Universe Shape: Flat or Hyperbolic? Cosmologists Weigh In

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Discussion Overview

The discussion centers on the shape of the universe, specifically whether it is flat or hyperbolic, exploring implications for cosmology and geometry. Participants examine theoretical models, observational data, and the relationship between curvature and the universe's topology.

Discussion Character

  • Debate/contested
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • Some participants assert that current measurements suggest the universe is flat with a small margin of error, referencing WMAP data.
  • Others propose that the hyperbolic nature of Einstein's equations could imply a negative curvature and an open universe, questioning the implications of flatness.
  • There is a discussion about the nature of closed and open universes, with some participants clarifying that closed universes are finite and have positive curvature, while open universes have negative curvature.
  • Participants note that flat geometries can be finite or infinite, challenging the assumption that flatness implies an infinite universe.
  • Some contributions highlight the complexities of manifold topology, suggesting that flat 3-manifolds can be compact or non-compact.
  • There is a mention of the implications of hyperbolic curvature on light cone distortions and geometric properties of triangles.
  • Participants discuss the global homogeneity of manifolds, noting that not all hyperbolic spaces are globally homogeneous.

Areas of Agreement / Disagreement

Participants express multiple competing views regarding the implications of curvature on the universe's topology, with no consensus reached on whether the universe is definitively flat or hyperbolic.

Contextual Notes

Some statements rely on specific definitions of curvature and topology, and there are unresolved mathematical steps regarding the implications of different geometries on cosmological models.

DiracPool
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It seems as though the contemporary consensus among cosmologists is that the universe is basically flat and Euclidean:

The recent Wilkinson Microwave Anisotropy Probe (WMAP) measurements have led NASA to state, "We now know that the universe is flat with only a 0.4% margin of error."[1] Within the Friedmann–Lemaître–Robertson–Walker (FLRW) model, the presently most popular shape of the Universe found to fit observational data according to cosmologists is the infinite flat model

http://en.wikipedia.org/wiki/Shape_of_the_Universe

However, Einsteins relativity equations describing events in space-time appear to be hyperbolic:

http://hermes.aei.mpg.de/1998/3/article.xhtml

Wouldn't the hyperbolic nature of Einstein's relativity equations suggest an Omega of less than 1, a negative curvature and a hyperbolic geometry to spacetime? What would be the argument against this conclusion?
 
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Space news on Phys.org
I've been looking around for that second article for quite some time now I read it a few years back.
Thanks for that.

The main implication of a hyperbolic curvature is mainly an indication of a closed universe. The flat geometry was at one time the only one considered as an open or infinite universe. This changed later on when it was realized that flat does not necessarily imply infinite, you can have a flat geometry that is finite.

However the geometry also affects light cone distortions a circle geometry will make a triangle have angles greater than 180 degrees. As we have a close to flat geometry the sum of angles is close to 180 degrees. You will notice the first link each geometry shows those grid lines if you look closely you can see how each curvature affects the shapes of the grid lines


The one problem with the second article is its age. There have been numerous advances in cosmology after that's been written however its still an excellent article in the (show the maths) element

edit: forgot to mention the flat geometry is the only possible geometry that is infinite, however flat does not necessarily mean infinite as it can be finite
 
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Mordred said:
I
The main implication of a hyperbolic curvature is mainly an indication of a closed universe.
I think you mean open universe -- closed universes have positive curvature.
 
yeah sorry good catch
 
Perhaps I misunderstand, but I thought (based on the FAQ, https://www.physicsforums.com/showthread.php?t=506986) that flat and closed geometries implied that the universe was infinite. Is that not the case?

Mordred said:
I've been looking around for that second article for quite some time now I read it a few years back.
Thanks for that.

The main implication of a hyperbolic curvature is mainly an indication of a closed universe. The flat geometry was at one time the only one considered as an open or infinite universe. This changed later on when it was realized that flat does not necessarily imply infinite, you can have a flat geometry that is finite.

However the geometry also affects light cone distortions a circle geometry will make a triangle have angles greater than 180 degrees. As we have a close to flat geometry the sum of angles is close to 180 degrees. You will notice the first link each geometry shows those grid lines if you look closely you can see how each curvature affects the shapes of the grid lines


The one problem with the second article is its age. There have been numerous advances in cosmology after that's been written however its still an excellent article in the (show the maths) element

edit: forgot to mention the flat geometry is the only possible geometry that is infinite, however flat does not necessarily mean infinite as it can be finite
 
dchartier said:
Perhaps I misunderstand, but I thought (based on the FAQ, https://www.physicsforums.com/showthread.php?t=506986) that flat and closed geometries implied that the universe was infinite. Is that not the case?
No. Closed universes, as the name suggests, are finite -- they are described by closed and bounded manifolds. A sphere is an example of a closed and bounded surface with a finite volume.

Flat universes may or may not be infinite. The Euclidean plane is an example of an infinite, flat surface; a torus is an example of a finite, flat surface.
 
Whoops, sorry, I miswrote -- I meant an open (negative curvature) universe, not a closed universe. Is an open universe necessarily infinite?

bapowell said:
No. Closed universes, as the name suggests, are finite -- they are described by closed and bounded manifolds. A sphere is an example of a closed and bounded surface with a finite volume.

Flat universes may or may not be infinite. The Euclidean plane is an example of an infinite, flat surface; a torus is an example of a finite, flat surface.
 
Infinite is used to refer to non-compact. It is not true that a flat 3-manifold must necessarily be non-compact. Flat just implies that each point on the manifold has a neighborhood isometric to euclidean space (which is non-compact of course) but flat does not imply that the manifold must itself be non-compact. I am referring specifically to 3-manifolds because the one-parameter family of space-like hypersurfaces of constant sectional curvature in the RW cosmological model are of course 3-manifolds embedded in space-time. See Joseph Wolf "Spaces of Constant Curvature" for a classification of flat compact 3-manifolds.
 
dchartier said:
Whoops, sorry, I miswrote -- I meant an open (negative curvature) universe, not a closed universe. Is an open universe necessarily infinite?
I don't think necessarily; in other words, I'm not aware of a proof that all spaces with negative Gaussian curvature are unbounded.

That said, the most common embeddings of hyperbolic manifolds result in unbounded, infinite surfaces.
 
  • #10
dchartier said:
Whoops, sorry, I miswrote -- I meant an open (negative curvature) universe, not a closed universe. Is an open universe necessarily infinite?

There are infi nitely many possible topologies for quotients of hyperbolic (negative curvature) space, but they can be either compact or not compact. However, if I remember correctly, none of these quotient manifolds is globally homogeneous except the infinite hyperbolic space H^3 itself. Of course global homogeneity may not be a necessary assumption for a realistic cosmology.
 
  • #11
a hyperbolic or negative curve is an open universe.

here is one link I like to use for beginners as it breaks down the main details in an easy manner

http://abyss.uoregon.edu/~js/cosmo/lectures/lec15.html

its not the best possible link but its nice short and sweet lol

edit just saw the other posts let's re qualify that as usually open
 
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  • #12
yenchin said:
However, if I remember correctly, none of these quotient manifolds is globally homogeneous except the infinite hyperbolic space H^3 itself. Of course global homogeneity may not be a necessary assumption for a realistic cosmology.
Good point. The standard categorization of global geometries as either flat, positively curved, or negatively curved is based on the assumption of global isotropy (homogeniety).
 
  • #13
DiracPool said:
It seems as though the contemporary consensus among cosmologists is that the universe is basically flat and Euclidean:
http://en.wikipedia.org/wiki/Shape_of_the_Universe

However, Einsteins relativity equations describing events in space-time appear to be hyperbolic:

http://hermes.aei.mpg.de/1998/3/article.xhtml

Wouldn't the hyperbolic nature of Einstein's relativity equations suggest an Omega of less than 1, a negative curvature and a hyperbolic geometry to spacetime? What would be the argument against this conclusion?

Hyperbolic in the case of PDEs has a very specific definition that basically amounts to the initial value problem being well-defined. It does not mean that solutions of the equation need to have hyperbolic geometry.
 
  • #14
One could argue that on aesthetic grounds, having a one-parameter family of space-like hypersurfaces with non-trivial quotient topologies is not as "simple" as simply taking the 3-hyperboloid for the case of negative sectional curvature but what is true/not true is of course up to experiment. You can deduce the necessity of constant sectional curvature of the hypersurfaces assuming only the isotropy property of the space-time with no need for homogeneity but homogeneity is assumed anyways.

Also, as fzero points out, the hyperbolic nature of the EFEs is a totally different concept. If that is what you are interested in then see chapter 10 of Wald (initial value formulation) for the relationship between the hyperbolic nature of the Einstein equations to GR being a theory with a well posed initial value formulation.
 
  • #15
OK, a related question...would it be a sound argument to state that prior to 7 bya, the dark matter dominated universe biased the universe (space-time) towards a positive curvature, and then the transition to a dark energy dominated universe flipped it towards a negative curvature, and that's why we now see these hyperbolic effects (motion) in einstein's (at least SR) formulations?

http://en.wikipedia.org/wiki/Hyperbolic_motion_(relativity )

Could it be that these hyperbolic effects were, in fact, elliptical effects prior to 7 bya? Furthermore, doesn't it make sense that if this transition did indeed occur and "open" the curvature of the universe, than the curvature now is indeed negative and will continue to expand indefinitely, not ever closing back on itself in a "big crunch?"

I know I'm waxing speculative here, but I'm reading Penrose's "Road to reality" and it got me thinking, he states on page 48...

...there is another role for this suberb (hyperbolic) geometry that is indisputably fundamental to our modern understanding of the physical universe. For the space of velocities, according to modern relativity theory, is certianly a 3-D hyperbolic geometry rather than a Euclidean one that would hold in the older Newtonian theory. This helps us to understand some of the puzzles of relativity.
 
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  • #17
here is one link I like to use for beginners as it breaks down the main details in an easy manner

http://abyss.uoregon.edu/~js/cosmo/lectures/lec15.html
I skimmed the information quickly and found this confounding statement:

...Our current technology allows us to see over 80% of the size of the Universe, sufficient to measure curvature.

80% ?? Since when?? Seems crazy. If true, I don't like our universe all that much anymore!
 
  • #18
Lol that's one of the many reasons I've been hunting for a better article. One that goes beyond the 3 classic shapes. Explains the open and closed. Shows the FLRW metrics of each and isn't misleading.

Sounds easy right? Unfortunately that's harder than one imagines particularly when your looking for one that a beginner can understand.

Another key point I'm looking for is how each affect measurements and light cones
 
  • #19
DiracPool said:
OK, a related question...would it be a sound argument to state that prior to 7 bya, the dark matter dominated universe biased the universe (space-time) towards a positive curvature, and then the transition to a dark energy dominated universe flipped it towards a negative curvature, and that's why we now see these hyperbolic effects (motion) in einstein's (at least SR) formulations?

http://en.wikipedia.org/wiki/Hyperbolic_motion_(relativity )

Could it be that these hyperbolic effects were, in fact, elliptical effects prior to 7 bya? Furthermore, doesn't it make sense that if this transition did indeed occur and "open" the curvature of the universe, than the curvature now is indeed negative and will continue to expand indefinitely, not ever closing back on itself in a "big crunch?"

I know I'm waxing speculative here, but I'm reading Penrose's "Road to reality" and it got me thinking, he states on page 48...

No. There are a lot of things that go by the name "hyperbolic" in mathematics, they do not refer to the same thing. As mentioned in previous post, a "hyperbolic" PDE has nothing much to do with "hyperbolic space" with negative curvature, and "hyperbolic" motion in SR refers to hyperbola drawn on spacetime diagram has nothing to do with either of them.
 
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  • #20
WannabeNewton said:
You can deduce the necessity of constant sectional curvature of the hypersurfaces assuming only the isotropy property of the space-time with no need for homogeneity but homogeneity is assumed anyways.
You mean isotropy of the spatial hypersurface, not the spacetime, right?
And of course homogeneity always follows from the global isotropy and thus it is also assumed.
 
  • #21
yenchin said:
No. There are a lot of things that go by the name "hyperbolic" in mathematics, they do not refer to the same thing. As mentioned in previous post, a "hyperbolic" PDE has nothing much to do with "hyperbolic space" with negative curvature, and "hyperbolic" motion in SR refers to hyperbola drawn on spacetime diagram has nothing to do with either of them.

So what was Penrose referring to in my other post?
 
  • #22
TrickyDicky said:
You mean isotropy of the spatial hypersurface, not the spacetime, right?
And of course homogeneity always follows from the global isotropy and thus it is also assumed.

Isotropy is a property of space-time and homogeneity is an independent property of space-time in their most general forms.

However for the FLRW cosmological model, we are interested in the following:

Let ##(M,g_{ab})## be a space-time then we say it is spatially homogenous if there exists a one-parameter family of space-like hypersurfaces ##\Sigma_{t}## that foliate the space-time such that for any ##t_0## and any ##p,q\in \Sigma_{t_0}## there exists an isometry ##\varphi## of ##g_{ab}## such that ##\varphi(p) = q##.

The space-time is spatially isotropic at every point if there exists a time-like congruence on all of ##M## with tangent field ##u^{a}## such that for any ##q\in M,s_{1}^{a},s_{2}^{a}\in T_{q}(M)## with ##g_{ab}s_{1}^{a}u^{b} = g_{ab}s_{2}^{a}u^{b} = 0## there exists an isometry of ##g_{ab}## that leaves ##q## and ##u^{a}## evaluated at ##q## the same but rotates ##s_{1}^{a}## into ##s_{2}^{a}##.

Spatial isotropy everywhere (as above) implies the first condition (spatial homogeneity) but things don't need to be that restrictive.

One can talk about homogenous anisotropic cosmologies (see section 7.2 of Wald) as well as inhomogeneous isotropic cosmologies: http://www.m-hikari.com/astp/astp2008/astp13-16-2008/zeccaASTP13-16-2008-2.pdf. Also see here: http://xxx.lanl.gov/pdf/gr-qc/9812046v5.pdf
 
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  • #23
DiracPool said:
So what was Penrose referring to in my other post?

It seems like he is talking about the importance of understanding hyperbolic geometry in the context of special relativity -- note that he refers to "space of velocities", he does not say anything about physical space being hyperbolic. :-)
 
  • #24
Mordred said:
Lol that's one of the many reasons I've been hunting for a better article. One that goes beyond the 3 classic shapes. Explains the open and closed. Shows the FLRW metrics of each and isn't misleading.

Sounds easy right? Unfortunately that's harder than one imagines particularly when your looking for one that a beginner can understand.

Another key point I'm looking for is how each affect measurements and light cones

You may enjoy this [however it is certainly not for beginner]：http://philsci-archive.pitt.edu/1507/1/Cosmology.pdf
 
  • #25
Diracpool:

I know I'm waxing speculative here, but I'm reading Penrose's "Road to reality" and it got me thinking, he states on page 48...

...there is another role for this suberb (hyperbolic) geometry that is indisputably fundamental to our modern understanding of the physical universe. For the space of velocities, according to modern relativity theory, is certianly a 3-D hyperbolic geometry rather than a Euclidean one that would hold in the older Newtonian theory. This helps us to understand some of the puzzles of relativity.

So what was Penrose referring to in my other post?

Yenchin:
It seems like he is talking about the importance of understanding hyperbolic geometry in the context of special relativity -- note that he refers to "space of velocities", he does not say anything about physical space being hyperbolic. :-)


Oh, good catch Yenchin!...I would have never figured that language out in a million years [maybe 2!] except Penrose references on page 48 you cited a later chapter, section 18.4, which turns out to be "Hyperbolic Geometry in Minkowski Space'...and he's got a diagram, figure 18.7...

So he goes on to discuss hyperbolic 'length' as 'rapidity'...[which relates back to Lorentz Transformations [ I think]...] ..I think all this means mass can't get to speed 'c'...
 
  • #26
  • #27
yenchin said:
You may enjoy this [however it is certainly not for beginner]：http://philsci-archive.pitt.edu/1507/1/Cosmology.pdf
You will not believe how long I've been trying to find this! I saw it once but forgot the name and where to find it on google. Now you've brought it back my way; I am forever indebted :smile:
 
  • #28
While we are at it, you guys may be interested in a paper of Linde [http://arxiv.org/abs/hep-th/0408164v2], which argues that "compact flat or open universes with nontrivial topology should be considered a rule rather than an exception."
 
  • #29
Those are both ecxellent links Yenchin as you mentioned the first is definitely not for beginners. However its a keeper.
Thanks as well to Naty that thread was entertaining to say the least. I never even considered observer dependancy until you pointed it out. So I certainly enjoyed that.
 
  • #30
Wannabe:
You may enjoy this [however it is certainly not for beginner]：http://philsci-archive.pitt.edu/1507/1/Cosmology.pdf

I do NOT like papers without introductory explanations and without "Discussion and Conclusions" at the hindquarters...Such formats mean I have to READ the whole thing...

anyway, the concluding section QUANTUM COSMOLOGY is rather full of zingers towards the Hawking-Hartle NO BOUNDARY PROPOSAL...Haven't seen anything like that before...
Ok, so that qualifies as enjoyable!

You are redeemed!
 
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