Lemma: Restricted sharp separation

If $B_f[k,\phi ]=0$ for every compact smooth metric direction $k$ supported in $V$ with $k_{00}=0$, and every smooth scalar $\phi$, then $f=0$.

Proof. For $\chi \in \mathcal{D}(V)$, let $H_{11}=\chi$ be the only nonzero component of $H$, and take $k=\square H$, where $\square$ is the chart operator. Then $k_{00}=0$ and

by Proposition prop:saturation. In particular $(CH{)}^0=\frac{1}{2}{\partial }_t\chi$. Choose a smooth scalar equal to $t$ near $\mathrm{supp}f$. The assumed vanishing gives

for every $\chi$, hence ${\partial }_{t}f=0$ in $V$. A time-independent smooth function compactly supported inside $(0,T_0)\times B_{r_0}$ is zero.