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The topology of spacetimes 
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#37
Jan313, 12:04 AM

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Another good reference is "Fundamental of differential geometry" by Serge Lang. He covers pseudoRiemannian metrics on page 175. It's a fun book to read, so I recommend it.



#38
Jan313, 12:11 AM

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According to wikipedia http://en.wikipedia.org/wiki/Neighbo...mathematics%29 neighborhood should contain an open set containing the point. Given spacetime properties neighborhood of any event in spcetime should include it's lightcones. But for any two distinct points there will be some place where their lightcones (future or past or future with past) will intersect. So they can't have disjoint neighbourhoods which is required to say they belong to Hausdorff space. 


#39
Jan313, 12:20 AM

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#40
Jan313, 12:51 AM

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All the usual spacetimes are of course Hausdorff. But just for interest, Hawking and Ellis mention one example of a nonHausdorff spacetime, and mention a paper by Hajicek.



#41
Jan313, 03:13 AM

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One shouldn't be able to draw conclusions about the global spacetime features from the purely local effect of the pseudoriemannian metric at a point, more so when the distance metric function that acts on the manifold doesn't coincide with the one that would be derived from the pseudoriemannian metric tensor, due to the smooth structure of the manifold. When I mention the global structure of the manifold I refer to things like its maximal extended form, its singularities or its Killing vector fields nature(timelike, spacelike,lightlike). 


#42
Jan313, 04:49 AM

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If we say that spacetime is Hausdorff then we can't include complete lightcones in the neighborhood of an event. But then we should relay on some concept of nearness that is positivedefinite and rather unrelated to spacetime distances.
It seems like a kind of double standard. 


#44
Jan313, 05:14 AM

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#45
Jan313, 05:15 AM

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#46
Jan313, 05:16 AM

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So what was the point? There is no analog of light cone on spacetime itself? And all spacetime distances are positivedefinite? Or what? 


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Jan313, 05:18 AM

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#48
Jan313, 05:21 AM

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#49
Jan313, 05:27 AM

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#50
Jan313, 05:29 AM

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#51
Jan313, 05:33 AM

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#52
Jan313, 05:35 AM

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#53
Jan313, 05:49 AM

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#54
Jan313, 06:41 AM

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Having said this, we have strayed far offtopic with respect to the original post. Physics Forums rules advises that, instead of posts that are offtopic, new threads should be started. 


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