Lemma: Constraint-compatible inverse

For the above conserved sources, $D(j)=0$. In this gauge $L_{\mathrm{E}}j=Q$.

Proof. The initial data give $D_{\nu }(j)(t_0)=0$. Also

Using ${{\partial }_t}^2\bar{j}=\Delta \bar{j}+2Q$ gives ${\partial }_t{D}_{\nu }(j)(t_0)=0$. Conservation implies $\square D_{\nu }(j)=0$. Wave uniqueness proves the first assertion; the second follows from (eq:LEidentities).