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How to determine extension of a pseudo-Riemannian manifold

  1. Jan 13, 2014 #1

    PAllen

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    Broad title, but really a specific question that I thought should be straightforward, but got stuck.

    Consider the geodesics of form t=contant, r>R, in exterior SC geometry in SC coordinates. These are spacelike geodesics. If we consider this geometry embedded in Kruskal geometry, it is easy to see that this family of geodesics, for each theta,phi, intersect at a particular point of the 2-sphere representing the V=0 (I use V as the timelike Kruskal coordinate) throat of the wormhole (this being the slice where the throat is largest, r < R doesn't exist in the manifold for this slice). It is not hard to see that a single point of this 2-sphere connects one family (a family for each theta, phi) of such geodesics in one sheet of the universe to a similar family in the other sheet.

    Fine. But what if I consider the SC geometry to be a manifold unto itself. I want to show that any extension of this manifold that preserves all of its geometry must have these geodesic families intersect. I believe this must be geometric argument, not a topological one, because this seems definitely not required if we deal in topological manifolds without metric (e.g. a plane missing a point is not topologically distinguishable from a plane missing a disk, I think). I tried a method that makes sense for Riemannian manifolds, but then convinced myself it showed nothing for psuedo-riemannian manifolds.

    The method for Riemannian manifolds was inspired by looking at a Euclidean plane minus a point in polar coordinates, where the missing point is the pole. Then one can look a the limit of circumference for circles of decreasing r, as r->0. For SC geometry, I derived a family of timelike curves intersecting all geodesics in a family as previously described, characterized by a parameter of maximum r along one of these curves. Then, I could show that as the parameter approaches R, the interval along such curve goes to zero. But this would be true for any family of geodesics intersecting a light like boundary. That is, if you have a manifold which can be extended such that its boundary in the extended manifold is lightlike, you would have this property. So I think this proves nothing at all about intersection. In a Riemannian manifold, you don't have this difficulty.

    So methods based on invariant interval seem a no go, and topological methods seem irrelevant, so where to go? Is it possible to say that any extension (or smooth extension??) of the SC exterior considered by itself, that preserves all of its geometry, must have these families of geodesics intersect? If so, how?

    Sorry for the long winded question, but this has been bugging me for a couple of days.
     
    Last edited: Jan 13, 2014
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  3. Jan 14, 2014 #2

    PAllen

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    Maybe my method is not so far off? I note that the family of timelike curves I described do not approach a lightlike curve - so the convergence to interval zero remains timelike. Is this enough to make the argument a valid demonstration that any geometry preserving extension will have these geodesics intersect?
     
  4. Jan 15, 2014 #3

    PAllen

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    I have a solution that satisfies me now. The duh realization is that for pseudo-riemannian manifolds, where you would use balls in a proof for riemannian manifolds, you use causal diamonds for a pseudo-riemannian manifold. Thus, I can show as follows:

    Pick a particular geodesic in the family I defined (say, the theta=phi=0 family), e.g. the t=0, r>R spacelike geodesic. Define the causal diamond defined by ingoing future directed null geodesic from r = R + δ and ingoing past null geodesic, and also the outgoing for R + ε, with ε < δ. If the geodesic for t=<large number> is not required intersect the t=0 geodesic, there would need to be some δ such that I can't find ε such that this causal diamond for the t=0 geodesic includes any of the t=k geodesic. I can show that for any δ I can always find such an ε. I conclude that any extension of SR exterior that preserves its geometry must have the whole t=k family of geodesics intersecting at one point.

    And this also establishes (or a slight generalization of the argument) that using SC coordinates alone, you can conclude that r->R, any t in (-∞,∞), really represent the same point in the manifold. Thus, that the coordinate singularity as r->R is extremely similar to the polar singularity in polar coordinates. Of course, you have a 2-sphere of such convergence points (one for each theta,phi).
     
    Last edited: Jan 15, 2014
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