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Can geodesics cross?

  1. Dec 19, 2008 #1

    I am currently doing a course in GR and have just gone over a proof of the focusing theorem..
    now this relied on the fact that geodesics do not cross. But I could not see clearly the contradiction if geodesics did happen to cross?

    any help would be greatly appreciated.

    Maki :smile:
  2. jcsd
  3. Dec 19, 2008 #2


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    I don't know the focusing theorem, but there can definitely be more than one geodesic between the same two events. A simple example is if you drill a hole through a planet with no atmosphere, along the rotational axis, and drop something into it. The path of the object can intersect the path of something in orbit on both sides of the planet.
  4. Dec 19, 2008 #3


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    ? Yes, of course, geodesics cross. At any point you can have geodesics in every direction. Surely there is more to this question? Are you asking if geodesics can cross twice? That would depend on the geometry.
  5. Dec 19, 2008 #4


    Staff: Mentor

    Consider geodesics on a sphere, they all cross twice.
  6. Dec 19, 2008 #5


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    Geodesics can cross, since at any point there is a different geodesic in each direction. However, for a given region in spacetime, one can choose a bunch of non-crossing geodesic wordlines (congruence of geodesics) that are all headed into the future. The focusing theorem says that under certain circumstances, at least some of these geodesics are going to cross anyway, because gravity is attractive.
  7. Dec 19, 2008 #6
    There is something similar in the proof that the area of black hole event horizons never decreases.
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