Theorem: Conditional linear bridge

Let $\mathcal{X}$ be a vacuum comparison space. Suppose $c\in {{\mathcal{D}}^{\prime }}_c(M;H)$ is symmetric and has a wave-shell repair

Then the following are equivalent:

1. $K^{\ast }c=0$;

2. $⟨c,K\xi ⟩=0$ for every compact smooth one-form $\xi$;

3. there exists a smooth compact conserved tensor $j$ satisfying

   $$⟨j,h⟩=⟨c,h⟩(h\in \mathcal{X}).$$

When these assertions hold, the representative in (eq:conserved-representative) is available.

For this representative,

In particular, $c_{\mathrm{sm}}$ and $j$ agree on ${\mathcal{S}}_{\mathrm{E}}\cap {\mathcal{S}}_{\square }$, but their equality on all of $\mathcal{X}$ requires the additional condition

Proof. The equivalence of (i) and (ii) is exactly the definition of the distributional formal adjoint.

If (iii) holds, compact conservation of $j$ gives

because $K\xi \in \mathcal{X}$. Hence (iii) implies (ii).

Assume (i). By Lemma lem:wave-equality,

is smooth. Thus the propagator-smoothness hypothesis of Theorem thm:current-smoothing follows from the wave-shell repair rather than being an additional unchecked assumption. That theorem supplies $j$, proving (iii).

Finally, on ${\mathcal{S}}_{\mathrm{E}}$ the representative satisfies $⟨j,h⟩=⟨c,h⟩$. Pairing (eq:bridge-wave-relation) with $h$ gives

which proves the remaining claims.