Time-dependent lapse can disguise flat spacetime ($d{s}^2=-N(t{)}^{2}d{t}^2+d{x}^2,T=\int N(t)dt$)

For positive lapse $N$, this time reparametrization gives a local Minkowski metric. If the new time coordinate ranges over the entire real line, it is a global isometry to Minkowski space. In particular $N=\cosh t$ gives $T=\sinh t$ and complete nonperiodic timelike geodesics. A lapse depending on the spatial coordinate instead can describe a different curved geometry; exchanging the arguments is not a harmless relabeling of the same metric.