Your question may contain more nuance than the number of words it contains.
However, I'll take it at face value and say the answer is contained within my previous link.
The overall geometry and, crucially, fate of the Universe are controlled by the density of the matter within it.
A universe of 'critical density' is said to be flat.
The critical density for the Universe is approximately 10^-26 kg/m^3 (or 10 hydrogen per cubic metre).
The Universe seems to be balanced on a knife edge according to this link: https://www.astronomy.swin.edu.au/cosmos/C/Critical+Density
So you would come back going in 2 different directions, the short tour or the long tour.
Geometry as per above definition - is it Euclidean geometry?
The three curvatures create universes that are = The three curvatures create Topologies that are .. OK?
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Geometry as per above definition - is it Euclidean geometry?
The three curvatures create universes that are = The three curvatures create Topologies that are .. OK?
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I find it hard to believe that something that is 13.8 billion yrs old is ‘balanced on a knife edge’. It appears to be on a knife edge but it’s probably more likely there is an underlying process driving it.
Your final question lacks clarity, TNT, but I'll give it a go!
You are probably asking something quite different from what I perceive to be the case.
And please note that I am an eager but ignorant student when it comes to topology!
The geometry of flat space is Euclidean since lines that set out parallel on its surface will stay parallel, unlike those on the surface of a sphere.
Mathematically, the 2-torus can also be represented by a flat rectangle like in the Asteroids computer game (see attachment).
The three plausible cosmic geometries are consistent with many different topologies.
Try searching for the topology of a Spherical Universe and that of a Hyperbolic Universe and let us know how you get on.
For Hyperbolic Universe, I've found this image...
...but I'm none the wiser! 🤓
Attachments
The three curvatures create universes that are = The three curvatures create Topologies that are .. OK?
Is Universes here, equal to Topologies? It was from your wording, i.e:
"Under the Standard Big Bang theory, the Universe could have one of three different curvatures: positive, negative and flat. The three curvatures create universes that are fundamentally different in their geometry. "
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This flatness of the Universe in higher dimensions subject is very troublesome.
I found this questionable statement a good starting point:
What is that about? Seems it is more correctly about the Euler Characteristic of a familiar convex polytope, here embedded in 3 dimensions:
OK, I think I get the idea. Draw it on a 2-sphere, do the sums, and it always comes out at 2:
Now a 2-sphere has positive Gaussian Curvature, which mathematically is the product of two orthogonal curvatures:
Now here is where it gets weird. Draw, say, a cube on a 2-torus and it comes out at 0, because your hypothetical ant crawling around on it loses two faces:
To a topologist that is 4 cubes laid end to end, you follow?
So is this French building in Paris:
To a geometrical algebra person it is of course a 4D tesseract, and I think that is what the architect is trying to convey, though IMO, most lay people wouldn't know a tesseract from a hole in the ground, CUE:
So it's Donuts all the way, since the Donut is commendably flat and has an Euler Characteristic of 0:
I thought it was all done and dusted at that point, but I am still disquieted. Two things.
1) The Donut has a variable Gaussian Curvature:
Now the Gauss-Bonnet Formula says that all averages out, but I really don't know. And amazingly relates the Euler Characteristic to the Gaussian Curvature:
https://mathworld.wolfram.com/Gauss-BonnetFormula.html
2) It all gets a bit weird in higher dimensions too according to ace 4D man, Greg Egan who is all buddy with John Baez:
https://www.gregegan.net/APPLETS/29/HypercubeNotes.html#EULER
So it alternates!
https://www.wired.com/story/the-hunt-for-ultralight-dark-matter/
Hope it helps. 🙂
I found this questionable statement a good starting point:
What is that about? Seems it is more correctly about the Euler Characteristic of a familiar convex polytope, here embedded in 3 dimensions:
OK, I think I get the idea. Draw it on a 2-sphere, do the sums, and it always comes out at 2:
Now a 2-sphere has positive Gaussian Curvature, which mathematically is the product of two orthogonal curvatures:
Now here is where it gets weird. Draw, say, a cube on a 2-torus and it comes out at 0, because your hypothetical ant crawling around on it loses two faces:
To a topologist that is 4 cubes laid end to end, you follow?
So is this French building in Paris:
To a geometrical algebra person it is of course a 4D tesseract, and I think that is what the architect is trying to convey, though IMO, most lay people wouldn't know a tesseract from a hole in the ground, CUE:
So it's Donuts all the way, since the Donut is commendably flat and has an Euler Characteristic of 0:
I thought it was all done and dusted at that point, but I am still disquieted. Two things.
1) The Donut has a variable Gaussian Curvature:
Now the Gauss-Bonnet Formula says that all averages out, but I really don't know. And amazingly relates the Euler Characteristic to the Gaussian Curvature:
https://mathworld.wolfram.com/Gauss-BonnetFormula.html
2) It all gets a bit weird in higher dimensions too according to ace 4D man, Greg Egan who is all buddy with John Baez:
https://www.gregegan.net/APPLETS/29/HypercubeNotes.html#EULER
So it alternates!
https://www.wired.com/story/the-hunt-for-ultralight-dark-matter/
Hope it helps. 🙂
Someone observing the torus from the outside (extrinsic observer) sees both positive and negative curvature.
However, someone living on the surface of a torus (intrinsic observer) sees zero curvature.
Here, we are using the surface of a torus to represent a flat Universe.
Geometry expresses the curvature of the Universe as permitted by the equations of general relativity. The geometry we are interested in is that of the flat Universe. Geometry looks at the Universe on the 'local' scale. Locally at least the Universe is flat.
Topology is also related to curvature, but looks at the Universe on the 'global' scale, providing a qualitative description of the actual shape of the Universe. Such a qualitative description would be that the Universe is shaped like a torus - or more accurately a 3-torus.
You can think about topology as determining whether the Universe contains holes which form closed loops that would enable you to travel out in the Universe without turning, yet end up back where you started.
Einstein described the Universe as 'finite' but 'unbounded'. If you live on the surface of a torus, you may walk around that torus as much as you like and never come to an edge. That is, your Universe is 'unbounded', despite the fact that it has a 'finite' area.
Jings, those few words took me a while! I hope my process of discovery helps your understanding as much as it has helped mine.
A hyperbolic shape or manifold is one in which a line of arbitrary length drawn on the surface always forms an open curve ie it is not closed and the ends will never meet. An example of this is the trajectory of Oumaouna which for an observer within the solar system was hyperbolic. The conclusion from this is that the Oumaouma is from another star system and was not an object orbiting the sun.
With respect to the shape of the universe, the saddle shape shown earlier forms a hyperbolic manifold and describes a universe with Omega>1 which means it is open and a photon will never ever return to its origin. For a closed universe, Omega <1 (spherical or toroidal) and a photon will eventually wrap back on itself. Current measurements (WMAP and Planck) indicate a flat universe with Omega ~1+- 0.0098. Whether our flat universe is ultimately hyperbolic or spherical is still being debated - it isn’t fully settled given the +-0.098 flatness spread.
With respect to the shape of the universe, the saddle shape shown earlier forms a hyperbolic manifold and describes a universe with Omega>1 which means it is open and a photon will never ever return to its origin. For a closed universe, Omega <1 (spherical or toroidal) and a photon will eventually wrap back on itself. Current measurements (WMAP and Planck) indicate a flat universe with Omega ~1+- 0.0098. Whether our flat universe is ultimately hyperbolic or spherical is still being debated - it isn’t fully settled given the +-0.098 flatness spread.
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If Ω>1 the geometry of the Universe is elliptic (includes spherical); if Ω<1 the geometry is hyperbolic; and if Ω=1 on the nose then it is Euclidean.
Naively, it seems unlikely that the geometry of the Universe is Euclidean considering it is balanced on the knife edge of Ω=1.
Just one hydrogen atom less than the critical amount and the lower mass-energy density would tip the flat Universe into hyperbolic geometry.
The current estimates for Ω leave the question of the geometry of the universe open. It is still possible that the geometry of our universe is hyperbolic or elliptical.
P.S. Each geometry type has possible universe topologies attached to it. Physicists describe the topology of the Universe as a 3-manifold. A 3-manifold will comply with either elliptic geometry, hyperbolic geometry or Euclidean geometry. For Euclidean geometry, I've mentioned the 3-torus. For elliptical geometry the simplest 3-manifold is the 3-sphere, or glome - a topology that Einstein assumed the Universe to have when he first solved his equations for general relativity.
EDIT: I should have defined Ω. It equals the ratio of the actual mass-energy density of the universe to the critical one.
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One reason why it is not so is that if it was, we would be drown in bright light from everywhere. Or the size is so big that it hasnt come back yet - but that day - wow - we will see BB replayed in front of our eyes - big screen ;-D
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The scenario does give one food for thought because if photons were to start going in one direction and keep that direction, they would end up in back at the spot where they started.
What astronomers might think is a distant galaxy could actually be the Milky Way - seen at a much younger age because the light has taken billions of years to travel around the universe!
Looks like I got my < and > mixed up. Apologies and thanks for correcting that Galu.
Re my post #4332 above, the flatness spread is not 0.098 but 0.0098 ie c 1% (from the Planck satellite data).
I think these terms like geometry and topology are best described as indicating how matter and radiation behave in the cosmos. So for example, the universe isn’t actually shaped like a hyperbolic saddle, or like a flat sheet or a toroid - it just describes how things behave as the move in the cosmos.
I think these terms like geometry and topology are best described as indicating how matter and radiation behave in the cosmos. So for example, the universe isn’t actually shaped like a hyperbolic saddle, or like a flat sheet or a toroid - it just describes how things behave as the move in the cosmos.