Basic postulate of GR is that locally spacetime looks flat i.e. any metric can be reduced to a local inertial metric at a point. The local inertial metric is exactly the flat Minkowski metric AT the point, and has zero derivatives AT the point but those are not true away from the point. Then Newtonian metric is the metric felt by a free fall observer. In it, the deviations from the Minkowski metric, to first order in 1/c^2, are described by a 'Newtonian gravitational potential'. I think this is the same thing as writing a general metric in 'Newtonian approximation', 'Newtonian gauge' or 'Weak field approximation'. I have a general metric and need a Mathematica notebook that will convert it to the Newtonian metric felt by a specified free fall observer. Does anyone know of something like that ?