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Is it like this:

if I'm in some laboratory I make some cartesian coordinate system (x,y,z) (maybe include time (t,x,y,z)), so that I can say that my particle is at p_0 = (x_0,y_0,z_0). Then my task is to find the metric in my laboratory coordinate system, so I can for example calculate the geodesic for my particle. But p=(x,y,z) should be functions to my manifold, that is p: R^3 -> M, and how should they look.

As you might see I have a bit trouble understanding, how to relate the physical coordinates to the manifold, so that the description of GR becomes useful.

Hope someone can help me understand it. Thanks in Advance, Anders Berthelsen.