Distance in GR

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What conditions do we have to put on a pseudo-Riemannian manifold in order for a unique and well defined concept of distance between events to be meaningful? I'm thinking about for example max. length of geodesic connecting two events. We have to require one such maximum or minimum length to exist (condition on number and lengths of geodesics between two points...). Anyone knows an exact way of dealing with this?
Thanx
Riccardo
 
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Nothing? Come on, someone must have something to say about this...
 

Stingray

Science Advisor
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The name for a region in which only a single geodesic connects any two points is a normal neighborhood. There are theorems that such things always exist around every point, although I don't know the precise statement. Kobayashi and Nomizu is the usual reference I see cited. Most differential geometry texts will probably work, though.

I do not know of any general results that allow an estimate of the "size" of a normal neighborhood. I can't even imagine how such a theorem might even look.

In a practical sense, geodesics are usually unique when connecting spacelike-separated points. Timelike geodesics "focus" much more easily, and overlap all the time.
 

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