Synchronous coordinates ($d{s}^2=c^{2}d{t}^2-h_{ij}(t,\mathbf{x})d{x}^{i}d{x}^j$)

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.

For relativistic dust with nowhere-zero density, stress-energy conservation implies zero four-acceleration. If its normalized four-velocity $u$ has hypersurface orthogonality, then $u^{♭}\wedge d{u}^{♭}=0$ and ${\iota }_{u}d{u}^{♭}=0$. The interior product of the first identity gives $d{u}^{♭}=0$. The Poincaré lemma supplies $u^{♭}=-dt$ locally; spatial coordinate functions constant along the smooth flow of $u$ give $u={\partial }_t$, $g_{tt}=-1$ and $g_{ti}=0$. The resulting synchronous coordinates describe the dust until the local coordinate construction fails.