Proposition: Bianchi obstruction to the coupled jet

If $(h,k,\varphi )\in \mathcal{C}$, then

where ${\varphi }_0$ is the free solution $P_m{\varphi }_0=0$ at order ${\kappa }^0$. For $m=0$, ${\varphi }_0=t$, and any nonconstant $g$, $({\partial }_{\mu }g)T^{\mu }{}_{\rho }[t]=({\partial }_{t}g)n_{\rho }-\frac{1}{2}{\partial }_{\rho }g$ does not vanish on $\mathrm{supp}\mathrm{d}g$; hence $\mathcal{C}=\varnothing$ for such data.

Proof. Insert $E^{\mu \nu }=0$, ${\mathsf{E}}_{\varphi }=0$ into (eq:noether): the right side vanishes identically in $\kappa$. At order ${\kappa }^0$, ${\mathfrak{L}}_{\eta }-{\mathfrak{L}}_q=O(\kappa )$, $\varphi ={\varphi }_0$, and ${\mathsf{E}}_{\varphi }=0$ at ${\kappa }^0$ is $P_m{\varphi }_0=0$, which kills the last term. What remains is (eq:bianchi-constraint). For ${\varphi }_0=t$, $T^{\mu \nu }[t]=n^{\mu }{n}^{\nu }-\frac{1}{2}{\eta }^{\mu \nu }$.