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Boundary of chronological future as a 3-d topological manifold.

  1. Sep 7, 2012 #1

    aleazk

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    This is basically proposition 6.3.1 of Hawking and Ellis (page 187).
    He proves that you can construct an injective map which goes from the boundary of the chronological future to R^3. It remains to show that this map is a homeomorphism (in the induced topology of the boundary). And here's where I'm confused. The injective map is constructed by assigning to each point of the boundary the 'spatial' coordinates of the timelike coordinate curve of some normal chart (x^0, x^1, x^2, x^3) which intersects that point in the boundary. Since each point on the boundary is characterized by the coordinates (x^1, x^2, x^3), you can see the '0 coordinate' (of the initial normal chart) of that point as a function of the (x^1, x^2, x^3), i.e., x^0(x^1, x^2, x^3). The book says that it is sufficient the proof that this function is Lipschitz for conclude that the map in question is a homeomorphism. That's the point I don't get. My timid guess is that this only shows the continuity of the inverse map, it would remain to show the continuity of the map. I have looked in other books, and all of them say the same thing that H and E, so I think I'm missing some very obvious step.
    Anyway, thanks in advance.
     
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