Theorem: Sourced local family

Let ${\varphi }_0$ be a real free Klein–Gordon solution with compact smooth auxiliary Cauchy data. There is a tuple in ${\mathcal{J}}_{R,b}$ with

The scalar ${\varphi }_2$ can be chosen with zero initial data. For nonzero scalar data, $Q\ne 0$.

Proof. A direct differentiation of the stress tensor gives

Thus $Q$ is conserved. Finite propagation makes it spatially compact on each bounded time slab.

At $0$, set

Prescribe the trace-reversed metric data

Solve

For example, its componentwise expression is

with $S_w(t)=\sin (t|D|)/|D|$. Local smooth existence follows also from Lemma lem:wave.

Write

The initial data give $d_{\nu }{|}_0=0$. Furthermore,

Using (eq:sourcedwave), these identities imply ${\partial }_t{d}_{\nu }{|}_0=0$. Conservation gives $\square d_{\nu }=0$, so uniqueness gives $D(j)=0$. Equation (eq:G1) now gives $G^{(1)}(j)=Q$.

Finally solve

with zero initial data. The source is smooth and is spatially compact on finite slabs because it contains derivatives of ${\varphi }_0$. All five jet equations hold.

For nonzero data,

so $Q\ne 0$.