# Flat or hyperbolic?

1. May 3, 2013

### DiracPool

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?

Last edited: May 3, 2013
2. May 3, 2013

### Mordred

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

Last edited: May 3, 2013
3. May 3, 2013

### bapowell

I think you mean open universe -- closed universes have positive curvature.

4. May 3, 2013

### Mordred

yeah sorry good catch

5. May 3, 2013

### dchartier

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?

6. May 3, 2013

### bapowell

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.

7. May 3, 2013

### dchartier

Whoops, sorry, I miswrote -- I meant an open (negative curvature) universe, not a closed universe. Is an open universe necessarily infinite?

8. May 3, 2013

### WannabeNewton

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.

9. May 3, 2013

### bapowell

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. May 3, 2013

### yenchin

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. May 3, 2013

### Mordred

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 lets re qualify that as usually open

Last edited: May 3, 2013
12. May 3, 2013

### bapowell

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. May 3, 2013

### fzero

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. May 3, 2013

### WannabeNewton

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. May 3, 2013

### DiracPool

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 [Broken])

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...

Last edited by a moderator: May 6, 2017
16. May 3, 2013

### Mordred

17. May 3, 2013

### Naty1

I skimmed the information quickly and found this confounding statement:

80% ?? Since when?? Seems crazy. If true, I don't like our universe all that much anymore!!

18. May 3, 2013

### Mordred

Lol thats 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 thats 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. May 3, 2013

### yenchin

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.

Last edited by a moderator: May 6, 2017
20. May 3, 2013

### TrickyDicky

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. May 3, 2013

### DiracPool

So what was Penrose referring to in my other post?

22. May 3, 2013

### WannabeNewton

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

Last edited: May 3, 2013
23. May 4, 2013

### 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. :-)

24. May 4, 2013

### yenchin

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

25. May 4, 2013

### Naty1

Diracpool:

Yenchin:

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'....