Proposition: Pure-trace counter-propagating pairing for the F1 data

Let $(f,{\varphi }_m)$ be F1 data, ${k_1}^{\mu }=(1,\mathbf{n})$, ${k_2}^{\mu }=(1,-\mathbf{n})$, and ${{\gamma }_a}^{\lambda }=\eta e^{i\lambda k_a\cdot y}$, $\lambda >0$. Both ${{\gamma }_a}^{\lambda }$ are free metric configurations, ${\varphi }_m$ is a free scalar, and

In particular there is $C$ with $|⟨u,{{\gamma }_1}^{\lambda }\otimes {{\gamma }_2}^{\lambda }\otimes {\varphi }_m⟩|\ge \lambda |{\Lambda }_f|-C$ for all $\lambda >0$.

Proof. $\square {{\gamma }_a}^{\lambda }=0$ since $k_a$ is null; $P_m{\varphi }_m=0$. By (eq:F1data), ${\bar{\beta }}_{ij}=0$, ${\bar{\beta }}_{0i}=0$ and ${\bar{\beta }}_{00}=B$; by (eq:qcounter), ${\mathsf{q}}^{00}=-2$. Insert in (eq:planewave): $-2{\lambda }^2(-B(0)/(2i\lambda )+O({\lambda }^{-2}))=\lambda B(0)/i+O(1)=-i\lambda {\Lambda }_f+O(1)$.