Lemma: Cauchy reduction

If $\square {\gamma }_1=\square {\gamma }_2=0$, then ${J_3}^{\mu \nu }({\gamma }_1,{\gamma }_2){|}_{\Sigma }=Q^{\mu \nu }(j_1,j_2)$ is a bidifferential form in the Cauchy data $j_a=({\gamma }_a{|}_{\Sigma },{\partial }_t{\gamma }_a{|}_{\Sigma })$, obtained from the jet expression of $J_3$ by replacing every ${{\partial }_t}^2{\gamma }_a$ by $\Delta {\gamma }_a$. Hence $B_{\Sigma }={\int }_{\Sigma }\beta {|}_{\Sigma }\cdot Q(j_1,j_2)$, with the scalar slot entering only through the smooth weight $\beta {|}_{\Sigma }=\beta [f,g,\varphi ]{|}_{\Sigma }$, linearly.

Proof. $J_3$ contains at most second derivatives, and on the shell ${{\partial }_t}^2{\gamma }_a=\Delta {\gamma }_a$. Mixed derivatives ${\partial }_t{\partial }_i{\gamma }_a$ are spatial derivatives of the second datum. The scalar enters $\beta$ linearly through $\alpha [f,g,\varphi ]$.