Lemma: Microcausal admitted linear kernels have zero charge

If $\ell \in {\mathcal{S}}_1$ has no nonzero nonspacelike wavefront covector, then $\mathsf{b}(\ell )=0$.

Proof. Write $\ell =m+q_a$. The kernel $m$ has only spacelike wavefront covectors by Assumption ass:carrier. Therefore

also has no nonspacelike covector. The source $a$ is compact smooth, so $b={G_{\mathrm{A}}}^{0}a$ is smooth. Lemma lem:surfaceWF forces $b{|}_{\Sigma }=0$. Equation (eq:charge) now gives the conclusion.