Lemma: A nonzero surface coefficient is not microcausal

Let $b$ be smooth near $\Sigma$. If $b{|}_{\Sigma }\ne 0$, then $C^{\ast }(\theta b)$ has a nonzero timelike wavefront covector.

Proof. By (eq:Bsurface), some component of $\beta =\mathcal{B}(b{|}_{\Sigma })$ is nonzero on a boundary patch. That component has local form

with $c$ smooth.

Choose compact localizers $\chi (t),\rho (\mathbf{x})$, with $\chi (0)=1$, so that $\int \rho (\mathbf{x})\beta (\mathbf{x}){\mathrm{d}}^3\mathbf{x}\ne 0$. The Fourier transform of the localized delta term at $(\tau ,0)$ is this nonzero constant. The transform of the localized Heaviside term is $O(|\tau {|}^{-1})$, by one-sided integration by parts. Thus the localized distribution is not rapidly decreasing along the normal rays as $|\tau |\to \infty$.

If the distribution had no timelike wavefront covectors on that compact patch, the local wavefront estimates, after a finite partition of unity, would give rapid decay in a cone around those normal rays. This contradicts the preceding asymptotic. Hence a timelike covector occurs.