Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Flat or hyperbolic?

  1. May 3, 2013 #1
    It seems as though the contemporary consensus among cosmologists is that the universe is basically flat and Euclidean:


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


    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. jcsd
  3. May 3, 2013 #2
    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
  4. May 3, 2013 #3


    User Avatar
    Science Advisor

    I think you mean open universe -- closed universes have positive curvature.
  5. May 3, 2013 #4
    yeah sorry good catch
  6. May 3, 2013 #5
    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?

  7. May 3, 2013 #6


    User Avatar
    Science Advisor

    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.
  8. May 3, 2013 #7
    Whoops, sorry, I miswrote -- I meant an open (negative curvature) universe, not a closed universe. Is an open universe necessarily infinite?

  9. May 3, 2013 #8


    User Avatar
    Science Advisor

    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.
  10. May 3, 2013 #9


    User Avatar
    Science Advisor

    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.
  11. May 3, 2013 #10
    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.
  12. May 3, 2013 #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


    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
  13. May 3, 2013 #12


    User Avatar
    Science Advisor

    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).
  14. May 3, 2013 #13


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    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.
  15. May 3, 2013 #14


    User Avatar
    Science Advisor

    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.
  16. May 3, 2013 #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 [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
  17. May 3, 2013 #16
  18. May 3, 2013 #17

    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!!
  19. May 3, 2013 #18
    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
  20. May 3, 2013 #19
    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
  21. May 3, 2013 #20
    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.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Discussions: Flat or hyperbolic?
  1. The universe is flat? (Replies: 0)

  2. Space flatness (Replies: 8)

  3. Flat Universe (Replies: 5)

  4. Space is flat? (Replies: 27)

  5. Flatness of the Universe (Replies: 15)