I have taken a relativity course, and am not quite clear as to whether the notion of a frame of referene is independent of the coordinate system. At the moment, I think that any frame of reference (of which a special subset are intertial- i.e. those related by lorentz boosts or free falling in GR) can choose any coordinate system, i.e. any way of labelling spacetime events, that they like.
Specific examples of where this difference (?) might beimportant:
I found myself arging that $\nabla_a u^a$ must always vanish for dust because if one goes into the local rest frame of dust and chooses local minkowki coordinates so that the components of the metric connection vanish, then this clearly vanishes. However I am not sure if I have applied too many restrictions here.
Secondly, In the Schwarzschild geometry, whose coordinates are those that are used in the scwarzschild line element? I'm not sure if such a question even makes sense? I think: they are the coordinate system of an observer 'at rest' at infinity who decides to have local Minkowski coords i.e. the line elemnt at infinity reduces to local cartesian coords.