Manuscript VII-A@1 · VII-A.tex · proposition · 11da3fe6

Proposition: Stability of the coupling jet

Restriction to a named Einstein–scalar coupling jet › Coupling expansion versus a scalar background · lines 1471–1503 · proof included or linked

VII-A@1#prop:jetstable

Back to manuscript

The transformations (eq:hprime)–(eq:kprime) and (eq:phi1transform)–(eq:phi2transform) preserve $J_{2}^{g}$ for the formal transformations admitted by $G_{U}$ .

Proof. The scalar equations are preserved by Proposition prop:intertwiners.

For the metric equations, naturality of the Einstein tensor gives

G [F_{κ}^{*} g] = F_{κ}^{*} G [g] .

At first order,

G^{(1)} [h^{'}] = G^{(1)} [h] .

At second order,

G^{(1)} [k^{'}] + G^{(2)} [h^{'}, h^{'}] = G^{(1)} [k] + G^{(2)} [h, h] + L_{ξ} G^{(1)} [h] .

The last term vanishes on (eq:Ein1). The leading scalar coefficient remains $ϕ_{0}$ , so the source $T^{(0)} [ϕ_{0}]$ in the order- $κ^{2}$ equation is unchanged. Thus (eq:Ein2) is preserved.

Displayed formulas in this passage

unlabeled · line 1493
$G^{(1)} [k^{'}] + G^{(2)} [h^{'}, h^{'}] = G^{(1)} [k] + G^{(2)} [h, h] + L_{ξ} G^{(1)} [h] .$
unlabeled · line 1485
$G [F_{κ}^{*} g] = F_{κ}^{*} G [g] .$
unlabeled · line 1489
$G^{(1)} [h^{'}] = G^{(1)} [h] .$

Proposition: Stability of the coupling jet

Displayed formulas in this passage

Nearby in this section