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

Lemma: Explicit zero-data solution

Explicit witness response and traceless channel · lines 819–851 · proof included or linked

XVI-D@1#lem:R

The fields $V_{λ}$ and $R_{λ}$ are smooth and vanish flatly for $t \leq 0$ . They satisfy

□ V_{λ} = S_{λ}, L R_{λ} = S_{λ},

and have zero initial position and velocity.

Proof. For $t \geq 0$ , differentiation gives

\partial_{t} V_{λ} (t) = \int_{0}^{t} S_{λ} (s) d s, \partial_{t}^{2} V_{λ} (t) = S_{λ} (t) .

Spatial derivatives vanish. The initial traces are zero. Higher time derivatives at $0$ are zero because $S_{λ}$ is flat there. This also proves smoothness of the past-zero extension.

Using (eq:P2),

L R_{λ} = (- □ P) (- P V_{λ}) = □ V_{λ} = S_{λ} .

The constant pointwise map $P$ preserves smoothness, flatness, and the initial data.

Displayed formulas in this passage

unlabeled · line 823
$□ V_{λ} = S_{λ}, L R_{λ} = S_{λ},$
unlabeled · line 833
$\partial_{t} V_{λ} (t) = \int_{0}^{t} S_{λ} (s) d s, \partial_{t}^{2} V_{λ} (t) = S_{λ} (t) .$
unlabeled · line 844
$L R_{λ} = (- □ P) (- P V_{λ}) = □ V_{λ} = S_{λ} .$

Lemma: Explicit zero-data solution

Displayed formulas in this passage

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