 Synchronous coordinates

In synchronous coordinates, fixed spatial-coordinate worldlines are timelike geodesics, and $t$ is their proper time. The metric has unit lapse function and zero shift vector in time units consistent with the displayed factor $c^2$. Indeed $g_{tt}=c^2$ and $g_{ti}=0$ give ${\Gamma }^{\mu }{}_{tt}=0$. Locally, an irrotational freely falling congruence permits this construction before crossings of the coordinate worldlines. Comoving observers in a Friedmann-Lemaître-Robertson-Walker metric provide the standard cosmological example.