Manuscript III-B@1 · III-B-v1.tex · theorem · d2c98f62

Theorem: Local first-order jet descent

The retained first-order jet and its retarded section › The equation space and its action · lines 999–1042 · proof included or linked

III-B@1#thm:jet-descent, III-B@1#eq:jet-observable

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Each orbit of $J_{R} / G_{R}$ has exactly one representative with $D h = 0$ , namely

S_{R} (h, ϕ_{0}, ϕ_{1}) = (Π_{R} h, ϕ_{0}, ϕ_{1} - X_{w_{R} (h)} ϕ_{0}) .

For every real compact $f$ in the laboratory,

F_{f} (h, ϕ_{0}, ϕ_{1}) = \int f [ϕ_{0} + ε (ϕ_{1} - X_{w_{R} (h)} ϕ_{0})] d^{4} x

descends to $J_{R} / G_{R}$ . Its scalar coefficient satisfies

P (ϕ_{1} - X_{w_{R} (h)} ϕ_{0}) = - L_{Π_{R} h} ϕ_{0} .

Proof. Apply $T_{ξ}$ with $ξ^{♯} = - w_{R} (h)$ . Lemma lem:section puts the metric in the required slice, and Proposition prop:jet-action preserves the equations.

The section is constant on each orbit because the two changes $X_{ξ} ϕ_{0}$ and $X_{ξ} ϕ_{0}$ cancel. Two representatives in the slice differ by a past-trivial homogeneous wave, which is zero.

The integrand in each coefficient of (eq:jet-observable) is therefore orbit invariant. For the final identity, writing $w = w_{R} (h)$ ,

P (ϕ_{1} - X_{w} ϕ_{0}) + L_{h - K w} ϕ_{0} = P ϕ_{1} + L_{h} ϕ_{0} - X_{w} P ϕ_{0} = 0.

Displayed formulas in this passage

III-B@1#eq:jet-observable · line 1009
$F_{f} (h, ϕ_{0}, ϕ_{1}) = \int f [ϕ_{0} + ε (ϕ_{1} - X_{w_{R} (h)} ϕ_{0})] d^{4} x$
unlabeled · line 1003
$S_{R} (h, ϕ_{0}, ϕ_{1}) = (Π_{R} h, ϕ_{0}, ϕ_{1} - X_{w_{R} (h)} ϕ_{0}) .$
unlabeled · line 1019
$P (ϕ_{1} - X_{w_{R} (h)} ϕ_{0}) = - L_{Π_{R} h} ϕ_{0} .$
unlabeled · line 1038
$P (ϕ_{1} - X_{w} ϕ_{0}) + L_{h - K w} ϕ_{0} = P ϕ_{1} + L_{h} ϕ_{0} - X_{w} P ϕ_{0} = 0.$

Theorem: Local first-order jet descent

Displayed formulas in this passage

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