Broad title, but really a specific question that I thought should be straightforward, but got stuck.(adsbygoogle = window.adsbygoogle || []).push({});

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.

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# How to determine extension of a pseudo-Riemannian manifold

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