- #26

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I think it is useful to make the distinction I made earlier between flat and curved manifolds. The example of Minkowski space is relevant, being a flat manifold you can cover the whole manifold with a 4 coordinate metric, the same happens with Euclidean space that can be covered with three coordinates or the plane with two. However in the case of a two sphere for instance the line elements expressed in 2 coordinates cover patches of the total manifold and there are always points like at the equator or at the poles dpending on the specific coordinates where one finds coordinate singularities, where you have to transform coordinates to a different overlapping patch of the manifold.(To avoid confusion: space in the following is short for spacetime.)

Global coordinates for me are coordinates that cover the whole manifold, p.ex. cartesian coordinates for Minkowski space. Every point of Minkowski space is described by one coordinate tuple (x,y,z,t), and to every coordinate tupel there belongs a point of Minkowski space. Coordinates which are not global for Minkowski space are for example Rindler coordinates: every tuple of Rindler coordinates corresponds to a point in Minkowski space, but there are points in Minkowski space which do not have a corresponding tuple.

So in curved manifolds, (unless some hidden symmetry is exploited) I would say global coordinates need one more coordinate to cover the whole manifold.

Yes, those are de sitter global coordinates, and are a Minkowkian 5-dimensional ambient space restricted to dS^4 (see http://www.bourbaphy.fr/moschella.pdf equations 3-8)An example of global coordinates for de Sitter space is given by ds² = - dt² + H^{-2} cosh²(H t) dΩ², where dΩ² is the metric of the 3-sphere. We don't need an embedding in a higher-dimensional space for global coordinates; but there are spaces which don't admit global coordinates (I think the Kerr black hole falls in this class).

Note de sitter geometry is very special, being maximally symmetric.