Manuscript XVI-D@1 · XVI-D.tex · proposition · b73c6f4a

Proposition: Explicit coordinate solution

Coordinate candidate and a restricted auxiliary extension · lines 1140–1174 · proof included or linked

XVI-D@1#lem:Xcand, XVI-D@1#eq:Xcand, XVI-D@1#eq:boxX

Define

X_{cand}^{μ} (t) = - \int_{0}^{t} σ (s) V_{λ}^{0 μ} (s) d s .

Then $X_{cand}$ is smooth, flatly zero for $t \leq 0$ , has $X_{cand}^{i} = 0$ , and satisfies

□ X_{cand}^{μ} = - F_{co}^{μ}, X_{cand} (0) = \partial_{t} X_{cand} (0) = 0.

It is the unique smooth solution of this Cauchy problem on $M_{T}$ .

Proof. The integrand is smooth and flatly zero in the past. Thus the integral has the asserted smoothness and past extension. It is spatially independent, and

\partial_{t} X_{cand}^{μ} = - σ V_{λ}^{0 μ}, \partial_{t}^{2} X_{cand}^{μ} = - \partial_{t} (σ V_{λ}^{0 μ}) = - F_{co}^{μ} .

The initial traces vanish. The spatial components vanish by Proposition lem:coord. Uniqueness follows componentwise from Lemma lem:uniqueness.

Displayed formulas in this passage

XVI-D@1#eq:Xcand · line 1143
$X_{cand}^{μ} (t) = - \int_{0}^{t} σ (s) V_{λ}^{0 μ} (s) d s .$
XVI-D@1#eq:boxX · line 1150
$□ X_{cand}^{μ} = - F_{co}^{μ}, X_{cand} (0) = \partial_{t} X_{cand} (0) = 0.$
unlabeled · line 1163
$\partial_{t} X_{cand}^{μ} = - σ V_{λ}^{0 μ}, \partial_{t}^{2} X_{cand}^{μ} = - \partial_{t} (σ V_{λ}^{0 μ}) = - F_{co}^{μ} .$

Nearby in this section

section: Coordinate candidate and a restricted auxiliary extension