Lemma: The coordinate vector fields in the chart w +

On $V_{+}$, as operators on ${\mathcal{D}}^{\prime }(V_{+})$ expressed in $w_{+}$,

with $d_{\mu }={\eta }_{\mu \nu }{d}^{\nu }$, $d^0=\sqrt{q+|\zeta {|}^2}$, smooth on $V_{+}$. Each is a first-order operator with smooth coefficients; its normal component is $\pm 2d_{\mu }{\partial }_q$ for ${\partial }_b$, $-{\partial }_{r^{\mu }}$ for ${\partial }_z$, and $-2d_{\mu }{\partial }_q+{\partial }_{r^{\mu }}$ for ${\partial }_a$.

Proof. Chain rule for $u=a$, $d=b-a$, $q={\eta }_{\nu \kappa }{d}^{\nu }{d}^{\kappa }$, ${\zeta }^i=d^i$, $r=a-z$: $\partial q/\partial d^{\mu }=2d_{\mu }$, $\partial {\zeta }^i/\partial d^{\mu }={{\delta }^i}_{\mu }$, and $b$ enters only through $d$ (sign $+$), $z$ only through $r$ (sign $-$), $a$ through $u$ ($+$), $d$ ($-$), $r$ ($+$).