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The topology of spacetimes 
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#1
Jan113, 10:49 AM

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Mod note: This thread contains an offtopic discussion from the thread http://www.physicsforums.com/showthread.php?p=4216768



#2
Jan113, 11:24 AM

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#3
Jan113, 11:28 AM

C. Spirit
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EDIT: dextercioby beat me to it while I was typing =D 


#4
Jan113, 12:07 PM

P: 123

The topology of spacetimes
Of course, thank you both.
But then, aha, the tangent space is indeed a topological vector space with a topology induced by the notion of distance induced by the norm induced by the inner product, itself defined by the metric of the Riemannian manifold! 


#5
Jan113, 01:35 PM

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why I do not regret leaving mathematical interpretations to others! {LOL}



#6
Jan113, 02:38 PM

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But I don't think there's anything that prevents you from doing that if you want. 


#7
Jan113, 02:59 PM

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[itex]g(X,Y)[/itex], but then we have to check that the triangular inequality is still true. Hmm, [itex]g[/itex] doesn't even define a distance in M, as M is pseudoRiemannian, the distance is usually defined as [tex] \int_a^b ds=\int_a^b\sqrt{g_{\mu\nu}dx^{\mu}dx^{\nu}}[/tex] and this is always positive. 


#8
Jan113, 04:03 PM

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In any case, even if that can be overcome the topology of a pseudoRiemannian manifold is inherited from the underlying manifold, not from the metric nor from the inner product of vectors in the tangent spaces. 


#9
Jan113, 04:33 PM

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This is equivalent to introducing a (positivedefinite) norm on [itex]T_p(M)[/itex]. 


#10
Jan113, 04:34 PM

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Yes, that is true, a manifold is a topological space before any other thing you may define on it. Can that be used on the tangent bundle? 


#11
Jan113, 06:27 PM

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#12
Jan213, 03:41 AM

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It's high time I threw in the reference for the one interested
Naber G.L.  The geometry of Minkowski spacetime (Springer, 1992)(271p) 


#13
Jan213, 04:17 AM

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So we ask  do we actually use the Minkowskii notion of distance to define the topology of our 4d space time? (Forget about the tangent space, for the moment, I'm talking about how spacetime is a 4d manifold). For instance, if we are now observing a distant event in the andromeda galaxy, and it's Lorentz interval is zero, does that mean it's close to us, in our neighborhood? THe answer is no, and no. 


#14
Jan213, 05:44 PM

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IOW, if the Lorentzian metric tensor only has a local significance, why are its pseudometric features extended to determine the global features of the manifold? 


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Jan213, 06:37 PM

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#16
Jan213, 07:03 PM

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#17
Jan213, 07:12 PM

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[tex] ds^2=g_{\mu\nu}dx^{\mu}dx^{\nu} [/tex] It is the form of [itex]g_{\mu\nu}[/itex] that will therefore determine all the odd properties of the manifold that we're used to see as nice in euclidean space. [tex] \frac{d\gamma}{ds}(0) = X [/tex] and therefore the path [itex]\gamma(s)[/itex] will give you a line 'along' X, on the manifold. The metric tensor only has local significance, but then you can imagine that its local features compose the global properties as seen locally, so that in the end the whole ensemble of local features will 'add up' to form the manifold. You can find a lot of these things in mathematics, e.g. the Cantor set, which is just a set defined by very simple rules on euclidean space with the usual distance, but then in the end you get an ensemble which has completely different properties and very strange ones indeed. You may also think about Minkowski space (flat spacetime), and about how you may define this light cone lat an event, and this is not just a drawing on paper, space itself gets different properties on different regions. You may want to view this as only locally defined, but there's nothing that contradicts the fact that you can extend this (relational) properties to the whole space. 


#18
Jan213, 09:17 PM

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Distances on a coffee cup and a donut are different even though they are the same topologically. Do you have an explanation how that works? 


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