Local coordinates, physical coordinates

[mrandersdk]

As far as I can understand space is af manifold with some metric on it. A manifold is described with some charts (coordinates), but how do I relate these coordinates with ex. physical coordinates of some particle.

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.

 

[Dale]

Well, the first thing is to realize that there is no such thing as “physical coordinates”. Coordinates are simply a smooth set of numbers assigned to events in the manifold. They are completely arbitrary and have no inherent physical significance.

Now, as to how you can assign coordinates, the answer is essentially any way you like. Coordinates have only two requirements: they must smoothly map an open subset of the spacetime manifold to an open subset of R4 and they must be one-to-one. Any method you devise that satisfies those two constraints is valid.