Proposition: Nonzero de Donder diagnostic

Let $\lambda \ne 0$ and ${\gamma }_{\mu \nu }=\lambda {\eta }_{\mu \nu }$. It solves the homogeneous gauge-fixed free equation and $D\gamma =0$. Its coordinate coefficient is

In particular, for $t\ge \epsilon$,

Proof. The constant tensor has zero derivatives, and ${a_1}^{\mu \nu }=\lambda {\eta }^{\mu \nu }$. For its spatially constant source, the zero-frequency wave formula gives

The ramp satisfies $\chi (1-s)=1-\chi (s)$, so ${{\int }_0}^{\epsilon }{\sigma }_{\epsilon }(s)\mathrm{d}s=\epsilon /2$. This yields (eq:layerexample).