have an edge?kimbyd said:The spatial curvature of the observable universe isn't necessarily related. With the exception of a large positive curvature, any curvature value permits either finite or infinite solutions.

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ergospherical

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no, for example a toruselcaro said:Wouldn't a flat finite universe have an edge?

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PeterDonis

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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:With the exception of a large positive curvature, any curvature value permits either finite or infinite solutions.

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kimbyd

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If the positive curvature was a local effect only, it could still be infinite. If it was sufficiently large, it would be hard for it to be a purely local effect.PeterDonis said: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).

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elcaro

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That is not exactly flat, unless you deform it into a pancake with a hole in it...ergospherical said:no, for example a torus

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PeterDonis

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Ah, so you are considering models that are not homogeneous.kimbyd said:If the positive curvature was a local effect only

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PeterDonis

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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:That is not exactly flat

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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:That is not exactly flat, unless you deform it into a pancake with a hole in it...

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Yes, it can.PeterDonis said:A 2-D torus cannot be flat

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PeterDonis

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Ah, yes, I was forgetting the "Asteroids" arcade game.Orodruin said:Yes, it can.

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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.

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elcaro

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I can't curve my head around that...Orodruin said:Not true. You are thinking of the geometry on the torus induced by its typical embedding in three-dimensional Euclidean space. He is not.

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A simple example of the difference betweenelcaro said:I can't curve my head around that...

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jbriggs444

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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:I can't curve my head around that...

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.

An infinite universe is the idea that the universe has no boundaries or limits in terms of space and time. This means that the universe is constantly expanding and has no end or edge.

A finite universe has a limited size and would eventually reach an end, while an infinite universe has no boundaries and is constantly expanding.

One piece of evidence is the observation that the universe is expanding at an accelerating rate, suggesting that it has no boundaries and is constantly growing. Another is the cosmic microwave background radiation, which is uniform in all directions and supports the idea of a homogeneous and infinite universe.

One implication is that there could be an infinite number of planets and galaxies, potentially increasing the chances of finding extraterrestrial life. Another is that the laws of physics may be different in different parts of the universe, leading to a greater understanding of the fundamental laws of the universe.

Some theories, such as the cyclic model of the universe, propose that the universe goes through cycles of expansion and contraction, suggesting that it is not infinite. Other theories, such as the multiverse theory, suggest that there could be multiple universes with different properties, challenging the idea of a single infinite universe.

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