Proposition: Three gauge tests

For a fixed $f,\varphi$:

1. annihilation of every $\xi \in {C_c}^{\infty }((0,\infty )\times {\mathbb{R}}^3)$ is equivalent to $\gamma =0$;

2. annihilation of every $\xi \in {C_c}^{\infty }(M)$ is equivalent to $\mathbf{H}=0$, equivalently $\gamma =a=b=0$;

3. annihilation of every smooth spacelike-compact vector wave $\square \xi =0$ is equivalent to ${\mathsf{E}}_1\mathbf{H}=0$.

Proof. The first two assertions follow from (eq:gaugepairing) and the disjoint bulk and slice supports in (eq:H). Arbitrary values and first normal derivatives of a compact test at the slice separate $a$ and $b$.

For necessity in (iii), test with $\xi ={\mathsf{E}}_{1}v$, for arbitrary compact smooth vector tests $v$. The causal propagator is skew-adjoint, so

If ${\mathsf{E}}_1\mathbf{H}=0$, (eq:compactprimitive) gives $\mathbf{H}=\square A$ with $A$ compact. Therefore $⟨\mathbf{H},\xi ⟩=⟨A,\square \xi ⟩=0$ for every residual wave.