Proposition: Nonempty sourced second-order family

Let ${\varphi }_0$ be any real Klein–Gordon solution with compact smooth Cauchy data. There is a tuple in ${\mathcal{J}}_T$ with

and with $j$ satisfying

For nonzero scalar data the source is nonzero.

Proof. Put

The identity

shows that $Q$ is conserved. Finite propagation makes $Q$ spatially compact on every bounded time interval.

At $t=0$, set

and

Prescribe trace-reversed metric data by

Solve

The Newton data are smooth, and the finite-time wave Cauchy problem therefore gives a smooth solution. Existence can also be obtained directly from the cosine and sine propagators. The multipliers $|p{|}^{-2}$ in the position data are locally integrable in three spatial dimensions.

Write $d_{\nu }=-{\partial }_t{\bar{j}}_{0\nu }+{\partial }_i{\bar{j}}_{i\nu }$. The data give $d_{\nu }(0)=0$. Since

equation (eq:sourced-reduced) gives ${\partial }_t{d}_{\nu }(0)=0$. Conservation of $Q$ implies ${\square }_{\eta }{d}_{\nu }=0$. Wave uniqueness gives $D(j)=0$, and (eq:G1) then gives $G^{(1)}(j)=Q$.

Finally, solve

with zero Cauchy data. The forcing is smooth and spatially compact on finite slabs because it contains derivatives of ${\varphi }_0$. The resulting tuple satisfies all equations of Definition def:jet-space.

For nonzero scalar data,

Thus $Q\ne 0$.