Theorem: Nonzero second-order sourced coefficient

There is a tuple in ${\mathcal{J}}_T$ and a nonnegative compact smearing $f$ such that

The metric coefficient $j$ is sourced by $T(\eta ,{\varphi }_0)/4$, not prescribed independently.

Proof. Choose a real nonnegative $F\in {C_c}^{\infty }({\mathbb{R}}^3)$ equal to a positive constant $A$ near $x=0$. Take

Use the construction of Proposition prop:sourced-family.

Here $J_i=Q_{0i}(0)=0$, so $V_i=0$ and $B_{ij}=0$. Also

It is nonnegative and nonzero. Therefore

Trace reversal of the prescribed initial data gives

Because $D(j)=0$, $h=0$, and ${\varphi }_1=0$, the coordinate coefficients satisfy

Thus the relational second coefficient is simply

on this particular family.

Near the spatial origin, the initial spatial Hessian of ${\varphi }_0$ vanishes and

Using $D(j)=0$,

The equation $P{\varphi }_2=-L_j{\varphi }_0$ and the zero Cauchy data of ${\varphi }_2$ imply

Consequently,

For sufficiently small positive $t$, the solution is positive in a spacetime neighborhood. A nonzero nonnegative $f$ supported there gives the assertion.