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I am confused about the relationship between coordinate systems and reference frame in GR.

I understand the coordinate systems can be used to describe reference frames, for example, Local inertial frames in GR can be defined by Riemann Normal Coordinates.

However, take the Schwarzschild Geometry for example, we have

[tex]ds^{2} = - (1 - \frac{2GM}{c^{2}r})(cdt)^{2} + (1 - \frac{2GM}{c^{2}r})dr^{2} + r^{2}(d\theta^{2} + sin^{2}\theta d\phi^{2})[/tex]

Obviously, when we write the Schwarzschild Geometry, we have already assume a coordinate in place, with coordinates (t, r, [itex]\theta, \phi[/itex])

My question is, does the coordinate system (t, r, [itex]\theta, \phi[/itex]) used in defining the line element happens to define a reference frame?

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# The relationship between coordinate systems and reference frame

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