Bondi-Sachs coordinates

Bondi-Sachs coordinates $(u,r,x^A)$ follow outgoing null hypersurfaces. The retarded time $u$ labels the hypersurfaces, the angles $x^A$ label their null generators, and the areal radius $r$ fixes the angular area element. The coordinate conditions include $g_{rr}=g_{rA}=0$ and $\det g_{AB}=r^4\det q_{AB}$, where $q_{AB}$ is the unit-sphere metric.

Flat spacetime already illustrates the construction: setting $u=t-r$ in units $c=1$ gives $d{s}^2=-d{u}^2-2dudr+r^{2}d{\Omega }^2$. Constant-$u$ surfaces are outgoing light cones, although $g_{uu}$ is nonzero. The null condition concerns the hypersurface normal and gives $g^{uu}=0$.^[ref]