What is the relationship between the differentiable manifold that is space-time and the physical space around us? How does one relate the three seemingly Cartesian coordinates around us, those which we can measure out with a ruler, to the coordinates of the Lorentzian manifold? If i say, measure out a length with a ruler, how would that relate to the three spatial coordinates of space-time? I'm just getting all confused thinking about this. Maybe this question doesn't make much sense, but I just want to see if anyone can help me with this.