I am trying to understand the derivation of a solution of Einstein's field equations for a static, spherically symmetric spacetime (chapter 10 of Schutz's 'A first course in general relativity'). The derivation defines a static spacetime as one satisfying two conditions, one of which is that the metric tensor g is independent of t (time). I am confused by this definition as whether something is independent of a given coordinate depends on the coordinate system. g may be independent of t in one coordinate system but dependent on it in another. This seems to imply that a spacetime may be static in some coordinate systems but non-static in others. Is that correct? If so, what do we mean by a static spacetime?