Theorem: Unique retarded harmonic section

The map ${\mathcal{S}}_{\mathrm{R}}:\mathcal{J}\to \mathcal{Y}$ obeys:

1. ${\mathcal{S}}_{\mathrm{R}}x=T_{-w_{\mathrm{R}}(h)}x$;

2. ${\mathcal{S}}_{\mathrm{R}}{T}_{\xi }x={\mathcal{S}}_{\mathrm{R}}x$;

3. ${\mathcal{S}}_{\mathrm{R}}{|}_{\mathcal{Y}}=\mathrm{id}$;

4. every ${\mathcal{G}}_{\mathrm{R}}$-orbit meets $\mathcal{Y}$ exactly once.

It induces a bijection

This is an orbit-space assertion, not an infinite-dimensional quotient-manifold theorem.

Proof. Because $w_{\mathrm{R}}(h)\in {\mathcal{G}}_{\mathrm{R}}$, part (i) follows from the action. The equations are preserved by Proposition prop:jet-action. The scalar correction remains spacelike compact because its support is contained in that of ${\varphi }_0$. The retarded section identities supply $\mathsf{D}{\Pi }_{\mathrm{R}}h=0$.

Under $T_{\xi }$, the scalar correction is

The metric projection is unchanged as well, proving (ii). On the slice, $w_{\mathrm{R}}(h)=0$, proving (iii).

Part (i) gives existence. If $h$ and $h+K\xi$ are harmonic with $\xi \in {\mathcal{G}}_{\mathrm{R}}$, then ${\square }_{\eta }\xi =0$, hence $\xi =0$ by Lemma lem:gauge-intersection. The scalar correction is consequently unchanged.