Time-Invariant Space: Metric ds^2 and Coordinates

[alejandrito29]

Question: have some sense that in a space time with metric $$ds^2 = g_{tt}dt^2+ g_{xx}dx^2+ g_{yy}dy^2+g_{zz}dz^2$$, the coordinates $$x,y,z \in ]-\infty, \infty[$$ , but $$t \in [0, \infty[$$ ?

 

[bcrowell]

In flat spacetime (Minkowski space), t ranges over the whole real line. In other spacetimes, there can be all kinds of different cases. In general, you can't even cover a whole manifold with a single coordinate chart.