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Are all lines through space also loops?

  1. Aug 30, 2007 #1
    Given the curvature of space-time, are all lines through space also loops? If so, are the loops always circular? If the loops are not always circular, what determines the shape of the loops?
  2. jcsd
  3. Aug 31, 2007 #2


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    I'm really not sure what you're asking here. If all worldlines (lines in spacetime) were "loops" then this would mean that one's life would end at the same point in spacetime as it began, which is clearly not true. There are models of GR with so called closed timelike curves, but these are not physical solutions.
  4. Aug 31, 2007 #3

    George Jones

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    I think redskies is asking about loops in space, not spacetime, i.e., do there exist worldliines that begin and end at the same spatial comoving coordinates?

    redskies, the answer is maybe. The situation depends on the spatial curvature of the universe, and we haven't measured spatial curvature with enough accuracy to give a definitive answer. Space might like the surface of the Earth (with one more dimension added), or it might be like like an infinite, unending (possible curved) blackboard. In the first case the answer is yes, while in the second its no.

    Our measurements indicate that the spatial curvature of universe is near the borderline of these cases, and either case is consistent with the errors involved in the measurements.
    Last edited: Aug 31, 2007
  5. Aug 31, 2007 #4


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    Ahh, sorry I read the question wrong!
  6. Aug 31, 2007 #5
    Thanks, George. :smile: Is it correct that curvature would not be constant throughout the universe because at a local level it would be affected by massive objects? If so, when you speak of measuring spatial curvature, are you speaking of a measurement analogous to temperature in that it is an average of curvatures throughout the universe (in temperature's case an average of kinetic energies)?
  7. Sep 2, 2007 #6


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    There is some work going on in this area, for instance by Neil Cornish.

    http://www.citebase.org/abstract?id=oai%3AarXiv.org%3Agr-qc%2F9602039 [Broken]

    These particular models of a finite universe with "closed spacelike curves" should have an observational signature. If you do a google, you'll see that people for a while thought they had found such a signature, but I don't think it's held up under further analysis.
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  8. Sep 2, 2007 #7


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    This is correct.

    Right. Cosmological models are built taking into consideration average curvature at large scales, more than 100 Mpc. When you average the matter density over different regions greater than 100 Mpc you will find that there are nearly no deviations between these average values. Thus, at this scale the distribution of matter density is homogeneous. This determines a constant spatial curvature.
  9. Sep 2, 2007 #8
    I hope you gentlemen won’t mind some exploratory, if naïve, inquiries along these lines…

    I’ve assumed that within the context of GR, curvature entails both space and time deformations, as we see with gravitation. I’m unclear how we can speak of spatial curvature independent of temporal curvature, and this reply implies that the possible curvature of the universe is essentially synonymous with gravitational curvature:

    So proceeding on this assumption, then isn’t it possible that the cosmological redshift is simply an artifact of universal curvature? From the layman’s point of view, it would seem that a cosmic-scale curvature would be difficult to discern from the current ‘expanding spacetime’ model. And wouldn’t the large-scale spacetime curvature of a finite universe also offer an alternative explanation for ‘dark energy’ as merely an optically-apparent condition?

    I guess I’m just not clear on how we’re confident that the acceleration of cosmic expansion isn’t simply an apparent observational condition endowed by cosmic topological considerations.

    Since the lower limit for the cosmic radius of 39B LY, as determined by a recent analysis of WMAP data ( Link ) is still lower than the apparent 46B LY diameter of the universe, isn’t it possible that at 46B years’ distance, the curvature could be at a maximum value, perhaps that of an event horizon, for example?

    Now I don’t know if requisite degree of curvature suggested by this crude hypothesis might correlate with the large-scale curvature of the Hubble law, or if an additional large-scale spatial dimension might be required to reconcile this idea with the observations, but I’d certainly appreciate any expertise which you might be willing to offer on this subject.

    I'm also confused why the prospect of a closed-timelike-curve at the cosmic scale would engender any logical paradoxes, or why these might be considered 'nonphysical' as cristo seemed to suggest earlier. There seem to be quite a few learned theoretical physicsts who feel that the closed-timelike-curved solutions of GR might very well be physical; it's at least an open question isn't it?

    I’d love to have a better grasp of what we know we know, and what we think we know, on the subject of cosmological topology.
    Last edited: Sep 3, 2007
  10. Sep 2, 2007 #9


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    We know the universe is closer to dead flat than we can accurately measure. The more intriguing question is why? There is not nearly enough visible matter to explain why it appears to be so flat, hence the dark matter hypothesis [among other reasons supporting the DM hypothesis].
  11. Sep 2, 2007 #10
    A circle or loop would appear to approach dead flat as one examined increasingly smaller pieces of it (approaching size 0 relative to loop size). Perhaps this could be analogous?
  12. Sep 3, 2007 #11
    That seems like a good point to me.

    I guess I'll have to resort to the anthropic principle regarding the 'why' question: if omega -wasn't- close to 1, the universe would've either collapsed very quickly, or would've blown apart too fast...in either case we wouldn't be having this conversation, right?

    ps - I don't know if our current analytical methods rule out a large-scale curvature in an extra spatial dimension, but I've always been intrigued by higher-dimensional physics models. String theory seems to confine extra spatial dimensions to Planck-scale manifolds, but I've never understood why we'd rule out the appearance of extra dimensions at the large scale. Particularly if they might help explain otherwise insoluble mysteries like 'dark energy.'

    pps - Ok, I see that much of this subject has already been addressed in the 'Does Space Expand?' thread...I'll come back and reframe my questions once I've absorbed the previous discussions...
    Last edited: Sep 3, 2007
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