Proposition: General detection

Let $(f,g)$ be admissible, ${\varphi }_0$ free, $\beta =\beta [f,g,{\varphi }_0]$, and for $\mathbf{p}\in {\mathbb{R}}^3$ and unit $\mathbf{n}$ put

If $\mathsf{b}\not\equiv 0$, then $[u{]}_{\mathrm{sol}}\cap {\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}=\varnothing$ for the kernel $u$ of ${\mathcal{C}}_3(f,g)$. In particular, for data of F1 type built from ${\varphi }_m$, where ${\beta }_{ij}=0$, the class is obstructed whenever ${\beta }_{00}{|}_{\Sigma }\not\equiv 0$.

Proof. Take ${k_1}^{\mu }=(1,\mathbf{n})$ and ${k_2}^{\lambda ,\mu }=({\omega }_{\lambda },-\mathbf{n}+\mathbf{p}/\lambda )$ with ${\omega }_{\lambda }=|\mathbf{n}-\mathbf{p}/\lambda |$, both null and future; ${{\gamma }_a}^{\lambda }=\eta e^{i\lambda k_a\cdot y}$ are free. Then $\lambda (k_1+{k_2}^{\lambda })\cdot y={\Omega }_{\lambda }t-\mathbf{p}\cdot \mathbf{x}$, ${\Omega }_{\lambda }:=\lambda (1+{\omega }_{\lambda })=2\lambda +O(1)$, and

by one integration by parts in $t$, ${\lambda }^2/{\Omega }_{\lambda }\to \lambda /2$, and continuity of the polynomial $\mathsf{q}$ as ${k_2}^{\lambda }\to k_2=(1,-\mathbf{n})$. By (eq:qcounter) the leading coefficient is $\frac{i\lambda }{2}\mathsf{b}(\mathbf{n},\mathbf{p})$. If $v\in [u{]}_{\mathrm{sol}}$ satisfies (eq:mc), Lemma lem:decay applies uniformly on a compact neighbourhood of $(k_{1♭},k_{2♭})$ in ${\tilde{V}}_{+}^2\setminus 0$, which contains $(k_{1♭},{k^{\lambda }}_{2♭})$ for large $\lambda$, giving decay; choosing $(\mathbf{n},\mathbf{p})$ with $\mathsf{b}(\mathbf{n},\mathbf{p})\ne 0$ contradicts the growth.