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

Hyperbolic spacetime

  1. Nov 2, 2005 #1
    What is a globally hyperbolic spacetime?

    I'm reading birrel and davies 'quantum fields' in curved space and chapter 3 starts with this assumption...

    Thanks in advance.
  2. jcsd
  3. Nov 3, 2005 #2


    User Avatar
    Science Advisor

    It's basically a spacetime that admits that admits Cauchy surfaces. There's a theorem which states that all such spacetimes can be assigned continuous "time functions" [itex]t:M \rightarrow \mathbb{R}[/itex] where [itex]t^{-1}(s)[/itex] gives a Cauchy surface for any s. Also, if each Cauchy surface has topology [itex]\Sigma[/itex], the manifold has topology [itex]\Sigma \times \mathbb{R}[/itex]. Wald's GR book goes into these things extensively.
    Last edited: Nov 3, 2005
  4. Nov 3, 2005 #3


    User Avatar
    Staff Emeritus
    Science Advisor

    The next logical question is "What is a Cauchy surface"?

    One of the less technical definitions is that it is a space-like surface representing an "instant of time" in the universe, and has the property that the future state of the universe and the past state of the universe can both be predicted/retrodicted from the values of "conditions" on the Cauchy Surface alone. (Of course this arises from a classical, deterministic viewpoint, but then GR is a classical theory, not a quantum theory).

    The more technical defintion (also in Wald, as was this less technical defintion which I paraphrased a bit) involves a lot of discusion of achronal sets and domains of dependency.
  5. Nov 5, 2005 #4
    Ok thanks!

    The terminology sounds a bit confusing. 'Hyperbolic' makes me think if conic sections and the like - but pretty much it has nothing to do with geometry then?
  6. Nov 6, 2005 #5


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

  7. Nov 6, 2005 #6


    User Avatar
    Staff Emeritus
    Gold Member
    Dearly Missed

    Hey, it does too have to do with geometry. It's hyperbolic as in "hyperboloid of one sheet, non-euclidean geometry on", the 2_D hyperboloc space.
  8. Nov 9, 2005 #7
    It is a spacetime in which
    (a) no signals can come back arbitrarily close to themselves
    (b) in which for any two events a,b where b is in the future of a one has a compact set of events c to the future of a and in the past of b
    For more info, see Hawking and Ellis.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Hyperbolic spacetime
  1. Hyperbolic angle (Replies: 7)

  2. Hyperbolic motions (Replies: 5)

  3. Hyperbolic Triangles (Replies: 3)